The continuous rise in global sea levels poses a pressing threat to coastal regions, ecosystems, and economies. The Black Sea, bordered by six countries and connected to the Mediterranean via the Bosporus, represents a unique semi-enclosed basin where even modest changes in sea level can have disproportionate impacts (Fig. 1). Despite numerous global studies, long-term regional sea-level analyses for the Black Sea remain limited, especially those that integrate both in-situ tide-gauge records and satellite altimetry.
This study bridges that gap by implementing an Extended Kalman Filter (EKF) framework to analyze 100 years (1921–2020) of Permanent Service for Mean Sea Level (PSMSL) tide-gauge data and comparing it with GLORYS12V1 satellite altimetry (1993–2020). The EKF method is used not only for filtering and prediction but also for detecting linear and nonlinear (accelerating) components in sea-level variation, a major advancement over traditional regression techniques.
Study Area and Data Overview
The Black Sea, with an area of 436,400 km² and an average depth of over 2 km, receives significant freshwater inflows from major rivers like the Danube, Dnieper, and Dniester. Its basin connects 24 European countries, making it one of the most hydrologically complex inland seas in the world.
Data from 10 tide-gauge stations, including Amasra, Batumi, Bourgas, Constantza, Igneda, Poti, Sevastopol, Tuapse, Trabzon, and Varna, were analyzed (Fig. 1). These PSMSL datasets offer high-quality, century-scale records, while the GLORYS12V1 altimetry reanalysis (Copernicus Marine Service) provides complementary open-ocean coverage with 1/12° horizontal resolution and 50 vertical layers.
To ensure consistency, both datasets were converted into sea-level anomalies (deviations from the mean) to minimize datum differences. Because Vertical Land Motion (VLM) corrections from GNSS/InSAR were unavailable, no explicit adjustments were applied, a limitation that future studies are encouraged to address.
Methodology: The EKF Framework
The EKF is a recursive algorithm capable of handling nonlinear systems with uncertain measurements. It updates the system state (sea level, trend, and acceleration) based on new observations while filtering random noise. The model used:
xk=[a0, a1, a2]T
where:
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a₀: Sea-level anomaly (mm)
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a₁: Linear trend (mm/yr)
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a₂: Acceleration (mm/yr²)
The process assumes constant acceleration between observations, with each time step (Δt) corresponding to monthly measurements. The EKF equations, prediction, and update were implemented in MATLAB, using adaptive tuning of noise covariance matrices (Q, R) for stability and sensitivity testing.
By doing so, the EKF continuously refined the trend estimates as new data were assimilated, offering real-time adaptability to sea-level dynamics.
Results and Discussion
1. Long-Term Trends
Linear regression analysis revealed a persistent rise in sea level along all stations, consistent with global averages attributed to thermal expansion and melting of land ice. Quadratic fits identified an accelerating rise, implying that the rate of increase itself is growing, an indicator of intensifying climatic effects.
Such acceleration may result from complex regional mechanisms: changes in water exchange through the Bosporus, increased river discharge, or tectonic uplift/subsidence along coastlines. The findings align with previous Mediterranean and Adriatic studies but present the most extensive dataset yet for the Black Sea.
2. EKF-Predicted Variations
The EKF-predicted sea-level curves closely matched observed PSMSL data at all locations, demonstrating its ability to reproduce both long-term and short-term variations.
At the Poti and Tuapse stations:
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Correlation (PSMSL–EKF): ≈ 0.99
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RMSE: ≈ 0.01 m
These metrics highlight EKF’s precision and reliability. By filtering measurement noise and retaining temporal continuity, the method offers a clear improvement over conventional static regression or spectral decomposition techniques.
The filter’s predictive performance also makes it suitable for early-warning systems, enabling near-real-time monitoring of anomalous trends in coastal sea levels.
3. Comparison with Satellite Altimetry
Normalized time series of PSMSL and satellite altimetry (SA) were compared. Both datasets exhibit similar multi-decadal patterns but differ in phase and magnitude, especially near coastlines.
At Poti, PSMSL–SA correlation was just 0.38 (RMSE ≈ 7.04 m), and at Tuapse, 0.78 (RMSE ≈ 5.13 m). These discrepancies are attributed to radar altimetry’s reduced precision in coastal zones, where land contamination, atmospheric effects, and complex coastal topography degrade accuracy.
Consequently, EKF-refined tide-gauge data provide a superior reference baseline for assessing the reliability of satellite products in coastal monitoring applications.
4. Taylor Diagram and Statistical Insights
A Taylor diagram summarizing correlation, standard deviation, and RMSE among PSMSL, EKF, and SA confirms that EKF estimates cluster tightly around observed data, indicating near-perfect statistical agreement, while satellite-based estimates deviate significantly (Fig. 2). This visualization underscores EKF’s value in refining observational datasets, improving reliability for model validation and risk evaluation.
5. Scientific Significance and Implications
This research demonstrates that EKF-based sea-level analysis provides a robust, adaptive, and mathematically stable approach to coastal monitoring.
Its key strengths include:
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Handling nonlinearity and data noise,
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Producing recursive forecasts with quantified uncertainties,
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Integrating data from multiple sources (tide gauges, satellites, GNSS).
Beyond the Black Sea, this framework can be extended to other coastal and estuarine systems where long-term, noise-free records are essential for understanding hydrodynamic behavior, flood risk, and climate resilience. Furthermore, integrating GNSS/InSAR-based vertical land motion corrections would allow for truly geocentric sea-level assessment, critical for accurate regional projections.
6. Conclusions
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The Black Sea shows a clear and accelerating rise in sea level over the last century.
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The EKF effectively removes noise, stabilizes predictions, and enhances trend detection.
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Satellite altimetry, while useful for open-ocean studies, remains less accurate in nearshore applications.
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EKF-based tide-gauge processing offers a dependable, computationally efficient alternative for long-term regional monitoring.
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Future efforts should focus on multi-sensor data fusion and VLM corrections to improve geophysical interpretations.
By uniting historical records and advanced filtering algorithms, this study sets a precedent for regional-scale sea-level prediction and coastal adaptation planning.
Author Bio
Dr. Kutubuddin Ansari is a researcher in the Department of Geomatics Engineering at Karadeniz Technical University, Türkiye. His research interests include sea-level variability, GNSS geodesy, ocean modeling, and climate change impacts. He has published multiple works in Marine Geophysics Research, Frontiers in Marine Science, and IEEE Access, focusing on integrating geodetic and remote-sensing data for Earth-system monitoring.