L1 ⚛ L1-189 ⊙ Testnet

Richards Equation — unsaturated variably-saturated flow

Fluid Dynamics · Richards eqn for unsaturated flow

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⬇ L1-189.md

# ⚛ L1 Principle — Richards Equation — unsaturated variably-saturated flow **ID:** `L1-189` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Fluid Dynamics — *Richards eqn for unsaturated flow* > **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** saturation field in soil > **📡 Carrier:** mechanical · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41553977 --- ## 🧠 1. Introduction **Richards Equation — unsaturated variably-saturated flow** is a **nonlinear inverse problem** whose unknown lives in **saturation field in soil** space, within the **Richards eqn for unsaturated flow** sub-domain of **Fluid Dynamics**. Measurements consist of mechanical vibrations or strain signals via a **tdr tensiometer** sensing mechanism. The forward operator applies, in order: L · nonlinear conductivity operator; mass-balance: ∂_t ρ + ∇·(ρv) = 0; L · capillary pressure operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. Highly nonlinear; degenerate in dry limit. ## ⚙ 2. Forward Model Physical chain: **x** → L · nonlinear conductivity → Continuity equation → L · capillary pressure → Temporal integration → **y** (detector). ``` y = ∫_t dt `L.capillary_pressure` ∂_t ρ + ∇·(ρv) `L.nonlinear_conductivity` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.nonlinear_conductivity` | L · nonlinear conductivity operator | | `L.continuity` | Mass-balance: ∂_t ρ + ∇·(ρv) = 0 | | `L.capillary_pressure` | L · capillary pressure operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Fluid Dynamics | | Sub domain | Richards eqn for unsaturated flow | | Carrier | mechanical | | Problem class | nonlinear_inverse | | Solution space | saturation_field_in_soil | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 5 | | L dag | 3.3 | ## 📡 4. Measurement Model Highly nonlinear; degenerate in dry limit. | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `richards` · **Forward operator:** `richards_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 128 | | W | px | 128 | | Z | — | 32 | | Snr db | dB | 30 | | N times | — | 1000 | | Vg n error | — | 0 | | K sat error | — | 0 | | Vg alpha error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 16 – 1024 | | W | px | 16 – 1024 | | Z | — | 8 – 512 | | Snr db | dB | 0 – 40 | | N times | — | 10 – 100000 | | Vg n error | — | 0.0 – 0.3 | | K sat error | — | 0.0 – 0.3 | | Vg alpha error | — | 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 = `440` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x98cc59321efe5cf3c9f44934034166823240163b94319253c6c59291915eafe7` - **Chain tx hash:** `0xa53e52b7acdcce31bbd2e0f8c6cd5fa639841242a03589d2fe1181ccc696711f` - **Chain block:** `41553977` --- ## File Mapping This bundle consists of: `L1-189.md`, `L1-189.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-189.md` | Source of truth — edit this | Human or LLM | | `L1-189.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 `L1-189`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-189.md).
Regenerate L1-189.json so every field matches the Markdown.
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L2-189-001 Richards Equation — unsaturated variably-saturated flow Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41553977,
  "chain_hash": "0x98cc59321efe5cf3c9f44934034166823240163b94319253c6c59291915eafe7",
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  "domain": "Fluid Dynamics",
  "hardness_fn": {
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    "metric": "PSNR_dB",
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    "regime": "Highly nonlinear; degenerate in dry limit.",
    "secondary": "SSIM"
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  "physics_fingerprint": {
    "L_DAG": 3.3,
    "carrier": "mechanical",
    "difficulty_delta": 5,
    "domain": "Fluid Dynamics",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
    "primitives": [
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      "L.continuity",
      "L.capillary_pressure",
      "int.temporal"
    ],
    "problem_class": "nonlinear_inverse",
    "sensing_mechanism": "tdr_tensiometer",
    "solution_space": "saturation_field_in_soil",
    "sub_domain": "Richards eqn for unsaturated flow",
    "title": "Richards Equation \u2014 unsaturated variably-saturated flow"
  },
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      "W",
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    "allowed_problem_classes": [
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    "center_spec": {
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      "input_format": "measurement_only",
      "omega": {
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        "K_sat_error": 0.0,
        "N_times": 1000,
        "SNR_dB": 30,
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        "Z": 32,
        "vg_alpha_error": 0.0,
        "vg_n_error": 0.0
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    "epsilon_bounds": {
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      "K_sat_error": [
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      "N_times": [
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        100000
      ],
      "SNR_dB": [
        0,
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      ],
      "W": [
        16,
        1024
      ],
      "Z": [
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      ],
      "vg_alpha_error": [
        0.0,
        0.3
      ],
      "vg_n_error": [
        0.0,
        0.3
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Richards eqn for unsaturated flow",
  "title": "Richards Equation \u2014 unsaturated variably-saturated flow"
}

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