# ⚛  L1 Principle — Hyperspectral Spectral Unmixing

**ID:** `L1-472` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geodesy — *Remote sensing spectroscopy*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** abundance map
> **📡 Carrier:** photon · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555299

---

## 🧠 1. Introduction

**Hyperspectral Spectral Unmixing** is a **linear inverse problem** whose unknown lives in **abundance map** space, within the **Remote sensing spectroscopy** sub-domain of **Geodesy**.

Measurements consist of photons collected by an optical detector via a **hyperspectral reflectance unmixing** sensing mechanism.

The forward operator applies, in order: an unspecified linear measurement operator; O · nonneg ls · fcls operator; S · endmember · extraction operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered abundance_map is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); endmember_variability dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Generic linear operator → O · nonneg ls · fcls → S · endmember · extraction → **y** (detector).

```
y = `S.endmember.extraction` `O.nonneg_ls.fcls` A x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.linear_op` | An unspecified linear measurement operator |
| `O.nonneg_ls.fcls` | O · nonneg ls · fcls operator |
| `S.endmember.extraction` | S · endmember · extraction operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geodesy |
| Sub domain | Remote sensing spectroscopy |
| Carrier | photon |
| Problem class | linear_inverse |
| Solution space | abundance_map |
| Noise model | gaussian |
| Integration axis | spectral |
| Difficulty delta | 3 |
| L dag | 2.8 |

## 📡 4. Measurement Model

Existence of the recovered abundance_map is guaranteed within the declared Omega bounds. Uniqueness holds on the measurement-supported subspace; out-of-support modes are controlled by declared priors. Stability is conditionally stable (kappa_eff ~= 50); endmember_variability dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | abundance_RMSE |
| Secondary | endmember_SAM_deg |

## 📏 5. Operating Range (Ω)

**Center problem class:** `linear_inverse` · **Forward operator:** `hyperspectral_reflectance_unmixing`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| H | px | 100 |
| W | px | 100 |
| Snr db | dB | 30 |
| K endmembers | — | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| H | px | 20 – 500 |
| W | px | 20 – 500 |
| Snr db | dB | 10 – 60 |
| K endmembers | — | 2 – 20 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.05 abundance_RMSE

| Metric | Range |
|---|---|
| Abundance rmse | 0.01 – 0.2 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **abundance_RMSE**, with κ = `1000.0` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xecf047403de0792a114aee70830d2d3023179f19ea9ed1d1f12c55d780fef091`
- **Chain tx hash:** `0xf8363ef360ceda918f16cd61a16901ce95b84aaae4cc9fb2349880f29440510f`
- **Chain block:** `41555299`

---

## File Mapping

This bundle consists of: `L1-472.md`, `L1-472.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-472.md` | Source of truth — edit this | Human or LLM |
| `L1-472.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 `L1-472`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._