# ⚛  L1 Principle — Gaussian Plume Atmospheric Dispersion

**ID:** `L1-414` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Environmental Science — *Air quality modeling*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** source strength location
> **📡 Carrier:** N/A · **🌫 Noise:** lognormal
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555220

---

## 🧠 1. Introduction

**Gaussian Plume Atmospheric Dispersion** is a **parameter-estimation problem** whose unknown lives in **source strength location** space, within the **Air quality modeling** sub-domain of **Environmental Science**.

Measurements consist of N/A via a **atmospheric dispersion gaussian** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; O · least squares · source inversion operator; S · stability · pasquill gifford operator.

Observations are corrupted by log-normal multiplicative noise. Existence of the recovered source_strength_location 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 ~= 25); wind_direction_uncertainty_deg dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → Spatial derivative → O · least squares · source inversion → S · stability · pasquill gifford → **y** (detector).

```
y = `S.stability.pasquill_gifford` `O.least_squares.source_inversion` ∇ x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `O.least_squares.source_inversion` | O · least squares · source inversion operator |
| `S.stability.pasquill_gifford` | S · stability · pasquill gifford operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Environmental Science |
| Sub domain | Air quality modeling |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | source_strength_location |
| Noise model | lognormal |
| Integration axis | downwind_distance |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered source_strength_location 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 ~= 25); wind_direction_uncertainty_deg dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Lognormal sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | source_strength_RMSE_percent |
| Secondary | concentration_RMSE_ug_m3 |

## 📏 5. Operating Range (Ω)

**Center problem class:** `parameter_estimation` · **Forward operator:** `atmospheric_dispersion_gaussian`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N sensors | — | 20 |
| Distance km | km | 1 |
| Wind speed m s | s | 3 |
| Stability class int | — | 4 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N sensors | — | 3 – 100 |
| Distance km | km | 0.1 – 50.0 |
| Wind speed m s | s | 0.5 – 10.0 |
| Stability class int | — | 1 – 6 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 25 source_strength_RMSE_percent

| Metric | Range |
|---|---|
| Source strength rmse percent | 5 – 100 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **source_strength_RMSE_percent**, with κ = `500` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x88d1348556cbb501895c7b92d25f30ecb11f263bfadc08a1bac159057048fe95`
- **Chain tx hash:** `0xf8ac93b16b2b9cc39d15fb24c59781512f4fc292f187e9326f3c1782b0bf6f3f`
- **Chain block:** `41555220`

---

## File Mapping

This bundle consists of: `L1-414.md`, `L1-414.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-414.md` | Source of truth — edit this | Human or LLM |
| `L1-414.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 `L1-414`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._