⚡ electrical engineering

Groundwater Forecasting Under Missing Data Uncertainty: A Multi-Horizon Assessment Using Hyperbolic Deep Learning

This study proposes an integrated multi-horizon groundwater forecasting framework that combines optimal missing-data reconstruction with hyperbolic deep learning to effectively address observational gaps and reduce error accumulation in long-term predictions, demonstrating superior stability and accuracy compared to conventional deep learning architectures.

Original authors: Kübra Küllahcı, Kürşat Şekerci

Published 2026-07-04

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kübra Küllahcı, Kürşat Şekerci

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Predicting the Underground Ocean

Imagine groundwater as a massive, invisible ocean hidden beneath our feet. Farmers, cities, and industries rely on it, but we can't see it. We only know how high or low the water is by checking a few "thermometers" (monitoring wells) scattered around.

The problem? These thermometers sometimes break, or the people reading them miss a few days. This leaves gaps in the data. Also, predicting what the water level will be next month is easy, but guessing what it will be in five months is like trying to predict the weather a year in advance—it gets messy and uncertain very quickly.

This paper introduces a new "super-forecasting" tool designed to handle these broken records and make better long-term guesses.

Part 1: Fixing the Broken Records (Data Imputation)

The Problem: Before you can predict the future, you need a complete history. But the data from the wells in Eastern Turkey had holes in it (missing months).

The Analogy: Imagine you are trying to finish a jigsaw puzzle, but several pieces are missing.

Old Way: You might just guess the missing piece based on the color of the piece next to it (simple interpolation) or look at a similar puzzle and copy a piece from there (k-Nearest Neighbors).
The Paper's Approach: The researchers tested four different ways to "fill in the blanks." They treated this like a science experiment, checking not just if the numbers looked right, but if the shape of the water's movement (its rhythm and patterns) stayed true.

The Winner: They found that a method called KalmanUCM was the best "puzzle fixer." It didn't just guess a number; it understood the underlying story of the water (the seasonal ups and downs and the slow trends) and filled the gaps in a way that kept the natural rhythm intact.

Part 2: The New "Brain" (Hyperbolic Deep Learning)

The Problem: Once the data was fixed, they needed a computer model to predict the future. Most standard computer models (like the ones used for stock markets or weather) think in flat, straight lines (Euclidean space).

The Analogy: Think of a flat sheet of paper. If you try to draw a complex, branching tree on a flat sheet, the branches get squished and crowded as they get further out. You can't show the full complexity without the paper tearing or the branches overlapping messily.

Groundwater is like that tree: It has simple short-term changes (daily rain) and complex, deep long-term changes (years of drought or storage).
The Old Models: Tried to force this complex tree onto a flat sheet of paper. They got confused when looking far into the future.

The New Solution: The researchers used Hyperbolic Deep Learning.

The Metaphor: Imagine a saddle or a pringle chip. This shape curves outward. On a saddle, you have much more room to spread out branches without them crowding each other.
How it works: This new model "thinks" in a curved space (like the Pringle chip) rather than a flat one. This allows it to organize the complex, hierarchical layers of groundwater data much better. It can see the "big picture" of long-term trends while still noticing the small, immediate ripples.

Part 3: The Race (Testing the Models)

The researchers put their new "Hyperbolic Brain" in a race against two other popular computer models:

CNN-BiLSTM-GRU: A heavy-duty, hybrid model that tries to learn patterns by looking at data from both the past and future.
TimesNet: A modern model that looks at time series like a 2D image to find repeating patterns.

The Race Track: They tested the models on predicting groundwater levels for 1 month, 2 months, 3 months, 4 months, and 5 months ahead.

The Results:

Short Term (1 month ahead): All models did okay, but the Hyperbolic model was the fastest and most accurate.
Long Term (5 months ahead): This is where the others stumbled. As the prediction got further away, the "flat" models started to make bigger and bigger mistakes (like a car drifting off the road).
The Winner: The Hyperbolic Deep Learning model stayed on the road. Even at the 5-month mark, it kept its accuracy much better than the others. It didn't lose its "cool" as the prediction got harder.

Part 4: The Proof (Did it really work?)

The researchers didn't just say "it looks good." They ran strict statistical tests (like a referee checking the score):

The "Wilcoxon Test": This confirmed that the Hyperbolic model wasn't just lucky; its success was statistically significant. It was genuinely better.
The "Scatter Plot": Imagine a graph where the perfect prediction is a straight diagonal line. The Hyperbolic model's dots hugged that line tightly, even for the 5-month predictions. The other models' dots started to scatter away from the line.

Summary

This paper is about building a better crystal ball for groundwater.

Step 1: They fixed broken data records using the best "fill-in-the-blanks" method (KalmanUCM).
Step 2: They built a new AI brain that thinks in a curved, "saddle-shaped" space (Hyperbolic) instead of a flat one.
Result: This new brain is much better at predicting groundwater levels far into the future (up to 5 months) without losing accuracy, making it a powerful tool for managing water resources when data is incomplete.

