Statistical and deep learning approaches in estimating present and future global mean sea level rise

Recent years have seen a gradual rise in sea levels, causing significant concern among scientists. Average sea levels have been consistently recorded since 1800, and since 1880, they have risen dramatically by approximately 21–24 cm, with much of this increase occurring in just the last two and a half decades (Lindsey 2023). Some residential areas, built at low sea levels, are at risk of being affected by this rise. Scientists have warned that several islands could disappear by 2050 due to the sharp increase in sea levels. The primary factor contributing to this disaster is mostly the global concentration of carbon emissions. The greenhouse effect was first identified in 1859, as atmospheric carbon dioxide levels began to rise (Verma and Kumar Ghosh 2022). Consequently, a rise in the average sea level is inevitable due to the accelerated thermal expansion of oceans and the melting of land-based ice sheets and mountain glaciers—all driven by the rapid increase in global temperatures caused by climate change (Boumis et al. 2023). To minimize risks to communities, it is essential to be prepared for and mitigate coastal floods and inundation hazards (Adnan Ikram et al. 2022). In recent years, climate change has led to accelerated glacier melt, causing sea levels to rise by several inches within just a decade (Bahari et al. 2023; Ghosh et al. 2022). This situation has served to heighten awareness among engineers and scientists of the necessity to adapt to the occurrence of water surges in coastal and low-lying areas. Governments around the world are now under pressure to address the destruction caused by floods and to minimize the damage as much as possible (Ikram et al. 2022). A variety of factors, including cyclonic events, earthquakes, wind waves, and river discharge, frequently impact sea levels (Khojasteh et al. 2023; Kopp et al. 2017). The occurrence of catastrophic inundations can be attributed to the failure to monitor sea levels adequately or to neglect updates on surrounding conditions. This has resulted in significant damage or the destruction of numerous structures and properties. Therefore, the development of accurate predictive models for sea level rise is of paramount importance for the mitigation of potential damage to coastal regions, the protection of marine ecosystems, the promotion of economic growth, and the safeguarding of the environment. Modeling sea level variation (SLV) is a complex task due to the presence of numerous factors that occur across different spatial and temporal scales (Nanwani et al. 2023).

There are generally two techniques to estimate sea level rise (SLR) such as physically based modeling and artificial intelligence. While physically based models require extensive hydrological and geomorphological data, artificial intelligence techniques rely solely on historical water level data to forecast future levels. This makes artificial intelligence techniques more cost-effective and time-efficient compared to physically based models. Consequently, artificial intelligence techniques are increasingly used for estimating SLR (Adli Zakaria et al. 2023). Time series forecasting has been applied across numerous domains for quite some time (Ebrahimi and Rajaee 2017). Lately, there has been increasing focus on predicting nonlinear or chaotic time series, largely because of the significant fluctuations driven by dramatic climate changes (Aral and Guan 2016; Pala and Şevgin 2024). Among the primary methods employed to forecast sea level rise is the use of regression models. Despite its widespread use, traditional regression methods have struggled to accurately forecast sea level rise because of the complex nonlinear relationships between input and output variables, making it a persistent scientific challenge. Recently, researchers have turned to artificial intelligence to address the shortcomings of these conventional models (Huang et al. 2010). Recently, artificial intelligence methods have garnered significant interest from researchers and are being used to tackle the limitations of existing models. These techniques have emerged as preferred computational tools for predicting sea level rises due to their ability to deliver fast results with minimal input parameters (Li et al. 2015; Muslim et al. 2020; Chang and Lin 2006; Demuth et al. 2008).

