# ⚛  L1 Principle — River Hydraulics Shallow Water Inversion

**ID:** `L1-421` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Environmental Science — *Hydrology*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** Manning roughness bathymetry
> **📡 Carrier:** N/A · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555239

---

## 🧠 1. Introduction

**River Hydraulics Shallow Water Inversion** is a **parameter-estimation problem** whose unknown lives in **Manning roughness bathymetry** space, within the **Hydrology** sub-domain of **Environmental Science**.

Measurements consist of N/A via a **water level discharge measurement** sensing mechanism.

The forward operator applies, in order: gradient / divergence with respect to position; O · least squares · rating curve operator; S · fdm · flood routing operator.

Observations are corrupted by additive Gaussian noise. Existence of the recovered Manning_roughness_bathymetry 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 ~= 80); bank_overflow_uncertainty 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** → Spatial derivative → O · least squares · rating curve → S · fdm · flood routing → **y** (detector).

```
y = `S.fdm.flood_routing` `O.least_squares.rating_curve` ∇ x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `D.space` | Gradient / divergence with respect to position |
| `O.least_squares.rating_curve` | O · least squares · rating curve operator |
| `S.fdm.flood_routing` | S · fdm · flood routing operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Environmental Science |
| Sub domain | Hydrology |
| Carrier | N/A |
| Problem class | parameter_estimation |
| Solution space | Manning_roughness_bathymetry |
| Noise model | gaussian |
| Integration axis | river_reach |
| Difficulty delta | 3 |
| L dag | 3 |

## 📡 4. Measurement Model

Existence of the recovered Manning_roughness_bathymetry 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 ~= 80); bank_overflow_uncertainty dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | water_level_RMSE_m |
| Secondary | discharge_RMSE_percent |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| N gauges | — | 5 |
| Roughness n | N | 0.035 |
| Reach length km | km | 50 |
| Discharge peak m3 s | s | 1000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| N gauges | — | 2 – 50 |
| Roughness n | N | 0.01 – 0.1 |
| Reach length km | km | 1 – 1000 |
| Discharge peak m3 s | s | 10 – 100000 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.10 water_level_RMSE_m

| Metric | Range |
|---|---|
| Water level rmse m | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **water_level_RMSE_m**, with κ = `2000` and δ = `3`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x52599914468dae4784412df2a6ba1035a0b0b4c83e90b5818a9a0b7e13d04bdc`
- **Chain tx hash:** `0xd4ecbe5745af85f060529b3b07df29d4aba36cf21b9c0e4ae5aa46bb59214c26`
- **Chain block:** `41555239`

---

## File Mapping

This bundle consists of: `L1-421.md`, `L1-421.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-421.md` | Source of truth — edit this | Human or LLM |
| `L1-421.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 `L1-421`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._