Self-Aware Joint Inversion of Multidisciplinary Geophysical Data in Mineral Exploration Using Hyperparameter Self-Adjustment: A Preliminary Study

Abstract

This paper introduces a novel methodology for subsurface characterization in mineral exploration, through the simultaneous joint inversion of seismic and geoelectrical data. By combining complementary information provided by multidisciplinary geophysical data, the joint inversion yields a more accurate and consistent representation of subsurface properties. Furthermore, the joint inversion algorithm is empowered by dynamic hyperparameter self-adjustment. Hyperparameters are settings or configuration values that control the behavior of the inversion algorithm but are not directly learned from the data. Examples include regularization weights, coupling parameters, learning rates (if using gradient-based methods), and number of iterations. In traditional approaches, these values must be manually selected or tuned, often through trial and error, which is time-consuming and may lead to suboptimal results. Instead, in the approach here introduced, a self-adaptive mechanism monitors the evolution of the cost function and optimization performance, automatically tuning hyperparameters to enhance convergence toward an optimal (global) solution. For the purposes of this preliminary study, the method is tested on synthetic 2D geophysical scenarios featuring resistivity and seismic velocity anomalies representative of potential mineral targets. Results show the effectiveness of the approach in accurately identifying these subsurface anomalies. Finally, we show that this joint inversion technique holds significant promise for mineral exploration, particularly in detecting geological features such as ore bodies and mineralized zones, which can manifest as contrasts in seismic velocity and resistivity.

1. Introduction

Geophysical prospecting is a critical aspect of mineral exploration, where complementary techniques play a key role in detecting mineral deposits. The characterization of the subsurface often relies on the use of different methods, such as refraction/reflection seismic techniques (which measure seismic velocity and help define elastic properties of the rocks, as well as geological/structural features) and electrical resistivity tomography (which provides information on electric properties of the geological formations and fluids) [1,2]. These methods offer complementary information about the subsurface, yet traditional inversion techniques often treat them separately, leading to potentially suboptimal results [3,4]. A more integrated approach, where both data types are jointly inverted, can offer more precise insights into subsurface features, improving the identification of mineralized zones or other geological anomalies [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19].

Various geophysical inversion techniques have been developed to address the challenges of subsurface characterization. Early methods focused on single-type inversions, such as seismic tomography or resistivity inversion [20,21,22]. However, joint inversion, as well as other integration techniques [23], have gained attention in recent years as they allow for the integration of multiple datasets to improve model accuracy. These methods have been successfully applied in geophysical exploration, particularly for applications such as hydrocarbon exploration, environmental monitoring, and mineral exploration.

More recently, self-aware learning methods have emerged in machine learning and geophysics, offering adaptive algorithms that can adjust, without any external human intervention, their hyperparameters and architecture during inversion [24,25,26]. These methods allow for dynamic learning rate adjustments, enhancing model convergence and enabling better performance in complex and noisy data environments.

The motivation behind this study is to combine the strengths of joint inversion and self-aware learning to enhance the accuracy and robustness of subsurface characterization in mineral exploration. Joint inversion leverages the complementary information from both seismic and resistivity data to provide a more comprehensive view of the subsurface. The integration of self-learning and self-adjustment of hyperparameters allows the inversion algorithm to adapt to changes in the data and reduce the risk of overfitting or underfitting.

The next sections describe the methodology of the proposed joint inversion approach, including the implementation of the self-aware learning mechanism. We then demonstrate the application of this approach to simulated mineral exploration scenarios and provide insights into its potential applications for real-world exploration tasks.

