# ⚛  L1 Principle — Receiver Function — teleseismic crust imaging

**ID:** `L1-275` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Geophysics — *P-to-S converted phase receiver function*
> **🎯 Problem class:** linear inverse · **🧮 Solution space:** crust velocity discontinuities
> **📡 Carrier:** acoustic · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554051

---

## 🧠 1. Introduction

**Receiver Function — teleseismic crust imaging** is a **linear inverse problem** whose unknown lives in **crust velocity discontinuities** space, within the **P-to-S converted phase receiver function** sub-domain of **Geophysics**.

Measurements consist of acoustic pressure waves recorded by transducers via a **broadband seismometer** sensing mechanism.

The forward operator applies, in order: L · deconvolve operator; L · stack back azimuth operator; L · invert layers operator; detector accumulates flux over the exposure window.

Observations are corrupted by additive Gaussian noise. Conditional stability; mismatch parameters dominate at Omega bounds.

## ⚙ 2. Forward Model

Physical chain: **x** → L · deconvolve → L · stack back azimuth → L · invert layers → Temporal integration → **y** (detector).

```
y = ∫_t dt `L.invert_layers` `L.stack_back_azimuth` `L.deconvolve` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `L.deconvolve` | L · deconvolve operator |
| `L.stack_back_azimuth` | L · stack back azimuth operator |
| `L.invert_layers` | L · invert layers operator |
| `int.temporal` | Detector accumulates flux over the exposure window |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Geophysics |
| Sub domain | P-to-S converted phase receiver function |
| Carrier | acoustic |
| Problem class | linear_inverse |
| Solution space | crust_velocity_discontinuities |
| Noise model | gaussian |
| Integration axis | temporal |
| Difficulty delta | 5 |
| L dag | 3.3 |

## 📡 4. Measurement Model

Conditional stability; mismatch parameters dominate at Omega bounds.

| Metric | Value |
|---|---|
| Metric | PSNR_dB |
| Secondary | SSIM |

## 📏 5. Operating Range (Ω)

**Center problem class:** `receiver_function` · **Forward operator:** `receiver_function_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| Snr db | dB | 25 |
| N events | — | 50 |
| Bg noise | — | 0 |
| N samples | — | 1024 |
| Bandwidth error | — | 0 |
| P slowness error | — | 0 |
| Multiples contamination | — | 0 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| Snr db | dB | 0 – 40 |
| N events | — | 5 – 1000 |
| Bg noise | — | 0.0 – 0.5 |
| N samples | — | 128 – 65536 |
| Bandwidth error | — | 0.0 – 0.3 |
| P slowness error | — | 0.0 – 0.05 |
| Multiples contamination | — | 0.0 – 0.3 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** 22.0

| Metric | Range |
|---|---|
| Psnr db | 10.0 – 45.0 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0xb878ba8807d15203e611d9d9423e3ce60b4ab50a422e3c521f13f33121f401c0`
- **Chain tx hash:** `0x846f79c3fa40ae9dcbba180ee7064d2a0d86fc2f85aec26d3c2d9792920dca0a`
- **Chain block:** `41554051`

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## File Mapping

This bundle consists of: `L1-275.md`, `L1-275.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-275.md` | Source of truth — edit this | Human or LLM |
| `L1-275.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 `L1-275`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._