Bayesian Frameworks for Uncertainty in 1D Velocity Profiling

Seismic surface wave analysis is a foundational technique in geotechnical engineering and geophysics for determining the shear-wave velocity (Vs) structure of the near-surface. The process typically involves measuring Rayleigh wave dispersion—the phenomenon where different frequencies of seismic waves travel at different speeds through a layered medium—and then performing an inversion to estimate the underlying physical properties of the subsurface. In conventional practice, this inversion often relies on deterministic methods that search for a single ‘best-fit’ model of velocity versus depth. However, the non-linear and non-unique nature of the inversion problem means that multiple disparate velocity profiles can produce nearly identical dispersion curves, leading to potential inaccuracies in site characterization.

Bayesian frameworks address these limitations by treating the inversion problem as an exercise in probabilistic inference. Instead of identifying a single solution, Bayesian methods characterize the entire range of models that are consistent with the observed data and any prior knowledge of the site geology. This approach allows for the rigorous quantification of uncertainty, providing practitioners with a statistical distribution of possible velocity profiles rather than a solitary, potentially misleading estimate. The adoption of these frameworks in 1D velocity profiling has increased as computational power and algorithmic efficiency have improved, allowing for more complex and realistic subsurface interpretations.

At a glance

Methodology:Bayesian inference utilizes Bayes’ Theorem to update the prior probability of a model as new seismic data becomes available.
Objective:To generate a posterior probability distribution (PPD) that describes the likelihood of various shear-wave velocity profiles.
Key Parameter:Uncertainty quantification, which measures the confidence intervals for velocity and layer thickness at different depths.
Primary Tool:Markov Chain Monte Carlo (MCMC) algorithms, which sample the model space to find a representative set of solutions.
Application:Site classification for earthquake engineering (Vs30), foundation design, and the detection of subsurface anomalies.

Background

The characterization of the subsurface through surface wave analysis, such as Multichannel Analysis of Surface Waves (MASW) or Spectral Analysis of Surface Waves (SASW), is predicated on the dispersive nature of Rayleigh and Love waves in heterogeneous media. Because high-frequency waves (short wavelengths) are constrained to the shallowest layers and low-frequency waves (long wavelengths) penetrate deeper, the observed phase velocity at different frequencies contains information about the variation of elastic moduli with depth. The mathematical relationship between the layered earth model (thickness, density, shear-wave velocity, and P-wave velocity) and the dispersion curve is well-established through forward modeling codes.

Despite this established relationship, the inverse problem—moving from an observed dispersion curve back to the earth model—is inherently ill-posed. Deterministic inversion techniques, such as those utilizing the least-squares approach or various local search optimization algorithms, are highly sensitive to the initial starting model. If the starting model is far from reality, the algorithm may converge on a local minimum, resulting in a ‘best-fit’ profile that does not accurately reflect the true lithology. Furthermore, these methods do not naturally provide a measure of how much confidence can be placed in the resulting profile.

The move toward Bayesian frameworks in seismic problems, as highlighted by the work of Minson et al. (2013), represents a major change toward detailed uncertainty estimation. By framing the problem probabilistically, researchers can move beyond the constraints of a single solution and evaluate the trade-offs between parameters, such as the correlation between layer thickness and velocity. This is particularly vital in engineering contexts where an overestimation of soil stiffness can lead to structural failure or inefficient design.

The Bayesian Inversion Framework

The core of the Bayesian framework is Bayes’ Theorem, which relates the posterior probability of a model (m) given the observed data (d) to the likelihood of the data given the model and the prior probability of the model. This is expressed as:

P(m|d) ∑ P(d|m) P(m) / P(d)

In this equation,P(m|d)Is the posterior probability distribution, which is the final product of the inversion.P(d|m)Is the likelihood function, representing how well a specific model predicts the observed dispersion data, often assuming a Gaussian distribution for measurement errors.P(m)Is the prior distribution, representing what is known about the subsurface before the data is collected (e.g., that velocity must be positive or that it generally increases with depth).P(d)Is a normalization constant known as the evidence.

