Bayesian Frameworks for Quantifying Uncertainty in Surface Wave Velocity Models

The characterization of the shallow subsurface depends heavily on the interpretation of seismic surface waves, primarily Rayleigh and Love waves. While traditional deterministic inversion techniques provide a singular profile of shear-wave velocity, modern geophysical research at the Surface Wave Hub emphasizes the integration of Bayesian frameworks. These frameworks use Markov Chain Monte Carlo (MCMC) methods to address the inherent non-linearity and non-uniqueness of the seismic inversion problem. By treating model parameters as random variables, researchers can quantify uncertainty and provide probabilistic error bounds for lithological models.

Advanced seismic analysis involves the precise calibration of geophones and accelerometers to record subtle ground-motion signatures. These datasets are then subjected to spectral analysis to extract dispersion curves, which represent the change in phase velocity as a function of frequency. In complex geological stratigraphies or engineered environments like bridges and tunnels, the dispersion curve is rarely a simple function, necessitating strong mathematical frameworks to infer subsurface properties such as elastic moduli, density, and porosity. The shift toward Bayesian methodologies reflects a broader trend in geophysics toward more rigorous statistical validation of site investigations.

At a glance

• Primary Objective:To move beyond single-model estimates by mapping the entire posterior probability density function of subsurface properties.
• Core Methodology:Markov Chain Monte Carlo (MCMC) algorithms, including Metropolis-Hastings and Gibbs sampling, to explore the model space.
• Key Parameters:Shear-wave velocity (Vs), layer thickness, Poisson's ratio, and density.
• Critical Applications:Non-destructive testing (NDT) of infrastructure, urban void detection, and microzonation for seismic hazard assessment.
• Data Sources:Passive microtremor recordings and active controlled-source wavefields.

Background

Surface wave inversion is the process of estimating the physical properties of a medium from the observed dispersion of seismic energy. Rayleigh waves, which involve a combination of longitudinal and transverse motions, and Love waves, which involve horizontal shearing, travel along the interface of different materials. Because lower-frequency waves penetrate deeper into the earth than higher-frequency waves, their velocities reflect the properties of deeper layers. This frequency-dependent velocity is the cornerstone of surface wave analysis.

Historically, this inversion was performed using linearized least-squares optimization. While computationally efficient, these methods are sensitive to the initial starting model and often fail to capture the full range of possible solutions that fit the observed data. In heterogeneous media, such as urban fill or complex glaciated terrains, the relationship between the dispersion curve and the underlying geology is highly non-linear. Deterministic methods may converge on a local minimum, leading to an accurate fit of the data that does not represent the physical reality of the site.

The development of global search algorithms, specifically those based on Bayesian inference, was a response to these limitations. By incorporating prior knowledge—such as borehole logs or known geological constraints—into the inversion process, researchers can narrow the search space without sacrificing the ability to explore multiple potential solutions. This approach allows for the generation of "probability clouds" rather than single lines, showing where the model is well-constrained and where data remains ambiguous.

Bayesian Frameworks and MCMC Algorithms

A Bayesian framework for seismic inversion is built upon Bayes' Theorem, which relates the posterior probability of a model to the likelihood of the observed data and the prior probability of the model parameters. In mathematical terms, the posterior distribution represents the updated belief about the subsurface after considering new seismic observations.

“The power of the Bayesian approach lies in its ability to quantify what we do not know, as much as what we do know, providing a rigorous statistical basis for engineering decisions.”

Because the posterior distribution in seismic problems is often high-dimensional and analytically intractable, Markov Chain Monte Carlo (MCMC) methods are employed. These algorithms generate a sequence of models (a "chain") where each new model depends on the previous one. Through thousands or millions of iterations, the chain eventually samples the model space in proportion to the posterior probability. The most common variant used in surface wave studies is the Metropolis-Hastings algorithm, which uses a stochastic acceptance criterion to decide whether to move to a new candidate model.

Quantifying Uncertainty

Uncertainty in surface wave velocity models originates from several sources, including ambient noise, instrumental limitations, and the mathematical simplification of the Earth's structure. Bayesian methods distinguish between two primary types of uncertainty:

• Aleatory Uncertainty:The inherent randomness in the seismic source or environmental noise.
• Epistemic Uncertainty:The lack of knowledge regarding the true geological structure or the limitations of the forward modeling equations.

By producing an ensemble of accepted models, researchers can calculate the mean, median, and standard deviation of velocity at every depth. This allows for the creation of confidence intervals. For instance, an engineer may be presented with a profile stating that the shear-wave velocity at five meters depth is 250 m/s, but with a 95% confidence interval ranging from 220 m/s to 280 m/s. This information is vital for assessing the risk of soil liquefaction or the load-bearing capacity of a foundation.

Applications in Non-Destructive Testing (NDT)

In the context of infrastructure management, the Surface Wave Hub applies these algorithms to analyze bridges, tunnels, and deep foundations. Non-destructive testing using surface waves is particularly effective because it does not require drilling into the structure. For bridge foundations, the analysis of dispersion curves of induced waves can reveal the depth of piles or the presence of scour-induced voids around the abutments.

Bridge Foundation Integrity

When evaluating older bridge structures where original construction records may be missing, Bayesian inversion of Rayleigh waves can estimate the thickness and stiffness of the foundation elements. MCMC methods provide a probabilistic assessment of whether a foundation reaches a specific load-bearing stratum. If the inversion shows a high probability of low-velocity material beneath a pile tip, it indicates potential instability. The use of Bayesian priors—such as the known design specifications of similar bridges from the same era—significantly improves the reliability of these NDT results.

Challenges in Urban Environments

Urban geophysical surveys face unique challenges, including high levels of anthropogenic noise (traffic, machinery) and restricted space for laying out geophone arrays. In these settings, microtremor or ambient noise surface wave imaging is often preferred over active source methods. Bayesian frameworks are particularly advantageous in urban environments because they can effectively handle the high noise-to-signal ratios.

Research has shown that utilizing Bayesian priors can prevent the inversion from being misled by urban noise. For example, if a site is known to be situated on limestone bedrock, the prior can be set to exclude unrealistic models with soft clay at depth. This integration of local lithological characterization with seismic data helps in the detection of shallow subsurface anomalies, such as buried utilities or abandoned tunnels. Recent case studies in metropolitan areas have demonstrated that Bayesian inversion can successfully map the geometry of subterranean voids with much higher confidence than traditional methods, primarily by identifying the specific frequencies where the dispersion curve deviates from the expected regional trend.

Conclusion

The implementation of Bayesian frameworks and MCMC algorithms represents a significant advancement in the empirical study of acoustic wave propagation. By acknowledging the limitations of seismic data and the complexity of heterogeneous media, the Surface Wave Hub provides a more detailed and accurate picture of the subsurface. Whether applied to the characterization of geological stratigraphies or the non-destructive testing of critical infrastructure, the ability to quantify uncertainty ensures that engineering decisions are based on the most statistically sound interpretations of ground-motion signatures. Future developments in this field likely involve the optimization of MCMC algorithms to reduce computational overhead, allowing for real-time probabilistic inversion in the field.