📄 earth_science

Multi-Objective Gravity Data Inversion for Basement Depth Estimation in Sedimentary Basins

This study proposes and validates an adaptive multi-objective optimization algorithm (AMALGAM) that simultaneously minimizes data misfit and model smoothness to accurately estimate basement depth in sedimentary basins, eliminating the need for post-inversion smoothing or weighting schemes while producing a robust set of optimal solutions.

Original authors: Ahmad A. Reza, Silvia Lestari, S. Sungkono, S. Saifuddin, Firman Syaifuddin, Juan P.G.N. Rochman, Ilham Arisbaya, Khalid S. Essa

Published 2026-08-13

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CC BY 4.0

Original authors: Ahmad A. Reza, Silvia Lestari, S. Sungkono, S. Saifuddin, Firman Syaifuddin, Juan P.G.N. Rochman, Ilham Arisbaya, Khalid S. Essa

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

Imagine the Earth's crust as a giant, layered cake. The top layers are soft, fluffy sediments that have settled over millions of years, while the bottom layer is a hard, ancient "basement" made of rock. To find valuable treasures like oil, gas, or geothermal energy hidden inside that cake, explorers need to know exactly how deep the soft layers go before hitting the hard floor. But here's the catch: you can't just cut a slice of the Earth to look. Instead, scientists use a tool called gravity. Because the soft sediment is lighter than the hard basement rock, the Earth's gravitational pull changes slightly depending on how thick the soft layer is at any given spot. By measuring these tiny wiggles in gravity, scientists can try to "invert" the data—working backward from the pull to figure out the shape of the hidden floor. It's like trying to guess the shape of a hidden object inside a box just by feeling how heavy the box feels from different angles. The problem is that this puzzle is tricky; many different shapes could create the same gravity feeling, and the measurements are often messy with "noise" (like static on a radio).

This paper tackles that messy puzzle using a clever new strategy called the AMALGAM algorithm. The authors, a team of researchers from Indonesia, Egypt, and beyond, wanted to find the depth of the basement in sedimentary basins without making too many guesses or relying on subjective rules. Instead of trying to force the answer into a single "perfect" shape, they treated the problem like a game with two competing goals: making the model fit the gravity data as closely as possible, and keeping the model's shape smooth and realistic. They used a smart computer program that runs many different search strategies at once, letting them evolve a whole family of possible answers. These answers form a "Pareto front," which is like a menu of trade-offs where you can see exactly how much smoothness you lose if you want a tighter fit to the data.

The researchers tested this method first on fake, computer-generated gravity data. They created two different "basement" shapes—one simple and one very bumpy and complex—and added random noise to mimic real-world measurement errors. The results were promising: the AMALGAM algorithm successfully reconstructed the true shapes in both cases, even with the noise. It didn't just give one answer; it gave a range of likely solutions that showed exactly where the uncertainty lay. To prove it worked in the real world, they applied the method to gravity data from two actual locations: the Büyük Menderes Graben in western Türkiye and the Moghan sedimentary basin in northwestern Iran. In both places, the depths estimated by their new method matched up very well with previous studies and even with data from actual drill holes. The paper suggests that this multi-objective approach is a robust, accurate, and efficient way to map the hidden underground world, offering a clearer picture of where the basement floor lies without needing to smooth out the results after the fact.

Technical Summary: Multi-Objective Gravity Data Inversion for Basement Depth Estimation in Sedimentary Basins

Problem Statement
Accurate estimation of basement depth is critical for energy resource exploration (hydrocarbons, geothermal, groundwater) and mineral prospecting within sedimentary basins. This estimation is typically achieved through gravity anomaly inversion, an inherently non-linear, ill-posed, and non-unique problem. Conventional approaches often rely on single-objective optimization where data misfit and model regularization (smoothness) are combined into a composite function via a weighted sum. This methodology introduces significant subjectivity, as the selection of the regularization parameter (Lagrange multiplier) is typically determined through trial-and-error. Furthermore, local optimization methods are prone to entrapment in local minima, while global optimization methods formulated as single-objective problems still suffer from the subjectivity of weighting parameters. Additionally, standard regularization techniques may not fully eliminate solution equivalence, potentially leading to misinterpretation if prior geological assumptions are inaccurate.