Accarino et al. (2021) explored the use of LSTM neural networks for short-term sea level forecasting in the Southern Adriatic Northern Ionian (SANI) region of the Mediterranean Sea. The results demonstrated that sufficiently trained LSTMs yield better outcomes than standard approaches. Latif et al. (2024) tested the performance of Support Vector Machine (SVM) and k-Nearest Neighbors (kNN) models in predicting sea levels. Their findings revealed that SVM models performed well during training but revealed relatively poor results during testing. In contrast, the KNN model demonstrated consistent effectiveness in both the training and testing phases. In the investigation of Sorkhabi et al. (2023), CNN and LSTM models were used to estimate dynamic sea level variability. According to their findings, RMSE for forecasts of wind speed, precipitation, sea surface temperature, and mean sea level are detected as ± 0.84 m/s, ± 48.75 mm, ± 3.48 °C, and ± 24 mm, respectively. Karsavran (2024) used Random Forest (RF), SVR and KNN models to estimate seawater level on the Erdemli coast of Mersin. The results indicated that the RF model can estimate seawater levels for the 1st and 2nd days with R² values of 0.80 and 0.63, respectively. The KNN model achieves R² values of 0.80 and 0.64 for the same periods, while the SVR model predicts R² values of 0.77 and 0.60, respectively. Lai et al. (2020) stated that the regression support vector machine (RSVM) model yields accurate results during the SLR prediction of Kerteh, Tanjung Sedili, and Tioman Island with correlation coefficient values of R = 0.861, 0.825 and 0.857, respectively. Hassan et al. (2021) tested four different algorithms such as Linear Regression, Moving Average, Dense Neural Network (DNN) and Wavenet which is a kind of deep convolution neural network, to predict global sea level rise. According to the results, the Wavenet algorithm was observed as the most accurate model in estimating sea level rise with a MAE value of 2.626 mm in the training phase and 1.332 mm in the testing phase. The research by Yang et al. (2022) introduced an innovative approach that combines long short-term memory (LSTM) with moving average (MA) to predict flood inundation depths using observed data. The analysis showed that LSTM outperformed RNN and BPNN in forecasting accuracy due to its capability to retain both long-term and short-term memory. Guillou and Chapalain (2021) employed four input parameters including the French tidal coefficient, wind velocity, atmospheric pressure, and river discharge to estimate sea level through multiple regression and artificial neural network (ANN) models. The findings indicated that the constructed models are observed to slightly underpredict the peak sea level. In a previous study, Hazrin et al. (2023) developed six distinct machine-learning algorithms for estimating sea level on a daily basis. This study emphasizes the crucial necessity for comprehensive training and testing of machine learning models to assist decision-makers in formulating effective mitigation strategies for sea level rise due to climate change, thereby ensuring the models' reliability. Bahari et al. (2023) conducted an exemplary review related to the estimation of sea level rise using artificial intelligence. According to their findings, with advancements in technology, sea level measurements can now be predicted using various artificial intelligence techniques, such as machine learning and deep learning, which are capable of extracting insights and identifying patterns from the provided datasets. In the study of Balogun and Adebisi (2021) ARIMA, SVR and LSTM algorithms were employed for the prediction of sea level variation in West Peninsular Malaysia. The correlation coefficient was observed as R = 0.853, 0.7748 and 0.710 with LSTM, SVR and ARIMA models, respectively. Winona and Adytia (2020) presented sea level forecasting using the LSTM approach with two different modes including feedback and no-feedback. While LSTM with no-feedback yields worse estimation outcomes, outstanding results are attained by utilizing LSTM with feedback including R value of 0.999 and RMSE values between 0.019 and 0.028. Ozdemir and Yildirim (2023) explored the use of artificial neural networks (ANNs) alongside four different recurrent neural network (RNN) algorithms to forecast lake water levels across various time series intervals, including 1 day, 5 days, 10 days, 20 days, 1 month, 2 months, and 4 months. Their findings demonstrated that Gated RNN-based algorithms tend to show higher RMSE values as the prediction horizon extends, suggesting diminished performance during shorter prediction intervals. The study of Ahmed et al. (2022) involved developing six distinct machine learning algorithms to predict daily river water levels, using data collected from 1990 to 2019 for training and testing the proposed models. The comparison of various data-driven regression methods reveals that the exponential Gaussian Process Regression (GPR) model provided superior accuracy in predicting daily water levels across different evaluation criteria. This model demonstrated a high level of precision and minimal uncertainty in predicting river water levels, as evidenced by the 95% prediction uncertainty (95PPU) and the d-factor, which were calculated to be 98.276 and 0.000525, respectively. Tur et al. (2021) introduced a method for forecasting sea level variations by utilizing sea level height and meteorological data collected from a tide gauge in Antalya Harbor, Turkey. The findings revealed that incorporating meteorological factors as input parameters enhanced the performance accuracy of multiple linear regression (MLR) models by up to 33% for short-term sea level predictions. Furthermore, the results provided clearer evidence that ANFIS outperforms MLR for sea level prediction.

Sea level rise can cause serious threats to coastal communities, ecosystems and economies. Coastal flooding, increased storm damage, loss of coastal land and erosion and displacement of populations are some of the major climate disasters associated with sea level rise (SLR). So that reason, it is very important to predict current and future SLR and take necessary precautions. The objective of this study is to evaluate the efficacy of both traditional and deep learning time series methods, including the SARIMA, LSTM, and GRU models, in estimating current and future global mean sea level rise (SLR). A review of the literature reveals the use of a range of techniques, including conventional, machine learning, and deep learning, for the prediction of global and regional sea level rise. However, none of these studies have made predictions about the future state of sea level rise (SLR), despite its crucial importance for climate change and catastrophic natural disasters. Furthermore, this study includes a dataset encompassing a very long period of SLR values, such as 30 years. As a result, this study is unique and innovative due to its predictions of future sea level rise and its inclusion of data on SLRs over a very long period of 30 years.

