Regularization Methods

The objective function for full waveform inversion (FWI) typically combines a data misfit term with a regularization term to stabilize the inversion process and incorporate prior information about the model. This can be formulated as:

\[\mathbf{m}^* = \underset{\mathbf{m}}{\arg\min} \left( \mathcal{J}(\mathbf{m}) + \alpha \mathcal{R}(\mathbf{m}) \right),\]

where \(\mathcal{J}(\mathbf{m})\) represents the data misfit term, and \(\mathcal{R}(\mathbf{m})\) is the regularization term. The regularization parameter \(\alpha\) controls the trade-off between data fidelity and model regularization. The choice of \(\mathcal{R}(\mathbf{m})\) depends on the desired properties of the recovered model and can take various forms.

1. Tikhonov Regularization

Tikhonov regularization is a commonly used method to stabilize inverse problems by adding smoothness constraints to the model parameters. It is often applied in FWI to suppress high-frequency artifacts and enforce geological continuity.

1.1 Introduction

(1) First-Order Tikhonov Regularization

First-order Tikhonov regularization penalizes the magnitude of the first-order differences (gradients) of the model parameters, enforcing smooth transitions between neighboring grid points:

\[\mathcal{R}_{\text{Tikh}1}(\mathbf{m}) = \alpha_x \sum_{i=1}^{n_x-1} \sum_{j=1}^{n_z} (m_{i+1,j} - m_{i,j})^2 + \alpha_z \sum_{i=1}^{n_x} \sum_{j=1}^{n_z-1} (m_{i,j+1} - m_{i,j})^2\]

This regularization is commonly used to reduce high-frequency noise while preserving overall model trends.

(2) Second-Order Tikhonov Regularization

Second-order Tikhonov regularization penalizes the magnitude of the second-order differences (curvature) of the model parameters:

\[\mathcal{R}_{\text{Tikh}2}(\mathbf{m}) = \alpha_x \sum_{i=1}^{n_x-2} \sum_{j=1}^{n_z} (m_{i+2,j} - 2m_{i+1,j} + m_{i,j})^2 + \alpha_z \sum_{i=1}^{n_x} \sum_{j=1}^{n_z-2} (m_{i,j+2} - 2m_{i,j+1} + m_{i,j})^2\]

This regularization enforces smoothness by suppressing second-order variations, leading to a more continuous and geologically reasonable model with minimal artificial oscillations.

1.2 Pros and Cons

Advantages:

Stabilization of Ill-posed Problems: Helps to regularize the inversion process and prevent overfitting to noise.
Enforcement of Smoothness: Encourages geologically plausible models by reducing sharp variations in the model parameters.
Flexibility: Can be applied in different orders (e.g., first-order and second-order) to control different aspects of smoothness.

Disadvantages:

Loss of Resolution: Excessive smoothing can remove meaningful small-scale structures, reducing inversion accuracy.
Bias in Model Recovery: May bias the solution toward overly smooth models that do not capture sharp geological boundaries.
Tuning of Regularization Parameters: The selection of optimal regularization coefficients (\(\alpha_x, \alpha_z\)) requires careful calibration.

1.3 Comparison Between First-Order and Second-Order Tikhonov Regularization

Regularization Type	Effect	Typical Use Case
First-Order (\(\mathcal{R}_{\text{Tikh}1}\))	Penalizes sharp gradients, reducing noise and enforcing smoothness.	Suitable for suppressing high-frequency noise while preserving large-scale structures.
Second-Order (\(\mathcal{R}_{\text{Tikh}2}\))	Penalizes curvature, leading to a smoother and more continuous model.	Used when geological continuity is a priority, minimizing artificial oscillations.

The choice between first-order and second-order Tikhonov regularization depends on the desired balance between resolution and smoothness in the inversion process.

2. Total Variation (TV) Regularization

Total Variation (TV) regularization is widely used in inverse problems to preserve sharp boundaries while reducing noise. Unlike Tikhonov regularization, which enforces smoothness, TV regularization promotes piecewise-constant solutions, making it particularly useful in applications where geological structures contain sharp contrasts.

2.1 Introduction

(1) First-Order TV Regularization

First-order total variation (TV1) regularization penalizes the absolute differences between neighboring model parameters:

\[\mathcal{R}_{\text{TV1}}(\mathbf{m}) = \alpha_x \sum_{i=1}^{n_x-1} \sum_{j=1}^{n_z} |m_{i+1,j} - m_{i,j}| + \alpha_z \sum_{i=1}^{n_x} \sum_{j=1}^{n_z-1} |m_{i,j+1} - m_{i,j}|\]

This formulation enforces sparsity in the model gradients, allowing for sharp transitions between regions with different properties, making it well-suited for geological structures with clear discontinuities, such as faults and layering.

(2) Second-Order TV Regularization

Second-order total variation (TV2) regularization penalizes the absolute second-order differences (curvature) in the model parameters:

\[\mathcal{R}_{\text{TV2}}(\mathbf{m}) = \alpha_x \sum_{i=1}^{n_x-2} \sum_{j=1}^{n_z} |m_{i+2,j} - 2m_{i+1,j} + m_{i,j}| + \alpha_z \sum_{i=1}^{n_x} \sum_{j=1}^{n_z-2} |m_{i,j+2} - 2m_{i,j+1} + m_{i,j}|\]

By penalizing curvature, TV2 regularization reduces high-frequency noise while maintaining significant structural variations in the model, making it particularly useful when smoothness is required without losing important geological features.

2.2 Pros and Cons

Advantages:

Edge Preservation: TV1 regularization maintains sharp boundaries in the model, preventing excessive smoothing of significant transitions.
Robustness to Noise: TV regularization is effective in reducing small-scale noise without blurring geological structures.
Adaptive Smoothing: Unlike Tikhonov regularization, which uniformly smooths the model, TV regularization selectively smooths regions without strong discontinuities.

Disadvantages:

Non-Differentiability: The absolute value function in TV regularization makes gradient-based optimization more challenging and often requires specialized numerical methods.
Blocky Artifacts: TV1 regularization can produce staircasing effects, where the model appears as piecewise-constant regions rather than smooth transitions.
Computational Cost: Solving TV-regularized inverse problems can be computationally demanding due to the need for iterative solvers and total variation minimization techniques.

2.3 Comparison Between First-Order and Second-Order TV Regularization

Regularization Type	Effect	Typical Use Case
First-Order (\(\mathcal{R}_{\text{TV1}}\))	Preserves sharp edges by penalizing first-order differences.	Suitable for models with abrupt changes, such as faulted geological formations.
Second-Order (\(\mathcal{R}_{\text{TV2}}\))	Reduces noise while maintaining large-scale variations.	Useful when geological continuity is desired without excessive smoothing.

The selection between first-order and second-order TV regularization depends on the need for sharp boundaries versus smooth transitions in the inversion process.