# ⚛  L1 Principle — Satellite Radar Altimetry

**ID:** `L1-468` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geodesy — *Sea surface height*
> **🎯 Problem class:** parameter estimation · **🧮 Solution space:** 1D sea surface height
> **📡 Carrier:** RF · **🌫 Noise:** thermal gaussian
> **⚖ Difficulty (δ):** 3 · **⛓ Block:** 41555298

---

## 🧠 1. Introduction

**Satellite Radar Altimetry** is a **parameter-estimation problem** whose unknown lives in **1D sea surface height** space, within the **Sea surface height** sub-domain of **Geodesy**.

Measurements consist of RF coil signals (typical of MRI) via a **radar altimetry nadir** sensing mechanism.

The forward operator applies, in order: S · altimeter · retracker brown operator; operator inherits structure from a graph (mesh, network); O · least squares · retrack operator.

Observations are corrupted by thermally-driven Gaussian fluctuations. Existence of the recovered 1D_sea_surface_height 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 ~= 30); rain_contamination dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Thermal gaussian sets the irreducible data-fidelity floor.

## ⚙ 2. Forward Model

Physical chain: **x** → S · altimeter · retracker brown → Structured graph operator → O · least squares · retrack → **y** (detector).

```
y = `O.least_squares.retrack` G `S.altimeter.retracker_brown` x
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `S.altimeter.retracker_brown` | S · altimeter · retracker brown operator |
| `G.structured` | Operator inherits structure from a graph (mesh, network) |
| `O.least_squares.retrack` | O · least squares · retrack operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geodesy |
| Sub domain | Sea surface height |
| Carrier | RF |
| Problem class | parameter_estimation |
| Solution space | 1D_sea_surface_height |
| Noise model | thermal_gaussian |
| Integration axis | range_surface |
| Difficulty delta | 3 |
| L dag | 2.5 |

## 📡 4. Measurement Model

Existence of the recovered 1D_sea_surface_height 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 ~= 30); rain_contamination dominates the stability cliff; the remaining mismatch parameters contribute higher-order bias terms. Thermal gaussian sets the irreducible data-fidelity floor.

| Metric | Value |
|---|---|
| Metric | SSH_RMSE_cm |
| Secondary | SWH_bias_cm |

## 📏 5. Operating Range (Ω)

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

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Swh m | m | 2 |
| N gates | — | 128 |
| Mispointing deg | deg | 0.1 |
| Noise sigma percent | — | 5 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Swh m | m | 0.0 – 10.0 |
| N gates | — | 64 – 512 |
| Mispointing deg | deg | 0.0 – 1.0 |
| Noise sigma percent | — | 1.0 – 20.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 3.0 SSH_RMSE_cm

| Metric | Range |
|---|---|
| Ssh rmse cm | 0.5 – 20.0 |

## ⚖ 7. Hardness Function

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

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x9ad6cd94a55549fd9138ab92df0b45a76970484232800376df163cb6c8a8395a`
- **Chain tx hash:** `0xe7c4f2cab5f62505f2ec421104771d1695e61a991dd5ec0a53b7eee09d87e32d`
- **Chain block:** `41555298`

---

## File Mapping

This bundle consists of: `L1-468.md`, `L1-468.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-468.md` | Source of truth — edit this | Human or LLM |
| `L1-468.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 `L1-468`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._