There are four sections in this study. The introduction, found in Sect. 1, examines related research and methodology from other authors. The Sect. 2, is divided into four subsections: seasonal autoregressive integrated moving average (SARIMA) model, long short-term memory (LSTM) neural network, gated recurrent unit (GRU) and error analysis parameters. The problem description, problem domain, structure of the suggested models, necessary equations, and parameters are all explained in this section. The results and discussion section, found in Sect. 3, uses a number of figures and tables to present the investigation's findings. The conclusion, found in Sect. 4, restates the research problem before summarizing the main points and conclusions.

2.1 Seasonal autoregressive integrated moving average (SARIMA) model

The SARIMA (p,d,q)(P,D,Q)s model for time series analysis incorporates both seasonal and non-seasonal elements. Here, d represents the differencing degree, q indicates the moving average (MA) model order, and p specifies the autoregressive (AR) model order. Additionally, Q is the order of seasonal moving average (SMA), D stands for the order of seasonal differencing, s denotes the seasonal component order, and P is associated with the order of seasonal autoregressive (SAR). The model incorporates the error term within the autoregressive (AR) process and considers past observations up to a certain maximum lag. The differencing parameter, d, stabilizes the data by removing trends or seasonality. In contrast, the moving average (MA) component, q, accounts for the impact of recent error terms to enhance predictive accuracy. The SARIMA model is formulated as follows (Lairgi et al. 2023):

in which \(Y_{t}\) represents the time series data, \(\varepsilon_{t}\) denotes an innovation process with zero mean and no correlation, and B stands for the backward shift operator. The term \(\left( {1 - B} \right)^{d}\) is the non-seasonal differencing operator which addresses non-stationarity in the data across successive time periods. The seasonal differencing operator, \(\left( {1 - B^{s} } \right)^{D}\), addresses the issue of non-stationarity in data collected during the same time frame across various years. The AR operator’s order p and the MA operator’s order q are indicated by \(\varphi \left( B \right)\) and \(\theta \left( B \right)\), respectively. The order P of a seasonal AR operator is denoted by \(\phi \left( {B^{s} } \right)\), while the order Q of a seasonal MA operator is indicated by \(\Theta \left( {B^{s} } \right)\). The coefficients \(\varphi\), \(\theta\), \(\phi\) and \(\Theta\) represent the non-seasonal AR, non-seasonal MA, seasonal AR, and seasonal MA components, respectively. These coefficients are specified by the equations provided below (Lairgi et al. 2023; Box and Jenkins 1994):

In the current study, the constructed SARIMA model considered p value, q value and d value as 1, 1, and 1 respectively.

2.2 Long short-term memory (LSTM) neural network

LSTM is commonly used for analyzing time series data and it is a specific type of recurrent neural network (RNN) architecture (Sahoo et al. 2019). LSTM models function by analyzing a series of past observations to forecast future values. They excel at capturing long-term relationships within the data, allowing them to make precise predictions even when the input series is highly nonlinear (Abdel-Nasser and Mahmoud 2017; ArunKumar et al. 2022). LSTM cells utilize gating mechanisms to regulate which information is retained or discarded over time. Figure 1 shows how these cells operate at each time step. Essentially, an LSTM neural network can be viewed as a series of these identical cells, as depicted in Fig. 1. The input sequence, x is processed across n time steps, and the network generates corresponding y outputs. Figure 2 illustrates processes involved in a one time step of an LSTM cell. The diagram shows that the LSTM architecture is composed of multiple cells, each equipped with three types of gates. An LSTM cell generates its hidden state based on the current time step's input data, \(x^{\left\langle t \right\rangle }\) and the output from the previous cell, \(a^{{\left\langle {t - 1} \right\rangle }}\). Moreover, at every time step, a variable referred to as the 'cell state' (denoted as \(c^{\left\langle t \right\rangle }\)) is used to manage and refresh the knowledge within the cell. This variable helps to produce both the hidden state, \(a^{\left\langle t \right\rangle }\) and predictions for each time step, \(y^{\left\langle t \right\rangle }\) (Ozbek 2023).