Prior Knowledge and Parameterization

One of the strengths of the Bayesian approach is the explicit inclusion of prior information. In 1D velocity profiling, this might include borehole logs, known depths to bedrock, or regional geological trends. The choice of parameterization—deciding how many layers to include in the model—is also a critical component. Fixed-dimension inversions require the user to pre-specify the number of layers, which can introduce bias. In contrast, transdimensional Bayesian inversions allow the number of layers to be a variable itself, letting the data determine the appropriate level of model complexity. This prevents ‘over-fitting’ the data with too many layers or ‘under-fitting’ it with too few.

Sampling via Markov Chain Monte Carlo

Because the posterior distribution in seismic inversion is usually too complex to calculate analytically, numerical sampling methods are required. The Markov Chain Monte Carlo (MCMC) algorithm is the most common tool for this task. MCMC methods, such as the Metropolis-Hastings algorithm, explore the model space by taking a ‘random walk.’ At each step, a small change is made to the current model. If the change improves the fit to the data, it is usually accepted; if it worsens the fit, it is accepted only with a certain probability. Over thousands of iterations, the chain spends more time in high-probability regions, effectively mapping out the posterior distribution.

Uncertainty Estimation and Minson et al. (2013)

The contribution of Sarah E. Minson and her colleagues in 2013 to the field of seismic inversion emphasized the necessity of a fully Bayesian approach for strong uncertainty estimation. While their work often focused on finite fault models and source characterization, the principles they championed have direct relevance to surface wave profiling. They demonstrated that neglecting the full range of model possibilities can lead to a false sense of certainty in seismic results. Their research showed that Bayesian methods provide not just a mean or median solution, but also the variance and covariances between parameters, which are essential for understanding the non-uniqueness of the earth structure.

In the context of 1D velocity profiling, these frameworks allow for the generation of ‘cloud’ plots of velocity profiles. In these visualizations, hundreds of potential profiles are overlaid, with higher-density regions indicating the most probable velocity at a given depth. If the cloud is narrow, the data has high resolving power; if it is wide, the uncertainty is high, suggesting that the dispersion data alone is insufficient to constrain the subsurface at that depth.

Practical Applications in Geotechnical Engineering

The practical application of Bayesian uncertainty quantification is most visible in geotechnical risk assessment. The shear-wave velocity of the top 30 meters (Vs30) is a standard parameter for building codes and seismic site classification. A deterministic analysis might provide a Vs30 value of 355 m/s, placing a site in ‘Site Class C.’ However, a Bayesian analysis might reveal a 30% probability that the true Vs30 is actually 345 m/s, which would place the site in the more restrictive ‘Site Class D.’

Infrastructure and Foundation Design

For large-scale infrastructure like bridges and tunnels, understanding the uncertainty in the depth to a stiff bearing layer is critical. Bayesian frameworks can quantify the probability of bedrock depth within a specific range, allowing engineers to apply appropriate safety factors to foundation designs. This is particularly relevant in urban environments where seismic noise and buried utilities can contaminate data, increasing the uncertainty that must be accounted for during the inversion process.

Non-Destructive Testing (NDT)

Surface wave methods are also employed in the evaluation of engineered materials, such as concrete pavements and liners. By analyzing the dispersion of induced waves, researchers can detect voids or delamination. The Bayesian approach allows for the detection of these anomalies with a quantified confidence level, aiding in the prioritization of repairs and maintenance for critical infrastructure.

Challenges and Future Directions

The primary hurdle for Bayesian frameworks in 1D velocity profiling is computational demand. MCMC sampling requires the execution of thousands, if not millions, of forward models. While this was once a prohibitive cost, modern multi-core processors and parallel computing have made these methods increasingly accessible for routine engineering projects. Furthermore, developments in ‘surrogate modeling’ and machine learning are beginning to accelerate the forward modeling stage, potentially reducing the time required for Bayesian inversions from hours to minutes.

Another area of active research is the integration of multi-modal data. By combining Rayleigh wave dispersion with Love wave dispersion and horizontal-to-vertical spectral ratios (HVSR), Bayesian frameworks can further reduce uncertainty and break the trade-offs between parameters. The ability to integrate these diverse data types into a single probabilistic model remains one of the most promising frontiers for the Surface Wave Hub and the broader geophysical community.