2. Methodology

2.1. Joint Inversion Overview

Geophysical joint inversion refers to the integrated process of simultaneously inverting multiple geophysical datasets to derive a coherent and unified model of the subsurface. In this study, we focus on the joint inversion of reflection seismic and geoelectrical data to obtain a more accurate and geologically consistent estimation of subsurface properties. The fundamental concept underpinning joint inversion is the recognition that different geophysical methods are sensitive to different physical properties of the subsurface and thus provide complementary information. At the same time, it frequently happens that these physical properties show spatially correlated distributions, indicating the presence of significant geophysical anomalies in the subsoil. Seismic velocity, for example, is primarily sensitive to mechanical-elastic properties such as rock stiffness, density, porosity, and lithological composition. In contrast, electrical resistivity is influenced by factors such as fluid content, salinity, porosity, mineralization, and the connectivity of pore spaces. By combining seismic and electric (and/or electromagnetic) data within a joint inversion framework, it becomes possible to better constrain the resulting subsurface model and reduce the inherent non-uniqueness and ambiguity associated with single-method inversions. The process involves the following key steps and basic elements (Figure 1):

• Simulating data (forward modeling) from an initial guess of the subsurface model for both seismic and electrical methods. For that purpose, we used open source Python libraries and packages, like pyGIMLi (Geophysical Inversion and Modelling Library), as well as proprietary modeling codes.
• Comparing simulated data (commonly indicated as predicted response) with observed data (commonly indicated as observed response) for each method to calculate a joint cost functional (Φ_joint).
• Iteratively updating the model parameters to minimize the joint misfit (in Figure 1 this is indicated as Φ_mis). This includes a weighted combination of all the individual misfits for each dataset. Various optimization approaches can be used, such as gradient-based, stochastic, or hybrid methods.
• Optionally, we can impose structural or petrophysical constraints, like cross-gradient (Φ_X) or rock-physical relationships (Φ_analytic) to enforce consistency between the models belonging to different geophysical domains.
• A regularization term (Φ_reg) introduces additional constraints or prior knowledge to stabilize the inversion process and guide it toward physically meaningful solutions. For instance, smoothing of the model (e.g., Tikhonov regularization) is often applied assuming that physical properties change gradually in space.
• All these terms are commonly weighted by a user-defined weight parameter (λ_i).

The general framework of the joint inversion workflow has been extensively examined in the scientific literature, with various authors emphasizing the theoretical and practical implications of each component of the joint objective function. For example, De Stefano et al. [27] provided a detailed formulation of the individual terms comprising the objective function, highlighting their physical meaning and role in the inversion process. Gallardo and Meju [28] introduced and elaborated on the cross-gradient constraint, which has become a widely adopted technique for enforcing structural similarity between different physical property models. Furthermore, Dell’Aversana et al. [29] presented the mathematical underpinnings of a stochastic joint inversion strategy, including a probabilistic formulation of the objective function and the incorporation of uncertainty and model space exploration. These contributions collectively form the theoretical foundation upon which our implementation is built. An illustrative example of joint inversion formulation is provided in Appendix A for didactical purposes. This consists of retrieving the porosity-saturation model from sonic, resistivity and density logs. The mathematical formulation of this joint inverse problem can be faced using a Bayesian approach [30] and can be easily generalized to different cases of integration of independent measurements.

2.2. Self-Aware Joint Inversion

In our implementation, we employ a consolidated joint inversion algorithm with gradient-based optimization and self-aware hyperparameter tuning (explained below). The goal is to simultaneously invert seismic and geoelectric data to retrieve two physical property models, seismic velocity and electric resistivity. We used simulated (synthetic) geophysical data (reflection travel times and electrical potentials). Below is a schematic explanation of the joint optimization strategy:

(A)

Gradient-based update: The algorithm begins by predicting data from the current velocity and resistivity models. It then computes the misfit, which is the difference between the predicted and observed data for both seismic and resistivity domains. This misfit is used to compute the gradient of the objective function with respect to the model parameters. The models are then updated iteratively in the direction opposite to the gradient (gradient descent), which progressively reduces the overall data misfit. The final goal is to minimize the mismatch between “real” and predicted data (i.e., improve model fidelity).