2.2.1 Forget gate

The subsequent expression \(\left( {\Gamma_{{\text{f}}}^{\left\langle t \right\rangle } } \right)\) is generated by the forget gate of an LSTM cell:

The behavior of the forget gate is determined by weights, denoted as \(W_{f}\) and \(b_{f}\). Using Eq. (6), a vector with values between 0 and 1 is generated. This forget gate vector is then multiplied element-wise with the previous cell state, \(c^{{\left\langle {t - 1} \right\rangle }}\). If any of the \(\Gamma_{{\text{f}}}^{\left\langle t \right\rangle }\) variables is 0, the LSTM will discard the corresponding component of \(c^{{\left\langle {t - 1} \right\rangle }}\). Conversely, if any of the values is 1, the data will be retained.

2.2.2 Update gate

The update gate, \(\Gamma_{{\text{u}}}^{\left\langle t \right\rangle }\) selects the features of the candidate, \(\tilde{c}^{\left\langle t \right\rangle }\) to incorporate into the cell state, \(c^{\left\langle t \right\rangle }\). It regulates the elements transferred to the cell state, \(c^{\left\langle t \right\rangle }\) from a candidate tensor, \(\tilde{c}^{\left\langle t \right\rangle }\). The update gate, \(\Gamma_{{\text{u}}}^{\left\langle t \right\rangle }\) is used to decide which features of the nominee, \(\tilde{c}^{\left\langle t \right\rangle }\) should be included in the cell state, \(c^{\left\langle t \right\rangle }\). The equation of the update gate can be written as:

here \(W_{u}\) and \(b_{u}\) are the parameters that regulate the behavior of the update gate. Similar to forget gate, \(\Gamma_{{\text{u}}}^{\left\langle t \right\rangle }\) is a vector with values ranging between 0 and 1.

2.2.3 Output gate

The output gate, \(\Gamma_{{\text{o}}}^{\left\langle t \right\rangle }\) is expressed in terms of hidden state, \(a^{{\left\langle {t - 1} \right\rangle }}\) and previous input, \(x^{\left\langle t \right\rangle }\) as follows:

In Eq. (10), \(W_{o}\) and \(b_{o}\) represent the weights and bias of the output gate, respectively. Like the other gates, the output gate operates in a similar manner, but it specifically determines which information to pass on as the output. The values produced by the output gate range between 0 and 1.

2.2.4 Hidden state

The hidden state, \(a^{\left\langle t \right\rangle }\) which is carried forward to the next time step in the LSTM cell, is used to determine the three gates for the following time step. It is also employed in predicting \(y^{\left\langle t \right\rangle }\). This updated state is represented as:

2.3 Gated recurrent unit (GRU)

A gated recurrent unit (GRU) is a kind of recurrent neural network constituted as an alternative to traditional LSTM or RNN architectures. It was introduced by Cho et al. (Cho et al. 2014) and the GRU architecture has been extensively applied in different language processing tasks. The GRU provides a streamlined version of the LSTM memory cell, often delivering similar performance while offering the advantage of faster computation. The GRU features an enhanced gating mechanism that improves the analysis of long-term dependencies compared to standard recurrent neural networks, while avoiding the computational complexity of LSTM. Its structure primarily consists of gating mechanisms that update the network's hidden state based on the input at each time step (Uluocak and Bilgili 2023). Compared to the LSTM model, the GRU model, as shown in Fig. 3, has fewer gates. It utilizes a single hidden memory state to carry information among units, eliminating the need for a separate cell state. Similar to the LSTM, the GRU uses a sigmoid function to activate its gates, resulting in values ranging between 0 and 1. Reset gates are designed to determine the extent to which the previous state should be retained. Additionally, a gate that regulates the proportion of the new state derived from the previous state is useful (ArunKumar et al. 2022). Mathematically, at a given time step, t let X_t represent the input from the minibatch and \(a^{{\left\langle {t - 1} \right\rangle }}\) denote the hidden state from the preceding time step. The reset gate, \(\Gamma_{{\text{r}}}^{\left\langle t \right\rangle }\) and the update gate, \(\Gamma_{{\text{z}}}^{\left\langle t \right\rangle }\) are then calculated in this manner:

here \(W_{r}\) and \(W_{z}\) denote weight parameters whereas \(b_{r}\) and \(b_{z}\) are used for bias parameters. Furthermore, the candidate hidden state, \(\tilde{a}^{\left\langle t \right\rangle }\) at the time step, t can be expressed as follows:

