High-Fidelity Compression of Seismic Velocity Models via SIREN Auto-Decoders

This paper proposes a high-fidelity SIREN auto-decoder framework that compresses multi-structural seismic velocity models into compact latent vectors, achieving significant storage reduction while enabling high-quality reconstruction, smooth interpolation, and zero-shot super-resolution for geophysical applications.

The Gist

Imagine you are trying to store a massive library of ancient, detailed maps. These aren't just simple drawings; they are complex, 3D-like maps of the Earth's underground, showing exactly how fast sound waves travel through different rocks, faults, and oil pockets.

The Problem: The "Pixelated" Map
Traditionally, scientists store these maps like a giant spreadsheet or a low-resolution digital photo. They chop the underground into tiny squares (pixels).

• The Issue: To make the map look smooth and accurate, you need billions of these squares. This makes the files huge, taking up massive amounts of computer memory.
• The Glitch: When you zoom in on a fault line (a crack in the earth) in these grid-based maps, it looks jagged and blocky, like a pixelated video game from the 1980s. It misses the smooth curves and sharp edges that nature actually has.

The Solution: The "Magic Recipe" (SIREN Auto-Decoders)
This paper introduces a new way to store these maps. Instead of saving the picture itself, the scientists save a tiny, secret recipe that can recreate the picture perfectly whenever you need it.

Here is how their method works, using some everyday analogies:

1. The "Magic Recipe" (The Latent Vector)

Imagine you have a complex cake. Instead of taking a photo of the cake (which is heavy and loses detail), you write down a short, 256-word recipe.

• Old Way: Save the photo of the cake (4,900 data points).
• New Way: Save the 256-word recipe.
• The Result: You shrink the file size by 19 times! But here's the magic: when you give this recipe to a smart kitchen robot (the AI), it doesn't just print a low-res photo. It bakes the cake from scratch, perfectly smooth, at any size you want.

2. The "Smart Kitchen Robot" (SIREN)

The robot they use is called SIREN.

• The Old Robot (ReLU): Imagine a robot that can only draw straight lines and flat surfaces. If you ask it to draw a smooth wave or a sharp cliff, it struggles. It gets "confused" by high-frequency details (the wiggles and sharp edges).
• The New Robot (SIREN): This robot is like a master artist who loves sine waves (the smooth, wiggly lines of a pendulum). Because the Earth's underground is full of waves and sharp cracks, this robot is perfectly suited to draw it. It can capture the smooth curves of rock layers and the razor-sharp edges of faults without getting pixelated.

3. The "Infinite Zoom" (Zero-Shot Super-Resolution)

This is the coolest trick.

• The Old Way: If you want to zoom in on a map, you have to ask the computer to "guess" the missing pixels. The image gets blurry and blocky.
• The New Way: Because the "recipe" describes the math of the map, not the pixels, you can ask the robot to draw the map at 100x, 1,000x, or even 1,000,000x resolution. It just calculates the new points on the fly. The image stays crisp and sharp, no matter how much you zoom in. It's like having a map that never gets blurry.

4. The "Smooth Morphing" (Latent Interpolation)

Imagine you have a map of a flat plain and a map of a mountain range.

• The Old Way: You can't really turn one into the other smoothly.
• The New Way: Because the "recipes" are stored in a smooth, organized space, you can mix them. You can take 50% of the flat plain recipe and 50% of the mountain recipe, and the robot will generate a brand new, realistic map of a gently rolling hill. It creates "in-between" worlds that make perfect geological sense.

Why Does This Matter?

• Storage: It saves massive amounts of space. Instead of storing millions of numbers for every map, you store a tiny list of numbers.
• Speed: It makes analyzing the Earth faster. Scientists can zoom in on specific areas (like a fault line) without loading the whole massive file.
• Accuracy: It helps find oil, gas, and geothermal energy more accurately because the maps show the true, smooth shape of the underground, not a blocky approximation.

In a Nutshell:
The authors took a messy, heavy, pixelated way of storing underground maps and replaced it with a smart, mathematical recipe. This recipe is tiny, creates perfect images at any size, and even lets us smoothly morph between different types of underground landscapes. It's like upgrading from a stack of paper photos to a single, infinite-resolution hologram.

Technical Summary

1. Problem Statement

Seismic velocity models are critical for geophysical applications such as hydrocarbon exploration, earthquake early warning, and carbon capture monitoring. Traditionally, these models are represented on discrete Cartesian grids (e.g., $70 \times 70$ grids). This approach faces three major limitations:

• Storage and I/O Bottlenecks: Storage requirements scale cubically with resolution, making high-fidelity, large-scale simulations prohibitively expensive.
• Discretization Artifacts: Grid-based representations introduce artifacts at sharp geological interfaces (faults, layers), failing to accurately represent smooth physical field gradients.
• Spectral Bias: Standard neural networks (MLPs with ReLU activations) suffer from "spectral bias," meaning they preferentially learn low-frequency components and struggle to capture the high-frequency details essential for sharp discontinuities in seismic data.