(B)

Cross-gradient constraint: To ensure that the seismic and resistivity models evolve coherently, a cross-gradient constraint is applied. The cross-gradient is a vector product of the gradients of the two models and tends to zero when structural features (e.g., layer boundaries, faults) are aligned in both models. By penalizing misalignment between structures in velocity and resistivity, this term forces structural similarity during inversion. It enhances geological consistency by favoring common interfaces in both models, even if their physical properties are different. The final goal is to promote structural similarity between the two models to reflect realistic geology.

(C)

Model smoothing: To suppress noise and unrealistic sharp features, Gaussian smoothing is applied to both models after each iteration. A spatial smoothing operator with a Gaussian kernel is used to enforce lateral continuity and avoid spurious high-frequency artifacts. Importantly, this smoothing operator is progressively reduced over iterations, starting with strong smoothing (to stabilize early updates), and gradually allowing finer details to emerge in the models. The final goal is to stabilize inversion in early stages and allow high-resolution features to develop later.

(D)

Self-aware hyperparameter tuning: The key structural novelty of the workflow lies primarily in the self-aware mechanisms through which we aim to empower the consolidated cross-gradient joint inversion approach.

The learning rates, as well as other hyperparameters (for both inversion seismic and geoelectric domains), are dynamically adjusted based on the misfit norm. For instance, if the misfit decreases below a threshold, learning rates decrease to avoid overfitting problems. If the misfit is large, the learning rates increase to encourage faster convergence (see further details in the next subsection).

(E)

Inversion loop: This starts with perturbed true models (to simulate realistic initial guesses). The user can decide the percentage of such perturbations, to assume starting models that are arbitrarily different from the true models. Then, two separate forward modeling algorithms (one for the seismic domain and another one for the geoelectric domain) allow predicted responses (reflection travel times and electric potentials) to be generated. Both models are progressively updated using gradient terms, smoothing, and cross-gradient constraints.

(F)

Intermediate results are displayed every N iterations (for instance, N = 50) for visualization and diagnostics.

This method is a hybrid approach between classical gradient descent and more intelligent tuning strategies that mimic a kind of rudimentary self-awareness and architecture self-adjustment. It balances data fitting, structural coupling, and model smoothness through dynamic control of learning rates, smoothing, and other hyperparameters of the joint optimization algorithm.

2.3. Expected Benefits

As anticipated above, the method proposed in this paper incorporates a joint inversion approach (briefly described in the previous subsection) with a self-learning technique that adjusts the learning rate and the other hyperparameters during the inversion process (see also Appendix B). For instance, let us consider the learning rate that controls the step size of model updates. In traditional inversion methods, it is commonly kept constant or manually adjusted. However, in complex problems such as mineral exploration, it is beneficial to dynamically adjust the learning rate based on the convergence behavior of the cost functions. This is what our algorithm does. It allows optimizing the learning rate autonomously, without any human intervention, to avoid overfitting problems or convergence to local minima of the joint cost function. How is such a self-adjustment mechanism effectively obtained? It is based on continuously monitoring the behavior of the cost functions associated with both seismic velocity and resistivity models. If the cost functions show improvement, the learning rate is increased to speed up the convergence. Conversely, if the cost functions worsen, the learning rate is decreased to allow for more gradual model adjustments. This adaptive approach allows the algorithm to find an optimal balance between exploration (finding new solutions) and exploitation (fine-tuning the existing solution). Similar self-adjustment mechanisms are obtained by applying an analogous strategy to the other key hyperparameters of the joint optimization algorithm.

The primary advantage of self-learning in the context of joint inversion is its ability to dynamically respond to changes in the data and adjust its optimization strategy accordingly. This can lead to faster convergence and more accurate results, particularly when dealing with noisy or incomplete data. Additionally, this method helps prevent the algorithm from getting stuck in local minima, as often happens when using “conventional” optimization techniques. This problem arises in many complex cases of optimization problems, including frequent scenarios of geophysical data inversion. Other authors have faced the same problem, proposing alternative techniques to “globally” explore the model space [31,32,33]. As discussed in previous papers (mentioned earlier), our self-learning approach is designed to ensure that the inversion process remains flexible and capable of handling complex geological scenarios, reducing the problem of falling into local minima of the cost function.

