# ⚛  L1 Principle — Magnetotelluric Inversion — resistivity imaging

**ID:** `L1-281` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *Deep-Earth resistivity via natural-source EM*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D resistivity field
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 10 · **⛓ Block:** 41554064

---

## 🧠 1. Introduction

**Magnetotelluric Inversion — resistivity imaging** is a **nonlinear inverse problem** whose unknown lives in **3D resistivity field** space, within the **Deep-Earth resistivity via natural-source EM** sub-domain of **Geophysics**.

Measurements consist of electromagnetic field measurements via a **mt station** sensing mechanism.

The forward operator applies, in order: L · maxwell freq operator; L · impedance tensor operator; L · nlcg operator; integration over a band of frequencies.

Observations are corrupted by additive Gaussian noise. Non-linear ill-posed; static shift and galvanic distortion degrade images.

## ⚙ 2. Forward Model

Physical chain: **x** → L · maxwell freq → L · impedance tensor → L · nlcg → Frequency integration → **y** (detector).

```
y = ∫dω `L.nlcg` `L.impedance_tensor` `L.maxwell_freq` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.maxwell_freq` | L · maxwell freq operator |
| `L.impedance_tensor` | L · impedance tensor operator |
| `L.nlcg` | L · nlcg operator |
| `int.frequency` | Integration over a band of frequencies |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | Deep-Earth resistivity via natural-source EM |
| Carrier | em |
| Problem class | nonlinear_inverse |
| Solution space | 3D_resistivity_field |
| Noise model | gaussian |
| Integration axis | frequency |
| Difficulty delta | 10 |
| L dag | 3.7 |

## 📡 4. Measurement Model

Non-linear ill-posed; static shift and galvanic distortion degrade images.

| Metric | Value |
|---|---|
| Metric | resistivity_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `mt_inversion` · **Forward operator:** `mt_inversion_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 64 |
| Snr db | dB | 25 |
| N freqs | — | 30 |
| N stations | — | 50 |
| Static shift | — | 0 |
| Impedance error | — | 0 |
| Topography error | — | 0 |
| Galvanic distortion | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 16 – 1024 |
| W | px | 16 – 1024 |
| Z | — | 8 – 512 |
| Snr db | dB | 0 – 40 |
| N freqs | — | 1 – 200 |
| N stations | — | 5 – 10000 |
| Static shift | — | 0.0 – 0.5 |
| Impedance error | — | 0.0 – 0.2 |
| Topography error | — | 0.0 – 0.2 |
| Galvanic distortion | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 8.0

| Metric | Range |
|---|---|
| Res rel | 1.0 – 50.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **resistivity_rel_error_percent**, with κ = `1000` and δ = `10`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x7acdd2367e0c594ace3edfc190f8303065bb2b974a31c3f15552f9828a2c8e0e`
- **Chain tx hash:** `0x46c43d0ff167d13d9a1a0272948d3116920c593d62c0395a3e2bd8d054b73b81`
- **Chain block:** `41554064`

---

## File Mapping

This bundle consists of: `L1-281.md`, `L1-281.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-281.md` | Source of truth — edit this | Human or LLM |
| `L1-281.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 `L1-281`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._