Linearised versus Nonlinear Estimates of Uncertainty in Full Waveform Inversion

This study demonstrates that while both linearised and nonlinear methods accurately recover mean models in Bayesian Full Waveform Inversion, nonlinear approaches are essential for obtaining accurate uncertainty estimates and unbiased meta-property calculations, as linearised methods fail to capture complex uncertainty structures due to the inherent nonlinearity of wave physics.

The Gist

Imagine you are trying to draw a detailed map of a hidden cave system deep underground, but you can't go inside. Instead, you stand at the entrance and throw rocks (seismic waves) into the dark. You listen to the echoes bouncing back to figure out where the walls, stalactites, and open chambers are. This is essentially what Full Waveform Inversion (FWI) does for geologists: it uses sound waves to create a high-resolution image of the Earth's interior.

However, there's a catch. The echoes are messy, the rocks don't always hit the same spot, and the cave is incredibly complex. Because of this, there is always uncertainty in your map. You might be 90% sure a wall is there, but maybe it's actually a bit to the left, or maybe it's a different shape.

This paper asks a crucial question: How do we best measure that uncertainty?

The authors compare two ways of guessing the "fog of uncertainty" around their map:

1. The "Linearised" Method (The Simple Shortcut)
2. The "Nonlinear" Method (The Realistic Deep Dive)

Here is the breakdown of their findings using simple analogies.

1. The Two Approaches

The Linearised Method: "The Straight-Line Guess"

Imagine you are standing on a hill, trying to guess the shape of the valley below.

• How it works: You pick the best spot you can see (the most likely answer) and assume the ground slopes away from you in a perfectly straight line. You draw a neat, symmetrical oval around your spot to show where you think the ground might be.
• The Problem: Real valleys aren't straight lines; they curve, twist, and have hidden pockets. By assuming everything is a straight line, this method creates a "safe" but often wrong shape for the uncertainty. It thinks the uncertainty is uniform and simple, missing the complex curves of reality.

The Nonlinear Method: "The Reality Check"

• How it works: Instead of assuming a straight line, this method simulates thousands of different possible valleys that could fit the echoes you heard. It doesn't just look at the "best" spot; it explores the whole neighborhood, acknowledging that the ground might curve wildly.
• The Result: The "fog of uncertainty" it draws is messy, irregular, and complex. It might show that while you are very sure about the top of the hill, the bottom of the valley could be in three completely different shapes.

2. The Big Discovery: "The Loop" Effect

The authors found a specific place where these two methods disagree the most: Layer Boundaries (like the edge between a soft mud layer and a hard rock layer).

• The Nonlinear Reality: When sound waves hit a boundary between soft and hard rock, they interfere with each other in complex ways. The authors found that the uncertainty forms "loops" around these boundaries.
  • Analogy: Imagine trying to guess the exact edge of a shadow cast by a tree. The shadow isn't a sharp line; it's fuzzy. The nonlinear method realizes that the "fuzziness" is actually a loop: the edge could be slightly inside the soft rock or slightly outside it, and both fit the data equally well.
• The Linearised Mistake: The shortcut method misses these loops entirely. It draws a straight line through the fuzz.
  • The Consequence: Because the linear method gets the shape of the uncertainty wrong, it makes biased guesses about the size of underground objects. For example, it might tell a drilling company, "There is a huge pocket of oil here," when the nonlinear method says, "Actually, the pocket is much smaller, and we aren't sure where the edges are."

3. The "Fit" Test: Does the Map Match the Echoes?

To prove their point, the authors did a test:

1. They took a random guess from the "Linearised" map and simulated what the echoes should have sounded like.
2. They compared this to the actual echoes recorded.
3. The Result: The Linearised guesses produced echoes that sounded nothing like the real data. They were out of sync (phase differences).
4. The Nonlinear guesses, however, produced echoes that matched the real data almost perfectly.

The Metaphor:
Think of the Linearised method as a musician who knows the notes of a song but plays them on a straight, flat piano. They hit the right notes (the average model), but the feeling and rhythm (the uncertainty) are wrong.
The Nonlinear method is like a jazz musician who understands the complex improvisation of the song. Even if they don't know the exact next note, their range of possibilities (uncertainty) perfectly matches the rhythm of the original performance.