in which \(b_{a}\) and \(W_{a}\) are bias and weight parameters, respectively. As seen in Eq. (14), a tanh activation function is settled. A plain RNN is achieved when the values in the reset gate, \(\Gamma_{{\text{r}}}^{\left\langle t \right\rangle }\) are close to 1. Additionally, the impact of the update gate, \(\Gamma_{{\text{z}}}^{\left\langle t \right\rangle }\) must be taken into account. This determines how similar the new hidden state, \(a^{\left\langle t \right\rangle }\) is to the previous hidden state \(a^{{\left\langle {t - 1} \right\rangle }}\) compared to its similarity to the new candidate state, \(\tilde{a}^{\left\langle t \right\rangle }\). The update gate, \(\Gamma_{{\text{z}}}^{\left\langle t \right\rangle }\) can achieve this by forming element-wise convex combinations of the previous hidden state, \(a^{{\left\langle {t - 1} \right\rangle }}\) and the new candidate state, \(\tilde{a}^{\left\langle t \right\rangle }\). This leads to the final update formulation of the GRU.

If the update gate, \(\Gamma_{{\text{z}}}^{\left\langle t \right\rangle }\) is nearly equal to 1, the previous state will be retained. In this case, the data from x_t is disregarded, effectively omitting time step t from the dependency sequence. Conversely, if \(\Gamma_{{\text{z}}}^{\left\langle t \right\rangle }\) is close to 0, the new hidden state, \(a^{\left\langle t \right\rangle }\) is set to the potential hidden state, \(\tilde{a}^{\left\langle t \right\rangle }\).

In this study, GRU and LSTM neural networks were used to estimate SLR, with predictions tested by varying the hidden layer number ranging from 10 to 200. The ideal number of epochs was found to be 600. The initial learning rate, a crucial parameter for training the deep learning model, was chosen from a range between 0.001 and 0.1. Ultimately, the optimal LSTM neural network configuration was identified through performance metrics, with the best outcomes achieved using a network with 50 hidden layers. While the optimization algorithm was selected as Adam, the hyperbolic tangent (tanh) function was employed as an activation function. The size of the input layer was considered as 1 for the constructed LSTM and GRU models.

2.4 Error analysis parameters

Predictions made using deep learning and statistical techniques inherently contain errors. When there are several outliers that break an otherwise steady trend or when data points are far dispersed, estimation accuracy tends to decrease. This problem also occurs when fitting methods like polynomial approximation or linear regression are used on a dataset because machine learning or other predictive algorithms cannot precisely identify the location of each point in a data cloud. Predictions in machine learning and related techniques are produced by mapping the functional relationships between data elements. When the correlation coefficient between measured and predicted values gets closer to 1, statistical error rates between actual observed data and predicted data are reduced. The requirements for the best functional model are determined in part by this optimization. Setting precise statistical parameters for error comparison is therefore crucial.

The study evaluates model prediction performance using various error analysis metrics, including error (E), mean absolute error (MAE), root mean square error (RMSE), mean absolute percentage error (MAPE), and margin of deviation (MoD). These metrics are defined as follows (Mateus et al. 2021):

In these definitions, \(\hat{y}_{i}\) denotes the actual value, \(y_{i}\) represents the predicted data, and \(\overline{y}_{i}\) is the mean value. n is the number of samples used in the training or test set.

The monthly SLR values are obtained from Copernicus Climate Change Service (2018) for years between 1993 and 2023. A total of 366 monthly SLR data points were used in this study. Of these, 288 data points (78%) from January 1993 to December 2016 were employed for training, while the remaining 78 data points (22%) from January 2017 to June 2023 were reserved for testing. Then, future predictions have been made until 2050. In all the graphs, the term "SLR" indicates the increase in the average sea level for the specified year compared to the average sea level at the beginning of 1993. In other words, the sea level value at the beginning of 1993 is used as the reference point.

The first and most important prerequisite for developing SARIMA, LSTM and GRU models for SLR data is to examine the stationarity and seasonality of the time series data. A time series achieves stationarity when its statistical properties, such as mean and variance, remain consistent over time or are unaffected by the period during which the series is observed. Consequently, an assessment of the stationarity of the original monthly SLR time series data has been conducted. In this context, box plots of the SLR time series data, classified by the months of the year, have been constituted and are demonstrated in Fig. 4. As it can be observed from this figure, the SLR data inherently approach the highest value in winter and inherently possess lowest value in summer since high amount of rains increases the sea water level in winter. This situation indicates that seasonality is present in the SLR time series data throughout the year.

