# ⚛  L1 Principle — Gravity Field Inversion — density imaging

**ID:** `L1-279` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *3D gravity inversion for subsurface density*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D density anomaly
> **📡 Carrier:** mechanical · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554063

---

## 🧠 1. Introduction

**Gravity Field Inversion — density imaging** is a **linear inverse problem** whose unknown lives in **3D density anomaly** space, within the **3D gravity inversion for subsurface density** sub-domain of **Geophysics**.

Measurements consist of mechanical vibrations or strain signals via a **gravimeter** sensing mechanism.

The forward operator applies, in order: L · newton gravity operator; L · linear forward operator; L · regularize operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Non-unique (Coulomb's inverse potential); requires smoothness/sparsity priors.

## ⚙ 2. Forward Model

Physical chain: **x** → L · newton gravity → L · linear forward → L · regularize → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.regularize` `L.linear_forward` `L.newton_gravity` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.newton_gravity` | L · newton gravity operator |
| `L.linear_forward` | L · linear forward operator |
| `L.regularize` | L · regularize operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | 3D gravity inversion for subsurface density |
| Carrier | mechanical |
| Problem class | linear_inverse |
| Solution space | 3D_density_anomaly |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 5 |
| L dag | 3 |

## 📡 4. Measurement Model

Non-unique (Coulomb's inverse potential); requires smoothness/sparsity priors.

| Metric | Value |
|---|---|
| Metric | density_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `gravity_inversion` · **Forward operator:** `gravity_inversion_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 256 |
| W | px | 256 |
| Z | — | 64 |
| Snr db | dB | 30 |
| Tidal error | — | 0 |
| N observations | — | 1024 |
| Background error | — | 0 |
| Terrain correction error | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 32 – 2048 |
| W | px | 32 – 2048 |
| Z | — | 8 – 512 |
| Snr db | dB | 0 – 40 |
| Tidal error | — | 0.0 – 0.2 |
| N observations | — | 64 – 100000 |
| Background error | — | 0.0 – 0.3 |
| Terrain correction error | — | 0.0 – 0.2 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 4.0

| Metric | Range |
|---|---|
| Density rel | 0.5 – 30.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **density_rel_error_percent**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x15850d9e58b36d8250c531c5eb127e483c2e7eb40cdb7d431dcd9100df9310ba`
- **Chain tx hash:** `0x0e86fd111978b84a8c945285ec30fa19d47e8c817cc965e7f21caed0d504262f`
- **Chain block:** `41554063`

---

## File Mapping

This bundle consists of: `L1-279.md`, `L1-279.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-279.md` | Source of truth — edit this | Human or LLM |
| `L1-279.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 `L1-279`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._