4. Why Does This Matter?

If you are an oil company, a mining firm, or a government planning a nuclear waste dump, you need to know not just where the rock is, but how sure you are about its shape and size.

• Linearised Uncertainty: Gives you a false sense of security. It says, "We are 90% sure this rock is 100 meters wide."
• Nonlinear Uncertainty: Gives you the truth. It says, "We are 90% sure this rock is between 50 and 150 meters wide, and the shape is weird."

The paper concludes that while the "Straight-Line" (Linearised) method is faster and cheaper, it is often dangerously inaccurate for complex underground structures. If you care about making safe, smart decisions based on the data, you must use the "Reality Check" (Nonlinear) method, even if it takes more computing power.

Summary

• The Goal: Map the Earth's underground using sound waves.
• The Problem: The math is messy and non-linear (curved).
• The Old Way: Simplify the math to a straight line. It's fast but gets the "fog of uncertainty" wrong, especially at boundaries.
• The New Way: Do the hard math to capture the curves. It's slower but reveals the true, complex shape of the uncertainty.
• The Takeaway: Don't take the shortcut. If you want to know the true size and shape of underground treasures (or dangers), you need the nonlinear method.

Technical Summary

Here is a detailed technical summary of the paper "Linearised versus Nonlinear Estimates of Uncertainty in Full Waveform Inversion" by Xuebin Zhao and Andrew Curtis.

1. Problem Statement

Seismic Full Waveform Inversion (FWI) is a high-resolution imaging technique used to reconstruct subsurface velocity structures. However, FWI solutions are inherently non-unique due to:

• Imperfect acquisition geometries (limited surface observations).
• Data noise.
• The nonlinearity of the forward wave physics.
• The under-determined nature of the inverse problem (heterogeneity across all scales).

While deterministic methods find a single "best" model (Maximum A Posteriori, or MAP), they fail to quantify the full range of possible solutions. Bayesian inference addresses this by estimating the posterior probability density function (pdf).

• The Linearised Approach: Traditionally, the posterior is approximated by a Gaussian distribution centered on the MAP solution. This relies on linearizing the forward physics (Born approximation) around the MAP model. It fails to capture uncertainties arising from the nonlinearity of wave physics.
• The Nonlinear Approach: Fully nonlinear methods aim to estimate the true, complex posterior distribution without linearization. However, these are computationally expensive (e.g., Markov Chain Monte Carlo) or require sophisticated algorithms (Variational Inference).

The Core Question: How significantly do linearised uncertainty estimates differ from fully nonlinear estimates in realistic FWI scenarios, and does this difference impact the accuracy of derived geological properties?

2. Methodology

The authors compare three methods applied to 2D acoustic FWI problems:

A. Linearised Method (Gaussian Approximation)

• Process: A deterministic, gradient-based optimization (L-BFGS) finds the MAP solution ($m_{MAP}$).
• Uncertainty Estimation: The forward function is linearized around $m_{MAP}$. The posterior covariance matrix ($\Sigma_{post}$) is approximated using the Gauss-Newton Hessian ($H$) and prior covariance ($\Sigma_{prior}$).
• Implementation: Due to the high dimensionality ($>10,000$ parameters), the Hessian is approximated using Randomized Singular Value Decomposition (SVD) to compute the posterior covariance efficiently.
• Assumption: The posterior is locally Gaussian; it ignores nonlinear physics.

B. Nonlinear Method 1: Physically Structured Variational Inference (PSVI)

• Algorithm: A variational inference approach that optimizes a transformed Gaussian distribution to approximate the true posterior by minimizing the Kullback-Leibler (KL) divergence.
• Transformation: Uses a logit function to map unconstrained Gaussian variables to bounded physical parameters (e.g., positive velocities).
• Covariance: Constructs a sparse covariance matrix considering only spatially proximal correlations (within one dominant wavelength) to manage computational cost.
• Goal: Finds the optimal Gaussian approximation over the full parameter space, capturing global nonlinearities.