The paper aims to develop a resolution-independent, high-fidelity compression framework that overcomes grid limitations and spectral bias while enabling efficient storage and multi-scale analysis.

2. Methodology

The authors propose a SIREN Auto-Decoder framework. This approach combines Implicit Neural Representations (INRs) with an auto-decoder architecture.

Core Components

• Implicit Neural Representation (INR): Instead of storing a grid of values, the velocity field $v(x, z)$ is parameterized as a continuous neural function $f_\theta(x, z; z_i)$ that maps spatial coordinates $(x, z)$ and a latent code $z_i$ directly to velocity values.
• SIREN Decoder: The decoder network utilizes Sinusoidal Representation Networks (SIREN). Unlike ReLU, SIREN uses periodic activation functions ($\sin(\omega_0 \cdot)$).
  • Why SIREN? Periodic activations allow the network to learn high-frequency functions and complex spatial derivatives, effectively capturing sharp fault lines and subtle stratigraphic variations that standard MLPs miss.
  • Architecture: The network consists of an input layer concatenating 2D coordinates and a 256-dimensional latent code, followed by 4 hidden layers (512 units each) using sine activations, and a linear output layer.
• Auto-Decoder Architecture: Unlike standard autoencoders that use a separate encoder network, this framework optimizes a unique latent code ($z_i$) for each sample alongside a shared decoder.
  • Training Objective: The loss function combines Mean Squared Error (MSE) reconstruction loss with an $L_2$ regularization term on the latent codes. This encourages a smooth latent manifold, facilitating interpolation and preventing overfitting.
  • Optimization: The decoder parameters ($\theta$) and all latent codes ($\{z_i\}$) are jointly optimized using the Adam optimizer.

Compression Mechanism

• Input: A $70 \times 70$ velocity map containing 4,900 data points.
• Output: A compact 256-dimensional latent vector per sample.
• Ratio: This achieves a 19:1 compression ratio (reducing 4,900 floats to 256 floats per sample, plus a shared decoder).

3. Key Contributions

1. Novel Application: First application of SIREN auto-decoders to compress multi-structural seismic velocity models from the OpenFWI benchmark.
2. High-Fidelity Compression: Achieves a 19:1 compression ratio while maintaining high reconstruction quality (Average PSNR: 32.47 dB, SSIM: 0.956) across diverse geological families.
3. Smooth Latent Interpolation: Demonstrates that the learned latent space is continuous, allowing for the generation of physically plausible intermediate velocity structures between different geological models.
4. Zero-Shot Super-Resolution: Leverages the resolution-agnostic nature of INRs to reconstruct velocity fields at arbitrary resolutions (up to $280 \times 280$, or 4x original) without additional training.
5. Open Source: The authors provide open-source implementation and trained models to ensure reproducibility.

4. Experimental Results

The method was evaluated on 1,000 samples from the OpenFWI benchmark, covering five geological families: FlatVel, CurveVel, FlatFault, CurveFault, and Style.

• Overall Performance:
  • Average PSNR: 32.47 dB (Max: 38.92 dB).
  • Average SSIM: 0.956.
  • MSE: $4.23 \times 10^{-4}$.
• Performance by Geological Family:
  • Simple Structures (FlatVel, CurveVel): Achieved the highest quality (PSNR > 33 dB), as smooth layers align well with neural learning characteristics.
  • Faulted Structures (FlatFault, CurveFault): Presented greater challenges (PSNR ~30.5–31.3 dB) due to the difficulty of capturing sharp discontinuities, though SIREN outperformed standard ReLU-based approaches.
  • Complex Patterns (Style): Achieved intermediate performance (PSNR ~31.24 dB).
• Super-Resolution:
  • Reconstructing at $4\times$ resolution ($280 \times 280$) resulted in only a modest PSNR drop (to 30.95 dB), confirming the method's ability to resolve fine details without retraining.
• Storage Efficiency:
  • For a dataset of 100,000 samples, the framework reduces storage from 490M floats (raw grids) to ~26.6M floats (including decoder overhead), an 18.4x reduction.

5. Significance and Future Implications

• Geophysical Impact: This framework offers a paradigm shift from discrete grids to continuous representations, enabling more efficient storage, faster I/O, and higher-fidelity imaging of subsurface structures.
• Downstream Applications: The smooth latent space is ideal for Full Waveform Inversion (FWI), where optimization can be performed directly in the compressed latent space, significantly reducing dimensionality and incorporating learned geological priors.
• Generalizability: The success of SIREN in capturing sharp geological discontinuities validates periodic activation functions for other scientific domains involving high-frequency signals (e.g., fluid dynamics, climate modeling).
• Future Directions: The authors identify integrating latent-space FWI, scaling to larger datasets (40k+ samples), extending to 3D models, and uncertainty quantification as key next steps.

In summary, the paper demonstrates that SIREN auto-decoders provide a robust, high-fidelity solution for compressing and representing complex seismic velocity models, overcoming the resolution and spectral limitations of traditional grid-based methods.