The distribution of the predicted data can be visualized utilizing a boxplot. The median is indicated by a line in the center of each box, while the 25th and 75th percentiles are represented by the lower and upper edges of the box, respectively. The boundary lines extend to the most extreme data points within 1.5 times the interquartile range (IQR). Outliers within 1.5 times IQR are indicated by contour lines. The mean of the data is also shown separately using a specific symbol of °. Figures 5 and 6 display boxplot illustrations of actual and estimated SLR values during the testing and training phases, respectively. In Fig. 5, whereas median values are detected as 9.344, 9.349, 9.346, and 9.377 cm, the minimum values are observed as 7.912, 7.896, 7.897, and 7.902 cm for actual data, LSTM, GRU and SARIMA models during the testing phase, respectively. It is clear in Fig. 5 that the boxes for all developed models closely resemble the distribution of the actual quantities. However, the distribution of predicted SLR data with the LSTM model is stronger and more compatible than other generated models with the distribution of actual SLR data. Because it shows a closer distribution to the actual values with a very similar box shape. One can observe in Fig. 6 that whereas median values are detected as 3.580, 3.5733, 3.583 and 3.566 cm, the mean values are observed as 3.764, 3.764, 3.767 and 3.764 cm for actual data, LSTM, GRU and SARIMA models during the training phase. It can be interpreted that all developed models yield excellent performance but LSTM and GRU models give closer results to actual values compared to the SARIMA model.

Figures 7 and 8 present a comparison of SLR values derived from SARIMA, LSTM and GRU models with the actual SLR values during the testing and training phases respectively. To prevent any confusion during the analysis and observation of Figs. 7 and 8, the actual SLR values are displayed as continuous lines, while the forecasted SLR values are represented as discrete points in the graphs. In this representation, solid black lines indicate actual SLR values, while dashed blue, red, and green lines represent the prediction of SLR values from SARIMA, LSTM, and GRU models, respectively. Figure 7 reveals that the SLR values estimated with all developed models are well matched with actual SLR values in the testing phase. For example, for 50. month, while the actual SLR value is observed as 9.6762 cm, it is predicted as 9.6833 cm, 9.6770 cm, and 9.6800 cm with SARIMA, LSTM, and GRU models, respectively. It corresponds absolute error of 0.0070 cm with SARIMA, 0.0008 cm with LSTM, and 0.0038 cm with the GRU model. However, the accuracy level of predicted curves alters with respect to the constructed mode. For instance, whereas the SLR (cm)-Month curve obtained with LSTM nearly overlaps with the actual SLR (cm)-Month curve, there are some differences between actual values and SLR values acquired by SARIMA and GRU models in the testing case. This situation is also valid in Fig. 8 during the training phase especially for initial SLR values obtained with SARIMA and GRU models even though there is generally excellent agreement between actual and predicted SLR values with three constructed models. As it is clear in Figs. 7 and 8, SLR values are increasing month by month. For example, in Fig. 7, while actual SLR value is observed as 8.576 cm in 20. month, it is recorded as 9.142 cm in 30. month, 9.321 cm in 40. month, and 9.676 cm in 50. month. The primary factors contributing to the rise in global average sea levels are the melting of glaciers which adds water to the oceans, and the thermal expansion of seawater as it warms (Bilgili et al. 2024). The Intergovernmental Panel on Climate Change (IPCC) (2023) reports that the accelerated rise in sea levels is due to increased ice loss from both the Greenland and Antarctic ice sheets. Rises in the global average sea level can be explained by four main factors, changes in water mass, temperature, salinity, and ocean circulation patterns (AVISO+ 2023). The rise in sea levels poses an important threat to low-lying islands, coastal areas, and communities worldwide, leading to flooding, coastal erosion, and the contamination of freshwater supplies and agricultural lands.

Figures 9 and 10 display the regression plot illustrating the coefficient of determination (R²) for predicting actual SLR values during the training and testing phases using SARIMA, LSTM, and GRU algorithms. As shown in Figs. 9 and 10, the coefficient of determination (R²) is detected to be 0.9997 during training, and 0.9999 during testing with the SARIMA model, 0.9999 during training, and 1 during testing with the LSTM model and 0.9998 during training and 0.9997 during testing with GRU model. These values, being nearly 1 and aligning closely with the 45o line, indicate that the predicted SLR values are very accurate compared to actual ones when using SARIMA, LSTM and GRU models. The great mean values of R² demonstrate the effectiveness of the generated algorithms in predicting SLR values. The predicted results align closely with the actual SLR values, indicating strong compatibility. If it is necessary to make a comparison between developed models, it can be said that the LSTM model, with its slightly higher R² value, is more successful compared to other models.