Methodology
The study proposes a multi-objective optimization framework utilizing the Adaptive Multi-Algorithm Genetically Adaptive Multi-objective (AMALGAM) algorithm to invert gravity data for basement depth estimation.

Forward Modeling: The subsurface is parameterized as a series of juxtaposed two-dimensional rectangular prisms. The vertical gravitational attraction is calculated using the polygonal body discretization method, where the thickness of each prism represents the unknown basement depth.
Multi-Objective Formulation: Instead of combining objectives, the inversion simultaneously minimizes two competing functions:
1. Data Misfit ( $F_1$ ): The normalized L2-norm difference between observed and calculated gravity anomalies.
2. Model Constraint ( $F_2$ ): A roughness/smoothness regularization term defined by the L2-norm of the second-order difference of the model parameters.
  This approach eliminates the need for post-inversion smoothing, spectral filtering, or the subjective selection of a regularization weight.
AMALGAM Algorithm: AMALGAM is an evolutionary-based algorithm that integrates four constituent search strategies: Non-dominated Sorting Genetic Algorithm (NSGA-II), Particle Swarm Optimization (PSO), Adaptive Metropolis Search (AMS), and Differential Evolution (DE). It employs a self-adaptive mechanism where the contribution of each algorithm to the offspring population is dynamically adjusted based on its reproductive success in generating Pareto-optimal solutions. This allows the algorithm to balance exploration and exploitation without manual parameter tuning.
Solution Selection: The algorithm generates a Pareto-optimal set (POS) of non-dominated solutions. A "knee point" method is applied to identify the optimal trade-off solution between data fit and model complexity. Uncertainty is quantified using the interquartile range (IQR) and standard deviation of the POS.

Key Contributions

Decoupling of Objectives: The study demonstrates a framework where data misfit and model smoothness are treated as independent, competing objectives, removing the subjectivity associated with Lagrange multipliers in single-objective formulations.
Adaptive Multi-Method Search: By leveraging AMALGAM, the approach dynamically adapts to the topology of the objective function landscape, utilizing the complementary strengths of multiple evolutionary algorithms to navigate high-dimensional, non-linear parameter spaces.
Uncertainty Quantification: The generation of a Pareto-optimal ensemble allows for a rigorous statistical assessment of model uncertainty, providing a range of geologically admissible solutions rather than a single, potentially biased estimate.
Elimination of Post-Processing: The method produces geologically realistic models directly from the optimization process, obviating the need for post-inversion smoothing or filtering.

Results
The proposed framework was validated through synthetic studies and field applications:

Synthetic Studies: Two models (a simple symmetrical basin and a complex irregular basin) were tested under noise-free and 10% noise-added conditions.
- The algorithm successfully recovered basement geometries closely matching the true models in both scenarios.
- Statistical analysis showed low standard deviations in objective function values, indicating stable convergence.
- The POS ensemble provided accurate uncertainty bounds, with median solutions aligning well with true depths even in the presence of noise.
Field Application 1 (Büyük Menderes Graben, Türkiye): Inversion of a 50 km gravity profile yielded basement depth estimates consistent with independent Differential Evolution (DE) results and available borehole lithological logs. The estimated depth at the borehole location (0.61 ± 0.06 km) matched the borehole constraint (0.61 km) and the DE solution (0.61 km). Maximum basement depths were comparable to previous studies using DE, Deep Learning, and MJAYA algorithms.
Field Application 2 (Moghan Sedimentary Basin, Iran): Inversion of gravity data over the Moghan basin produced basement depth profiles that closely agreed with results from a combined Imperialist Competitive Algorithm and Firefly Algorithm (ICAFA) approach, with minor deviations attributed to edge effects.

Significance
The paper establishes the AMALGAM-based multi-objective inversion approach as a robust, accurate, and computationally efficient method for subsurface imaging in sedimentary basins. By treating data fit and model complexity as competing objectives, the method provides a more rigorous and systematic assessment of model uncertainty while mitigating the non-uniqueness inherent in gravity inversion. The results confirm that this framework is a promising alternative to conventional single-objective inversion techniques, offering reliable basement depth estimation without the need for subjective regularization parameter tuning or post-processing smoothing.

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