# ⚛  L1 Principle — Seismic Travel-Time Tomography — ray-based velocity imaging

**ID:** `L1-272` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *First-arrival travel-time tomography*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D velocity field
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554050

---

## 🧠 1. Introduction

**Seismic Travel-Time Tomography — ray-based velocity imaging** is a **linear inverse problem** whose unknown lives in **3D velocity field** space, within the **First-arrival travel-time tomography** 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 · eikonal operator; L · ray trace operator; L · linearize operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Ill-posed; null space for regions with no ray coverage.

## ⚙ 2. Forward Model

Physical chain: **x** → L · eikonal → L · ray trace → L · linearize → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.linearize` `L.ray_trace` `L.eikonal` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.eikonal` | L · eikonal operator |
| `L.ray_trace` | L · ray trace operator |
| `L.linearize` | L · linearize operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | First-arrival travel-time tomography |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | 3D_velocity_field |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Ill-posed; null space for regions with no ray coverage.

| Metric | Value |
|---|---|
| Metric | velocity_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `traveltime_tomo` · **Forward operator:** `traveltime_tomo_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 128 |
| Snr db | dB | 30 |
| N sources | — | 32 |
| N receivers | — | 128 |
| Picker error | — | 0 |
| Source error | — | 0 |
| Ray coverage gaps | — | 0 |
| Velocity prior error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 1024 |
| W | px | 32 – 1024 |
| Z | — | 16 – 512 |
| Snr db | dB | 0 – 40 |
| N sources | — | 4 – 1024 |
| N receivers | — | 16 – 8192 |
| Picker error | — | 0.0 – 0.05 |
| Source error | — | 0.0 – 0.05 |
| Ray coverage gaps | — | 0.0 – 0.5 |
| Velocity prior error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 5.0

| Metric | Range |
|---|---|
| Velocity rel error | 0.5 – 50.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **velocity_rel_error_percent**, with κ = `600` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xe50bd37c5b909a13a9270fb9c8572b77fc01a9306d0fb047238a19c66b51b67c`
- **Chain tx hash:** `0xedefd71993f70361daf8861ff945586658ec8565c58c64dd87ef7787796a526a`
- **Chain block:** `41554050`

---

## File Mapping

This bundle consists of: `L1-272.md`, `L1-272.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-272.md` | Source of truth — edit this | Human or LLM |
| `L1-272.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 `L1-272`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._