In the next section, we discuss synthetic tests aimed at showing the effectiveness of our joint inversion approach in retrieving multi-physics geophysical anomalies that could be representative of interesting scenarios for mining exploration.

3. Synthetic Tests

3.1. Introduction to the Tests

To evaluate the effectiveness of the proposed self-aware joint inversion method, a synthetic dataset was generated to simulate simplified mineral exploration scenarios. The scenario consisted of both seismic velocity and resistivity models, representing a subsurface two-parametric model including spatially consistent anomalies with different resistivity and seismic velocity characteristics.

In a real-world mining exploration scenario, such a pattern could represent, for example, a subsurface structure, such as an ore body, or a secondary mineral deposit formed due to geological processes of hydrothermal activity.

In previous works, we have already applied the concepts of “artificial self-awareness” to empower consolidated deep neural networks and reinforcement learning methods. In this research, we empowered the joint inversion algorithm with similar mechanisms of self-awareness. Rather than employing traditional deterministic or purely stochastic inversion algorithms, our method integrates adaptive, self-regulating feedback loops that adjust learning rates as well as other hyperparameters of the optimization algorithm, and update strategies based on the dynamic performance of the inversion process itself. The underlying algorithm is an iterative gradient-based inversion process where synthetic observed data are compared to simulated forward responses derived from a pair of evolving models: one for P-wave velocity and one for electrical resistivity. Gradients of the misfit function (differences between predicted and observed data) are smoothed and used to iteratively update the models. Unlike fixed learning rate gradient descent, the algorithm incorporates multiple layers of self-aware mechanisms (as briefly explained in Appendix B) that enhance convergence stability and model robustness.

3.2. Model and Acquisition Geometry

The subsurface domain was modeled as a 2D grid of dimensions 100 m × 100 m, discretized into 50 × 50 cells. The true subsurface properties included a background seismic velocity of 1500 m/s and a background resistivity of 50 Ωm, both perturbed by several Gaussian-shaped anomalies of varying amplitude, size, and location. The anomalies in velocity and resistivity were co-located in the same spatial positions, but were independent in intensity and structure, to mimic realistic subsurface heterogeneity. A synthetic seismic reflector was introduced at a depth of 100 m within the subsurface model. This is necessary to generate the seismic reflected arrivals and associated travel times at the receivers’ locations, providing sufficient seismic sensitivity to subsurface structures.

A total of 5 sources and 25 receivers were distributed on the surface to simulate seismic (travel times) and resistivity (electric potentials) responses. The electrical layout followed a Wenner–Schlumberger configuration (with electrodes located at the same positions of the seismic sources and receivers), which is widely used in geoelectrical prospecting. For each source–receiver pair, reflected travel times were computed for the seismic data, while a potential field response was calculated for the electrical data. Synthetic data were contaminated by Gaussian noise (10%–20%).

The inversion was initialized with randomly perturbed versions of the true models, incorporating up to ±100% variations. During the inversion process, both the velocity and resistivity models were simultaneously updated using gradient-based corrections derived from the respective data misfits. A structural cross-gradient constraint was applied to enforce spatial alignment between gradients in the velocity and resistivity models. This regularization term encourages co-location of subsurface features across the two physical properties, reflecting the assumption that geological boundaries often affect multiple geophysical parameters.

The inversion was run for 500 iterations (this number can be arbitrarily set by the user; for 2D inversion, this number is still reasonable to ensure relatively fast convergence, in the case of our simple tests). At each iteration, the predicted seismic travel times and electrical potentials were forward modeled, the misfits with respect to the “true” data (observed response) were computed, and the models were updated accordingly. A smoothing term was also included to stabilize the inversion.