C. Nonlinear Method 2: Stein Variational Gradient Descent (SVGD)

• Algorithm: An independent variational method that iteratively updates a set of particles (samples) to match the posterior density.
• Role: Used as a cross-validation tool. Unlike PSVI, SVGD is not constrained to a Gaussian form, providing an independent check on the true nonlinear uncertainty structure.

3. Key Contributions

1. Systematic Comparison: This is the first systematic comparison of linearised vs. nonlinear uncertainty estimates specifically within the context of Bayesian FWI.
2. Identification of "Loop-like" Uncertainties: The study demonstrates that nonlinear methods reveal specific uncertainty structures (loops around layer boundaries) caused by the nonlinearity of wave physics, which are completely absent in linearised estimates.
3. Quantification of Bias in Meta-Properties: The authors prove that linearised uncertainty estimates lead to biased interpretations of geological meta-properties (e.g., volume of geological bodies), whereas nonlinear methods provide unbiased estimates.
4. Data Fit Validation: The paper introduces a rigorous validation showing that samples drawn from linearised posteriors often fail to fit the observed data, whereas nonlinear samples do.

4. Results

A. Layered Model Experiments

The authors tested three scenarios: ideal data, higher frequency data, and sparse data.

• Mean Models: All three methods (Linearised, PSVI, SVGD) recovered the posterior mean velocity models with similar accuracy.
• Uncertainty Structures:
  • Linearised: Produced smooth, symmetric uncertainty maps that did not reflect the physics of wave interference.
  • Nonlinear (PSVI/SVD): Revealed high uncertainty loops around layer interfaces and velocity anomalies.
  • Specific Finding: Near interfaces, nonlinear methods showed higher uncertainty on the low-velocity side and lower uncertainty on the high-velocity side. This is due to wave interference in recorded seismograms (nonlinear physics) where travel times are dominated by the high-velocity side. The linearised method showed the opposite trend (high uncertainty on the high-velocity side), which is physically incorrect.
• Data Sensitivity: When data were highly informative (high frequency) or non-informative (sparse), linearised and nonlinear results converged. The divergence was most pronounced when data sensitivity was moderate but the problem remained nonlinear.

B. Modified Marmousi Model (Realistic Scenario)

A complex model with a high-impedance contrast interface was used.

• Data Fit: Samples drawn from the linearised posterior generated synthetic data with significant misfits (phase errors) compared to observed data. Samples from PSVI and SVGD fit the data well (residuals matched noise levels).
• Likelihood Analysis: The linearised posterior distribution had a low average log-likelihood, indicating that most samples in the estimated uncertainty range were actually poor fits to the data.
• Interrogation Theory (Meta-Properties): The authors used the uncertainty estimates to calculate the volume of a low-velocity body.
  • Linearised Result: Underestimated the uncertainty and provided a biased volume estimate that did not include the true value with high probability.
  • Nonlinear Result: Provided an accurate probability density for the volume, with the true value falling within the high-probability region.

5. Significance and Conclusions

• Failure of Linearisation: The linearisation of wave physics (Born approximation) fundamentally distorts the uncertainty structure in FWI. It creates a "false sense of security" by providing tight but incorrect uncertainty bounds, particularly around geological interfaces.
• Impact on Decision Making: Because linearised methods yield biased estimates of geological volumes and properties, they can lead to non-robust risk-based decision-making in exploration and reservoir management.
• Recommendation: While linearised methods are computationally cheaper, the authors argue that fully nonlinear inversion methods (like PSVI or SVGD) should be prioritized whenever accurate uncertainty quantification is required. The computational cost is justified by the elimination of bias in both data fitting and geological interpretation.
• Future Work: The study suggests extending these comparisons to 3D elastic FWI and investigating the impact of complex, non-Gaussian priors (e.g., deep learning-based priors) on these uncertainty estimates.

In summary, the paper establishes that while linearised methods can find the correct mean model, they fail catastrophically in describing the uncertainty of that model in nonlinear regimes, rendering them unsuitable for rigorous quantitative interpretation of subsurface properties.