The MAPE, MAE, and RMSE values for the prediction of SLR with developed SARIMA, LSTM, and GRU models are demonstrated in Table 1. As it is clear in the table, although performance evaluation parameters are within the acceptable range for all constructed models, the LSTM algorithm outperforms other generated models for the prediction of SLR data in both the training and testing phases. In the testing process, while MAPE, MAE, and RMSE values for the estimation of SLR data with the LSTM model are observed as 0.0631%, 0.0058 cm, and 0.0073 cm, respectively, they are detected as 0.0899%, 0.0084 cm, and 0.0109 cm with GRU model. On the other hand, in the testing stage, the worst estimation is made with the SARIMA model with a MAPE of 0.1335%, MAE of 0.0123 cm and RMSE of 0.0155 cm. Likewise, in the training stage, the LSTM model yields the best accurate results with a MAPE of 0.3817%, MAE of 0.0070 cm, and RMSE of 0.0141 cm. Therefore, it can be inferred that LSTM-based predictions exhibit a strong correlation with the real values, indicating highly impressive outcomes. Since deep learning techniques such as LSTM can capture intricate patterns, nonlinear dependencies, and long-term interactions, they frequently perform better than traditional methods such as SARIMA for time series analysis.

Analyzing the MoDs, which demonstrate the proportional deviation between estimated monthly SLR data and actual SLR values, is significant in showing the correctness and success of generated models. Figure 11 presents the computed MoDs for each of the 78 data utilized in developing SARIMA, LSTM, and GRU models during the testing phase. As it is clear in Fig. 11, the MoD values for the developed SARIMA, LSTM, and GRU models are rather low. The proximity of data displaying MoD values to the zero deviation line reveals the correctness of estimated values acquired from the developed models. As one can observe from Fig. 11 while MoD values acquired for LSTM and GRU models are usually concentrated in close proximity of the zero deviation line, the distribution of MoD values obtained with the SARIMA model is a little far away from the zero deviation line. For example, whereas the MoD values are generally distributed between + 0.5 and − 0.5% for the SARIMA model, they are observed within the range of + 0.25% and − 0.25% for the LSTM and GRU models. So, we can conclude that the accuracy and success of the LSTM and GRU models are higher than the SARIMA model for the estimation of SLR values in the testing phase. In particular, when the Fig. 11 is considerably examined, it can be said that the LSTM model yields more accurate results compared to the GRU model since MoD values are distributed between + 0.2 and − 0.2% for the LSTM model whereas they are observed within the range of + 0.25% and − 0.25% for GRU model. As a result, it can be interpreted that although the developed SARIMA, LSTM, and GRU models can be successfully used to estimate SLR data due to low MoD values, the LSTM model gives the best accurate outcomes compared to other generated models.

Furthermore, the computed MoDs for each of the 288 data utilized in developing SARIMA, LSTM, and GRU models during the training phase are displayed in Fig. 12. As it is clear in Fig. 12, although the MoD values are higher and dispersed for the initial data, it progressively gets reduced and concentrated around the zero deviation line for subsequent data obtained with developed SARIMA, LSTM, and GRU models. Here, like in the testing phase, the accuracy and success of LSTM and GRU models are higher than the SARIMA model since the predicted SLR data are distributed with lower MoD values and higher amount of data are closely distributed in close proximity to the zero deviation line for LSTM and GRU models.

Additionally, the differences between the actual and predicted SLR values were calculated in order to perform a more thorough examination of the accuracy of the SARIMA LSTM and GRU models. In Figs. 13 and 14, the error measurements for each developed model are demonstrated in the testing and training stages, respectively. As illustrated in Figs. 13 and 14, the SARIMA, LSTM, and GRU models yield low error levels. The forecasts appear to closely match the observed data, based on the small differences between the actual and estimated values. As a result, these models showed low error rates and great prediction accuracy. On the other hand, since the estimated SLR data are dispersed with lower error values and more data are closely distributed in near proximity to the zero-deviation line for the LSTM model, its prediction performance is higher than those of the SARIMA and GRU model. For example, in Fig. 13 which demonstrates the distribution of error values in the testing phase, the error values are concentrated between + 0.03 and − 0.03 cm for the SARIMA model, + 0.025 cm and − 0.025 cm for the GRU model while the LSTM model provides the concentration of error values between + 0.015 and − 0.015 cm. Similar comments can be made for Fig. 14 which demonstrates the distribution of error values in the training phase.