Technical Summary: Groundwater Forecasting Under Missing Data Uncertainty Using Hyperbolic Deep Learning

Problem Statement
Groundwater level (GWL) forecasting is critical for adaptive aquifer management but faces two persistent technical challenges: incomplete monitoring records due to observational gaps and the accumulation of prediction uncertainty at extended forecast horizons. Traditional time-series approaches (e.g., ARIMA) often fail to capture the nonlinear, multi-scale interactions inherent in groundwater systems. While deep learning (DL) models like LSTM and GRU have improved performance, they typically operate in Euclidean space, which may not efficiently represent the hierarchical and scale-free structures of hydrological dynamics. Furthermore, existing studies often treat data imputation as a routine preprocessing step without systematically evaluating how reconstruction strategies affect the structural integrity of the time series or downstream multi-horizon forecasting stability.

Methodology
The study proposes an integrated framework combining optimal missing-data reconstruction with hyperbolic deep representation learning. The methodology proceeds in four stages:

Data Reconstruction and Imputation:
Monthly GWL data from three monitoring wells in Elazig Province, Türkiye (1978–2015), were analyzed. Four imputation strategies were evaluated: Seasonal Interpolation, Kalman Unobserved Components Model (KalmanUCM), k-Nearest Neighbors (kNN), and Random Forest (RF). Unlike standard approaches relying solely on pointwise error, a multi-criteria evaluation framework was employed. This assessed reconstruction quality based on error magnitude (RMSE, MAE), variance preservation, autocorrelation distortion (lags 1 and 12), spectral consistency, and trend deviation. The KalmanUCM method was selected as optimal due to its superior ability to preserve both numerical accuracy and the intrinsic temporal structure (variance and autocorrelation) of the series.
Hyperbolic Deep Learning Architecture:
The core contribution is a forecasting model formulated within the Poincaré ball manifold (a model of hyperbolic geometry with negative curvature). Unlike Euclidean networks, this architecture leverages hyperbolic operations (Möbius addition, exponential and logarithmic maps) to embed temporal dependencies. The model is designed to capture hierarchical and multi-scale relationships inherent in groundwater dynamics more efficiently than flat-space representations. The network structure involves embedding input vectors into the hyperbolic manifold, applying hyperbolic linear transformations and pointwise nonlinearities, and projecting outputs back to the Euclidean space for prediction.
Benchmarking and Experimental Design:
The proposed Hyperbolic Deep Learning (HDL) model was benchmarked against two established Euclidean architectures: a hybrid CNN-BiLSTM-GRU and TimesNet. All models were trained on the same reconstructed datasets using a sliding-window approach for multi-horizon forecasting (lead times $t+1$ to $t+5$ ). Hyperparameters were optimized via k-fold cross-validation.
Statistical Validation:
Performance was evaluated using standard hydrological metrics (RMSE, MAE, NSE, KGE, PBIAS) and rigorous statistical tests. The Wilcoxon signed-rank test was used to assess the statistical significance of error differences between models, while Pearson correlation and residual diagnostics evaluated agreement and bias.

Key Results

Imputation Performance: The KalmanUCM method consistently outperformed other imputation techniques, achieving the lowest RMSE and MAE while maintaining near-perfect variance preservation (VarRatio $\approx$ 1.0) and minimal distortion of autocorrelation structures. In contrast, kNN imputation yielded negative NSE values, indicating poor suitability for this dataset.
Forecasting Accuracy: The HDL model consistently achieved the highest predictive accuracy across all three wells and all forecast horizons.
- At $t+1$ , the HDL model achieved NSE values of 0.989 (Kuyulu), 0.988 (Yurtbaşı), and 0.972 (Gözebaşı).
- Crucially, the HDL model demonstrated superior stability at extended horizons ( $t+4$ and $t+5$ ). While benchmark models (CNN-BiLSTM-GRU and TimesNet) exhibited significant error accumulation and performance degradation at $t+5$ , the HDL model maintained higher NSE values and lower RMSE. For instance, at $t+5$ for the Kuyulu well, HDL achieved an NSE of 0.838 compared to 0.677 for CNN-BiLSTM-GRU.
Statistical Significance: Wilcoxon signed-rank tests confirmed that the error reductions achieved by the HDL model were statistically significant compared to the benchmarks. Pearson correlation coefficients remained high (often >0.90) across all horizons, and residual analysis indicated controlled bias and dispersion.

Significance and Claims
The paper claims that this study represents one of the earliest applications of hyperbolic deep learning to groundwater level forecasting. Its primary significance lies in demonstrating that hyperbolic representation learning provides a structurally efficient mechanism for capturing the hierarchical temporal dependencies and delayed responses characteristic of groundwater systems.

The authors argue that the improved performance is not merely due to better parameter tuning but stems from the geometric properties of the hyperbolic space, which allows for lower-distortion embedding of complex, multi-scale temporal structures. By integrating structurally consistent data reconstruction with geometry-aware deep learning, the proposed framework offers a robust pathway for improving multi-step forecasting reliability in data-limited or incomplete observational conditions. The study concludes that this approach offers a promising, conceptually innovative direction for hydrological modeling, particularly where long-range dependencies and missing data are critical constraints.