# ⚛  L1 Principle — Earthquake Magnitude Classification (PWDR)

**ID:** `L1-522` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *Earthquake source-parameter inversion with moment-magnitude / Richter-scale categorical readout*
> **🎯 Problem class:** linear inverse with categorical readout · **🧮 Solution space:** 1D magnitude scale categorical
> **📡 Carrier:** elastic_wave · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554066

---

## 🧠 1. Introduction

**Earthquake Magnitude Classification (PWDR)** is a **linear inverse with categorical readout** whose unknown lives in **1D magnitude scale categorical** space, within the **Earthquake source-parameter inversion with moment-magnitude / Richter-scale categorical readout** sub-domain of **Geophysics**.

Measurements consist of elastic waves recorded by seismometers / geophones via a **seismograph with magnitude classifier** sensing mechanism.

The forward operator applies, in order: L · seismograph array operator; L · waveform acquisition operator; L · moment tensor inversion operator; L · fault dimension estimation operator; L · magnitude formula operator; L · magnitude threshold classifier operator; pixel-level spatial averaging on the detector.

Observations are corrupted by additive Gaussian noise. Existence inherited from L1-278. Uniqueness conditional on adequate station coverage (azimuthal gap < 180 degrees) and SNR > 10 dB. Stability dominated by velocity_model_uncertainty (~0.1-0.3 Mw bias) and magnitude_scale_saturation (ML saturates at ~7; Mw is preferred for great earthquakes). Joint Hadamard well-posedness for the source-inversion + magnitude-formula forward established by Hanks-Kanamori 1979 (foundational Mw formula), Richter 1935 (foundational ML), Kanamori 1977 (great-earthquake moment paradox), Aki-Richards 2002 (Quantitative Seismology textbook), Vallee 2013 (SCARDEC moment-tensor algorithm), Duputel et al. 2012 (W-phase moment-tensor inversion for tsunami warning).

## ⚙ 2. Forward Model

Physical chain: **x** → L · seismograph array → L · waveform acquisition → L · moment tensor inversion → L · fault dimension estimation → L · magnitude formula → L · magnitude threshold classifier → Spatial integration → **y** (detector).

```
y = ∫_A dA `L.magnitude_threshold_classifier` `L.magnitude_formula` `L.fault_dimension_estimation` `L.moment_tensor_inversion` `L.waveform_acquisition` `L.seismograph_array` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.seismograph_array` | L · seismograph array operator |
| `L.waveform_acquisition` | L · waveform acquisition operator |
| `L.moment_tensor_inversion` | L · moment tensor inversion operator |
| `L.fault_dimension_estimation` | L · fault dimension estimation operator |
| `L.magnitude_formula` | L · magnitude formula operator |
| `L.magnitude_threshold_classifier` | L · magnitude threshold classifier operator |
| `int.spatial` | Pixel-level spatial averaging on the detector |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | Earthquake source-parameter inversion with moment-magnitude / Richter-scale categorical readout |
| Carrier | elastic_wave |
| Problem class | linear_inverse_with_categorical_readout |
| Solution space | 1D_magnitude_scale_categorical |
| Noise model | gaussian |
| Integration axis | spatial_temporal |
| Difficulty delta | 5 |
| L dag | 6.3 |

## 📡 4. Measurement Model

Existence inherited from L1-278. Uniqueness conditional on adequate station coverage (azimuthal gap < 180 degrees) and SNR > 10 dB. Stability dominated by velocity_model_uncertainty (~0.1-0.3 Mw bias) and magnitude_scale_saturation (ML saturates at ~7; Mw is preferred for great earthquakes). Joint Hadamard well-posedness for the source-inversion + magnitude-formula forward established by Hanks-Kanamori 1979 (foundational Mw formula), Richter 1935 (foundational ML), Kanamori 1977 (great-earthquake moment paradox), Aki-Richards 2002 (Quantitative Seismology textbook), Vallee 2013 (SCARDEC moment-tensor algorithm), Duputel et al. 2012 (W-phase moment-tensor inversion for tsunami warning).

| Metric | Value |
|---|---|
| Metric | categorical_accuracy |
| Secondary | RMSE_Mw |

## 📏 5. Operating Range (Ω)

**Center problem class:** `earthquake_mw_pwdr` · **Forward operator:** `earthquake_magnitude_pwdr_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 25 |
| N stations | — | 30 |
| Sampling rate hz | Hz | 100 |
| Frequency band hz | Hz | 0.01 – 25 |
| Station coverage gap | — | 0 |
| Near field truncation | — | 0 |
| Station distance km range | — | 0 – 1000 |
| Magnitude scale saturation | — | 0 |
| Velocity model uncertainty | — | 0 |
| Hypocenter localization error | — | 0 |
| Instrument response uncertainty | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 3 – 50 |
| N stations | — | 3 – 500 |
| Sampling rate hz | Hz | 20 – 500 |
| Station coverage gap | — | 0.0 – 0.6 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 0.85_accuracy

| Metric | Range |
|---|---|
| Categorical accuracy | 0.5 – 0.99 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **categorical_accuracy**, with κ = `200` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x77766239d0ed2b70cfeaf020e6742d6bf253792347655b5353a3334248508d27`
- **Chain tx hash:** `0x94f5ce0765c0f54904044874b4053d57ca94613587f5e01eafe418156d4880ea`
- **Chain block:** `41554066`

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## File Mapping

This bundle consists of: `L1-522.md`, `L1-522.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-522.md` | Source of truth — edit this | Human or LLM |
| `L1-522.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 `L1-522`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._