Figures 15 and 16 display the error histogram for SLR prediction during the testing and training phases, respectively for the developed models. It is known that as the mean error between the actual dataset and the predicted dataset gets closer to zero, the overall success of the algorithm improves correspondingly. In other words, a greater concentration of error values near zero signifies a more accurate algorithm. Figure 15 shows that the error rate of LSTM and GRU algorithms is smaller than the SARIMA model for predicting SLR data in the testing phase. This is because the LSTM and GRU algorithms demonstrate a significantly greater frequency of error values close to zero (− 0.01 cm, 0, 0.01 cm) compared to the SARIMA model. For example, SLR estimations with error values of − 0.01 cm, 0, and 0.01 cm are observed at 9, 21, and 19 points, respectively for the SARIMA model, at 5, 26, and 40 points, respectively for LSTM model, and at 7, 31 and 23 points, respectively for GRU model. These observations support that LSTM model outperforms other generated models in prediction. Actually, similar comments can be made in Fig. 16 for the forecasting of the SLR dataset during the training phase. Again, it can be concluded that the LSTM algorithm yields the best accurate estimations while SARIMA is observed as the least successful model among them with respect to the closeness of error values to zero in the error histogram.

Up to now, the results demonstrated that all developed models such as SARIMA, LSTM, and GRU can effectively predict global monthly sea level rise (SLR) values although the LSTM model yields better outcomes compared to others. Thus, the forecasting abilities of the SARIMA, LSTM, and GRU models are evaluated to predict monthly SLR values for the upcoming years. These models are employed to produce SLR forecasts for 330 months, covering the period from the half of 2023 to 2050. The results are illustrated in Fig. 17. The forecasts indicate a consistent increase in monthly SLR values for all constructed models in the coming years. For example, in July 2022, the actual SLR value is observed as 10.315 cm while this value is estimated to reach 13.115 cm, 13.211 cm and 13.215 cm with SARIMA, LSTM and GRU models, respectively by July 2030. Furthermore, the LSTM algorithm which is observed as best accurate model, predicts the SLR values as 17.218 cm by July 2040 and 21.236 cm by July 2050. The rise in monthly and yearly SLR data between 1993 and 2050 is demonstrated in Fig. 18. The results from SARIMA, LSTM and GRU models indicate an upward trend in yearly SLR data extending up to 2050. For example, while average SLR values are detected as 5.17 cm in 2010 and 9.43 cm in 2020, they are expected to grow 13.22 cm in 2030, 17.22 cm in 2040 and 21.17 cm in 2050 with the developed best accurate model, LSTM.

This study aims to evaluate the effectiveness of both traditional and deep learning time series methods, such as SARIMA, LSTM and GRU in predicting current and future global mean SLR. The models were trained and tested using monthly sea level rise (SLR) data collected from 1993 to 2023, and future predictions were made up to 2050. A total of 366 monthly SLR data points were employed, with 288 data points (78%) from January 1993 to December 2016 used for training, and 78 data points (22%) from January 2017 to June 2023 used for testing. The outcomes show that while the SLR values estimated by all the developed models closely match the actual SLR values during the testing phase, the LSTM model produces more accurate results than the others. For monthly SLR data estimation, the LSTM model yields MAPE, MAE, and RMSE values of 0.0631%, 0.0058 cm, and 0.0073 cm, respectively, while the GRU model shows values of 0.0899%, 0.0084 cm, and 0.0109 cm. In contrast, the SARIMA model performs the worst in the testing stage, with a MAPE of 0.1335%, MAE of 0.0123 cm, and RMSE of 0.0155 cm. As seen here, deep learning time series methods are more effective than conventional ones due to their ability to automatically extract features and store memory. Furthermore, while the MoD values for the SARIMA model generally range between + 0.5 and − 0.5%, they are observed to fall within + 0.25% and − 0.25% for the LSTM and GRU models. While average sea level rise (SLR) values were measured as 5.17 cm in 2010 and 9.43 cm in 2020, they are projected to reach 13.22 cm in 2030, 17.22 cm in 2040, and 21.17 cm in 2050 using the most accurate developed model, LSTM.

The generated models have the potential to be effective tools for modeling and estimating current and future global mean sea level rise. This study will be highly beneficial for decision-makers in devising mitigation strategies for sea level rise linked to climate change by employing these models. Future work could be carried out by using innovative hybrid deep learning algorithms such as the integration of a convolutional neural network (CNN) with a long short-term memory (LSTM) neural network and a gated recurrent unit (GRU) to enhance the forecasting ability of sea level rise. Furthermore, future investigation could focus on improving the constituted methods for forecasting sea level rise by incorporating additional complex parameters such as precipitation, wind direction, ocean current, and sea surface temperature.