Source-linked AI summary
Data-driven rational function neural networks: a new method for generating analytical models of rock physics
Weitao Sun
TL;DR
Predicting seismic-wave velocity requires models that capture complex rock properties, but theoretical derivations are lengthy and empirical formulas may lack physical formulation. This paper introduces RafNN, which learns rational-function analytical models directly from observational data. It reconstructs Gassmann’s equation from noisy data and also produces expressions beyond standard theoretical models.
Problem
Rock-physics velocity modeling must handle complex material relationships, while theoretical models require lengthy derivations and empirical formulas may lack physical formulation.
Method
RafNN learns rational-function analytical expressions from data using polynomial terms, sparse coefficients, and neural-network training.
Results
Gassmann’s equation was reconstructed from data, with six of eight coefficients exact and the other two differing by 0.01% and 0.004%.
Takeaways & Limitations
RafNN can construct analytical rock-physics models directly from field or observational data, including expressions beyond standard theoretical forms.
Takeaways & Limitations
Without prior knowledge of model structure, the number of functional terms can grow rapidly and the resulting expression may take many forms.
Abstract
from arXiv · showhide
Seismic wave velocity of underground rock plays important role in detecting internal structure of the Earth. Rock physics models have long been the focus of predicting wave velocity. However, construction of a theoretical model requires careful physical considerations and mathematical derivations, which means a long research process. In addition, various complicated situations often occur in practice, which brings great difficulties to the application of theoretical models. On the other hand, there are many empirical formulas based on real data. These empirical models are often simple and easy to use, but may be not based on physical principles and lack a proper formulation of physics. This work proposed a rational function neural networks (RafNN) for data-driven rock physics modeling. Based on the observation data set, this method can deduce a velocity model which not only satisfies the actual data distribution, but also has a proper mathematical form reflecting the inherent rock physics. The Gassmann's equation, which is the most commonly used theoretical model relating bulk modulus of porous rock to mineral composition, porosity and fluid, is perfectly reconstructed by using data-driven RafNN. The advantage of this method is that only observational data sets are required to extract model equations, and no complex mathematical and physical processes are involved. This work opens up for the first time a new avenue on constructing analytical expression of velocity models using neural networks and field data, which is of great interest for exploring the heterogeneous structure of the Earth.
1 Introduction
Seismic velocities help locate internal Earth discontinuities, but heterogeneous underground structures and velocity gradients remain incompletely resolved. Existing rock-physics models must relate velocity to complex material properties, while rational-function relationships are not well covered by common learned function forms.
- Seismic-wave arrival times and velocities help determine the depth of internal Earth discontinuities.
- Underground wave velocities depend on mineral composition, compressibility, density, anisotropy, porosity, and fluid properties.
- Actual underground velocity gradients and seismic heterogeneity remain unresolved despite established large-scale Earth structure.
- Common data-driven candidate functions were limited to polynomial and trigonometric terms, whereas rock-physics relationships often take rational-function form.
- Rational functions represent ratios of polynomial functions and can describe linear and nonlinear systems with arbitrary precision.
2 Rational functional neural networks
RafNN represents a target relationship as a ratio of polynomial functions and learns its coefficients from data. Its sparse, layered architecture supports analytical rock-physics expressions, including multivariable formulations such as Gassmann’s equation.
- RafNN represents nonlinear functions as ratios of numerator and denominator polynomial functions.
- Gassmann’s volumetric modulus depends on dry-skeleton, porosity, solid-material, and pore-fluid bulk moduli.
- For Gassmann modeling, the independent-variable vector is (K_d, K_m, K_f, φ), with numerator order p = 4 and denominator order q determined by the model.
- The network uses polynomial-function inputs, separate numerator and denominator hidden-layer combinations, and an output ratio.
- The cost function combines approximation error with coefficient norms to encourage sparse network representations.
- Training alternates forward propagation and error back propagation to update numerator and denominator coefficients across samples and iterations.
3 Model structure and parameter determination for data-driven RafNN
RafNN parameter determination can use prior knowledge of a known model’s functional terms to accelerate convergence. Without such structural knowledge, the functional search space grows rapidly and concise accurate expressions become difficult to obtain.
- A Gassmann reconstruction uses 69 numerator and 34 denominator function terms before identifying the relevant coefficient structure.
- Without prior model-structure knowledge, nonlinear models require searching rapidly expanding functional spaces, producing potentially many final expressions.
- Finding concise and accurate rational-function expressions is therefore important for practical rock-physics modeling.
4 Reconstruction of the Gassmann’s equation from logging data
RafNN reconstructs Gassmann’s equation from generated rock-physics data while also producing alternative analytical expressions for data not fully explained by existing theoretical models. In the reported experiment, the recovered coefficients closely match Gassmann’s equation despite noisy data and repeated training.
- Method: RafNN models velocity or bulk modulus as a ratio of numerator and denominator polynomial functions.The method represents rock-physics relationships through a rational-function neural network whose output is the ratio of learned numerator and denominator terms.
- Experimental setup: The training and test data were generated from Gassmann’s equation with 1% amplitude noise and porosity ranging from 0.01 to 0.4.The synthetic setup used sandstone skeleton properties and water as the pore fluid.
- Training results: After 35 independent trainings, the numbers of numerator and denominator terms converged to 4 as training error decreased, matching Gassmann’s equation.Among networks with cost-function mean square error below 1, the model with the smallest mean square error was selected.
- Coefficient recovery: Six of eight rational-function coefficients were exactly equal to Gassmann’s coefficients, while the other two differed by 0.01% and 0.004%.Although the learned coefficients differed before normalization, their relationships remained consistent with Gassmann’s equation.
- Training results: The predicted bulk moduli of water-saturated rock were perfectly consistent with Gassmann’s theoretical values.The comparison used predicted bulk modulus as a function of porosity.
- Coefficient recovery: Across additional trainings, coefficient errors remained below 0.03%, while alternative expressions were also obtained.These results support RafNN’s ability to recover the target theoretical form while exploring other expressions from the same data.
5 Conclusion
The paper establishes RafNN as a data-driven method for constructing analytical rock-physics models from field data. It perfectly reproduces Gassmann’s equation and also yields data-derived models with mathematical forms different from standard models.
- RafNN constructs analytical rock-physics equations directly from field data without theoretical assumptions or mathematical derivation.The method is presented as fast and adaptable for different kinds of rocks.
- Gassmann’s equation is perfectly reproduced from data satisfying that equation.
- The method also obtains data-derived models with mathematical forms different from standard models such as Gassmann’s equation.The paper states that these expressions may contain previously undiscovered physical implications.
Appendix
The appendix reports RMSE values and fitted numerator and denominator coefficients for RafNN modeling. The listed coefficients are generally close to ±1, while the reported RMSE values span approximately 5.36e-07 to 5.36e-05.
- The appendix tabulates the RMSE of RafNN modeling together with coefficients of numerator rational terms.
- 1.024647e-05 is one listed RMSE paired with coefficients approximately 1, -1, -1, and 1.
- Table 2 reports the RMSE of RafNN modeling and coefficients of denominator rational terms.