# ⚛  L1 Principle — Underwater Acoustic Ray Tracing (Bellhop)

**ID:** `L1-262` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Acoustics — *Range-depth ocean-acoustic ray / gaussian beam tracing*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** transmission loss range depth
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554049

---

## 🧠 1. Introduction

**Underwater Acoustic Ray Tracing (Bellhop)** is a **linear inverse problem** whose unknown lives in **transmission loss range depth** space, within the **Range-depth ocean-acoustic ray / gaussian beam tracing** sub-domain of **Acoustics**.

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

The forward operator applies, in order: isotropic point-source emission model; L · refract · sspe operator; L · bottom reflect operator; detector accumulates flux over the exposure window.

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

## ⚙ 2. Forward Model

Physical chain: **x** → Source emission → L · refract · sspe → L · bottom reflect → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.bottom_reflect` `L.refract.sspe` δ_source x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.source_emit` | Isotropic point-source emission model |
| `L.refract.sspe` | L · refract · sspe operator |
| `L.bottom_reflect` | L · bottom reflect operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Acoustics |
| Sub domain | Range-depth ocean-acoustic ray / gaussian beam tracing |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | transmission_loss_range_depth |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.2 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | TL_error_dB |
| Secondary | arrival_time_MAE_ms |

## 📏 5. Operating Range (Ω)

**Center problem class:** `bellhop_ray` · **Forward operator:** `bellhop_ray_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N rays | — | 10000 |
| Snr db | dB | 30 |
| N depth | — | 256 |
| N range | — | 512 |
| Freq hz | Hz | 1000 |
| Ssp error | — | 0 |
| Range error | — | 0 |
| Bathymetry error | — | 0 |
| Bottom properties error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N rays | — | 1000 – 1000000 |
| Snr db | dB | -10 – 40 |
| N depth | — | 32 – 1024 |
| N range | — | 64 – 4096 |
| Freq hz | Hz | 10 – 100000 |
| Ssp error | — | 0.0 – 0.02 |
| Range error | — | 0.0 – 50.0 |
| Bathymetry error | — | 0.0 – 10.0 |
| Bottom properties error | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 3.0

| Metric | Range |
|---|---|
| Tl db | 0.5 – 20.0 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x6b27bd1b0fd0e67b1aa427bac1373576e9dd06c3f39cb6f6c050ba1d840ca301`
- **Chain tx hash:** `0x63b415cef453dc802d7f1c23afaea26358b183876297bf2d4aa424099e12bb92`
- **Chain block:** `41554049`

---

## File Mapping

This bundle consists of: `L1-262.md`, `L1-262.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-262.md` | Source of truth — edit this | Human or LLM |
| `L1-262.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 `L1-262`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._