# ⚛  L1 Principle — Induced Polarization — chargeability imaging

**ID:** `L1-284` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *Time- or frequency-domain IP inversion*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** 3D chargeability field
> **📡 Carrier:** em · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554064

---

## 🧠 1. Introduction

**Induced Polarization — chargeability imaging** is a **nonlinear inverse problem** whose unknown lives in **3D chargeability field** space, within the **Time- or frequency-domain IP inversion** sub-domain of **Geophysics**.

Measurements consist of electromagnetic field measurements via a **dc electrodes transient** sensing mechanism.

The forward operator applies, in order: L · current injection operator; L · relaxation time operator; L · cole cole 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** → L · current injection → L · relaxation time → L · cole cole → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.cole_cole` `L.relaxation_time` `L.current_injection` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.current_injection` | L · current injection operator |
| `L.relaxation_time` | L · relaxation time operator |
| `L.cole_cole` | L · cole cole operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | Time- or frequency-domain IP inversion |
| Carrier | em |
| Problem class | nonlinear_inverse |
| Solution space | 3D_chargeability_field |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | chargeability_rel_error_percent |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `ip` · **Forward operator:** `ip_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 128 |
| W | px | 128 |
| Z | — | 32 |
| Snr db | dB | 30 |
| N timesteps | — | 20 |
| N quadrupoles | — | 500 |
| Cc params error | — | 0 |
| Topography error | — | 0 |
| Electromagnetic coupling | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 16 – 1024 |
| W | px | 16 – 1024 |
| Z | — | 4 – 256 |
| Snr db | dB | 0 – 40 |
| N timesteps | — | 5 – 500 |
| N quadrupoles | — | 10 – 100000 |
| Cc params error | — | 0.0 – 0.3 |
| Topography error | — | 0.0 – 0.2 |
| Electromagnetic coupling | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 8.0

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

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **chargeability_rel_error_percent**, with κ = `560` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x2c7d49af6c6dd8bbc96bdeaba256d265bfdaf464f142ca2f244a340e8fcc3445`
- **Chain tx hash:** `0x7f2c8b14cd3bb2a8015a8fbf8252c051f8652e4d374f48a373022d63ef60fb39`
- **Chain block:** `41554064`

---

## File Mapping

This bundle consists of: `L1-284.md`, `L1-284.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-284.md` | Source of truth — edit this | Human or LLM |
| `L1-284.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 `L1-284`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._