# ⚛  L1 Principle — Surface Wave Dispersion — shallow Vs profile inversion

**ID:** `L1-274` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *Rayleigh/Love wave phase-velocity dispersion*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 1D shear velocity
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554051

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## 🧠 1. Introduction

**Surface Wave Dispersion — shallow Vs profile inversion** is a **nonlinear inverse problem** whose unknown lives in **1D shear velocity** space, within the **Rayleigh/Love wave phase-velocity dispersion** sub-domain of **Geophysics**.

Measurements consist of acoustic pressure waves recorded by transducers via a **geophones** sensing mechanism.

The forward operator applies, in order: L · mass stiffness 1D operator; L · eigenvalue operator; L · dispersion match operator; integration over a band of frequencies.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · mass stiffness 1D → L · eigenvalue → L · dispersion match → Frequency integration → **y** (detector).

```
y = ∫dω `L.dispersion_match` `L.eigenvalue` `L.mass_stiffness_1D` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.mass_stiffness_1D` | L · mass stiffness 1d operator |
| `L.eigenvalue` | L · eigenvalue operator |
| `L.dispersion_match` | L · dispersion match operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | Rayleigh/Love wave phase-velocity dispersion |
| Carrier | acoustic |
| Problem class | nonlinear_inverse |
| Solution space | 1D_shear_velocity |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | Vs_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `masw` · **Forward operator:** `masw_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 25 |
| N freqs | — | 60 |
| N layers | — | 10 |
| Vp prior error | — | 0 |
| Density prior error | — | 0 |
| Higher mode contamination | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N freqs | — | 10 – 500 |
| N layers | — | 2 – 100 |
| Vp prior error | — | 0.0 – 0.2 |
| Density prior error | — | 0.0 – 0.1 |
| Higher mode contamination | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 4.0

| Metric | Range |
|---|---|
| Vs rel error | 0.5 – 30.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **Vs_rel_error_percent**, with κ = `400` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x7601231eefd5fa4fb5049dd796a20ce6098d58c50d7aca10ae839cfdcdc640fe`
- **Chain tx hash:** `0x1d67b90b95c8e1dc970adcb7495b08364611afc2a4c8e6f79779e9b50df4d84b`
- **Chain block:** `41554051`

---

## File Mapping

This bundle consists of: `L1-274.md`, `L1-274.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-274.md` | Source of truth — edit this | Human or LLM |
| `L1-274.json` | Structured metadata for the registry | LLM regenerates from the sections above |

**Prompt for your LLM after editing this Markdown:**

> Read the attached Markdown. Regenerate the sibling `.json` so every field matches.
> Preserve the schema documented in the rows above.
> Output each file in its own fenced code block tagged with the filename.
> Output only the JSON object.

_This Markdown was auto-synthesized from the catalog row for `L1-274`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._