3.3. Cross-Gradient Constraint and User Control

As anticipated, to enhance the structural consistency between the inverted seismic velocity and electrical resistivity models, the inversion incorporated the cross-gradient constraint method. This constraint minimizes the angle between the gradients of the two models, promoting alignment of interfaces, faults, and geophysical boundaries across the models. Importantly, the strength of the cross-gradient coupling term is user-adjustable, allowing the interpreter to control how strongly the models should conform to one another. This flexibility is particularly useful when the expected geological relationships between seismic and resistivity properties are known to vary, for instance, in the case of highly fractured but resistive mineralized zones.

3.4. Results

We ran many tests using the self-aware joint inversion approach, collecting variable results depending on the initial hyperparameter settings and on starting models. Figure 2, Figure 3, Figure 4 and Figure 5 show some representative examples of results. Visual comparisons between true and inverted models, in correspondence with four inversion moments (after 0, 50, 200 and 499 iterations), confirm the effectiveness of the method in resolving progressively complex subsurface heterogeneities through integrated geophysical interpretation.

Each figure shows, for a given iteration step, the comparison between the true and inverted model and their difference in percentage terms, for both velocity and resistivity models. Finally, Figure 6 shows the trend of the joint inversion cumulative cost function for the whole inversion run.

The joint inversion framework, with cross-gradient regularization, successfully recovers the three anomalous features present in both models and preserves their structural correspondence. Model errors range between 0 and ±25% and are caused mostly by the Gaussian noise added to the synthetic data and the starting models.

This setup provides stable measurements and good depth sensitivity while minimizing sensitivity to near-surface heterogeneity.

As said earlier, the seismic/geoelectric layout used the same coordinates for source and receiver positions, allowing for a tight coupling between datasets in the joint inversion process. This common acquisition grid ensures that both datasets sample the same subsurface volume, which is essential for applying the cross-gradient constraint method in the joint inversion approach. Such a configuration is highly applicable in mineral exploration, where resolving both structural and compositional variations is crucial. The inclusion of a deep reflector (not visible in the figures) mimics typical scenarios found in hard rock geology and ore body detection.

The configuration and methodology employed in this simulation reflect conditions often encountered in mineral exploration campaigns, where high-resolution imaging of resistive (or conductive) and/or fast seismic anomalies is necessary to identify ore deposits, alteration zones, or fluid-filled fractures. The joint use of seismic and electrical methods enhances interpretability, especially when geophysical contrasts are subtle or ambiguous when using only a single technique. This simulated scenario represents a typical synthetic benchmark for subsurface prospecting in mineral-rich regions, especially in the early stages of exploration when drilling data are sparse.

4. Discussion

4.1. A Comparative Test

To quantify the advantages offered by the self-aware joint inversion approach with respect to a “standard” joint inversion approach (which means using the same optimization algorithm without any self-aware mechanism), we performed a comparative study by evaluating the different results, together with several performance parameters for both approaches. First, we ran the joint inversion test on the same (synthetic) data, comparing the inverted sections at the same iteration number. Next, we focused the analysis on quantitative (normalized) metrics, including the root mean squared error (RMSE) and the L2 norm, calculated between the true and inverted models.

Figure 7 shows the same panels as shown in Figure 5 (at iteration 499), obtained this time by fixing the hyperparameters (without applying any self-adjustment mechanism). We can see that, in this specific test, the reconstructed velocity–resistivity models do not show the same accuracy of the self-aware joint inversion approach. The anomalies are reconstructed only partially and appear too smoothed with respect to the true model. We performed many tests like this, with variable results, generally showing the superiority of the self-aware joint inversion approach. To summarize the global results of many comparative tests,

Table 1. Comparison of performance parameters (average values calculated on several comparative tests).

Model	Final RMS Misfit	|%| Difference Velocity	|%| Difference Resistivity
Standard JI	0.0895	38.21	35.74
Self-Aware JI	0.0721	24.52	20.38

shows a quantitative comparison in terms of average performance parameters for the two joint inversion methods.

In summary, the comparative results highlight the superior performance of the self-aware approach. The findings show that this technique generally provides a more accurate and robust reconstruction of subsurface properties, significantly reducing the misfit and improving convergence toward the true model. However, we expect that the advantages of the self-aware method would become even more pronounced in real-data inversions, where the observations are affected by additional noise and unpredictable uncertainties.

4.2. Benefits and Limitations

The proposed framework demonstrates a pathway for robust and adaptive joint inversion, capable of operating without tight manual supervision. In mineral exploration contexts, such an algorithm could be deployed to identify targets where complex lithological contrasts coexist. The integration of seismic and electrical data provides enhanced imaging capacity, and the algorithm’s self-regulation reduces the need for fine-tuning, making it well suited for automated prospecting platforms, drone-based surveys, or large-scale regional screening.

This preliminary synthetic validation provides a foundation for future applications on real datasets, potentially including magnetotellurics, gravity, or EM methods, all within the same self-aware computational framework. Of course, our proposed approach includes both benefits and limitations, which can be summarized as follows:

• Benefits:
  • Improved subsurface accuracy through joint inversion: By simultaneously inverting seismic and resistivity data, the proposed method leverages complementary sensitivities of the two geophysical techniques. In the synthetic tests, this joint inversion led to a more accurate reconstruction of key geological features and anomalies. The cross-gradient constraint ensured structural coherence between the velocity and resistivity models, resulting in a more geologically plausible solution (see below for a quantitative comparison with standard joint inversion).
  • Dynamic self-aware learning for hyperparameter optimization: The integration of a self-aware learning mechanism allows the algorithm to autonomously adjust key hyperparameters, such as step sizes, regularization weights, and smoothing levels, during the inversion process, based on the evolving misfit landscape and model behavior. This adaptive behavior significantly reduces the need for manual tuning, which is typically time-consuming and uncertain. In the synthetic benchmarks, the self-aware version of the joint inversion consistently achieved faster convergence and better final model quality.
  • Adaptability: The self-adjustment of hyperparameters ensures that the algorithm adapts to changing data characteristics, preventing overfitting and ensuring robust results.
  • Flexibility: The method can be applied to a wide range of geophysical datasets and can incorporate additional geophysical data types in the future.
• Limitations:
  • Computational complexity: The joint inversion process, combined with self-aware learning, can be computationally intensive, particularly for large-scale 3D datasets.
  • Noise sensitivity: While the self-aware learning mechanism helps prevent overfitting, extreme levels of noise in the data may still pose challenges.
  • Limited real-world testing: While 2D inversions in simple scenarios, such as those discussed in the current tests, require only a few minutes to reach a satisfactory level of convergence, significantly higher computational demands are expected in the case of 3D models and more geologically realistic settings. At present, we have not conducted extensive tests on complex 3D models. However, preliminary simple tests indicate that substantially greater computational resources will be needed to achieve satisfactory inversion results within reasonable timeframes, typically on the order of several hours.

5. Conclusions and Future Work

This preliminary study demonstrates the potential of self-adaptive geophysical joint inversion for subsurface characterization in mineral exploration, with an emphasis on the integration of seismic velocity and resistivity data. The key methodological innovation in our work lies in the self-aware learning mechanism, which enables adaptive hyperparameter tuning and network restructuring based on internal performance metrics such as misfit feedback. This approach allows the network to modulate its learning behavior in a “biologically inspired way” (based on rudimental self-awareness mechanisms), promoting faster and more stable convergence.

Future work will focus on applying the method to real-world datasets to validate its performance in actual mineral exploration settings. Additionally, the methodology could be expanded to include other geophysical methods, such as magnetics or gravity, to further enhance subsurface characterization. Computational optimization techniques will also be explored to improve the scalability and efficiency of the algorithm for larger datasets.