{
  "version": 1,
  "topic": "RF11",
  "title": "Two equally good permeability fits with different noisy data",
  "createdAt": "2026-10-24",
  "case": {
    "title": "Two equally close fits with different slopes",
    "premise": "Porosity grid 10,15,20,25,30%; generating log10(k/mD) = 0.2φ − 2. Two explicit deterministic perturbation arrays are supplied in JSON.",
    "formula": "log10(kA) = -2.08 + 0.204φ; log10(kB) = -1.92 + 0.196φ",
    "variables": "Both use unweighted log-space least squares. The residual pattern is shared, while opposite linear perturbations shift the fitted slopes.",
    "steps": [
      {
        "label": "Define the baseline",
        "calculation": "Evaluation porosity (%) = 20",
        "explanation": "Porosity grid 10,15,20,25,30%; generating log10(k/mD) = 0.2φ − 2. Two explicit deterministic perturbation arrays are supplied in JSON."
      },
      {
        "label": "Select the relation",
        "calculation": "log10(kA) = -2.08 + 0.204φ; log10(kB) = -1.92 + 0.196φ",
        "explanation": "Both use unweighted log-space least squares. The residual pattern is shared, while opposite linear perturbations shift the fitted slopes."
      },
      {
        "label": "Substitute the values",
        "calculation": "Fitted slopes A = 0.204, B = 0.196; both predict 100 mD at 20% porosity, with identical log-space residual RMSE but different predictions elsewhere",
        "explanation": "Retain intermediate precision. The displayed values are rounded only after evaluation."
      },
      {
        "label": "Read the result",
        "calculation": "Dataset A fitted k (mD) = 100; Dataset B fitted k (mD) = 100",
        "explanation": "Equal residual RMSE does not imply identical fitted parameters or predictions. These deterministic perturbations are not a probability model."
      }
    ],
    "result": "Dataset A fitted k (mD) = 100; Dataset B fitted k (mD) = 100",
    "interpretation": "Equal residual RMSE does not imply identical fitted parameters or predictions. These deterministic perturbations are not a probability model.",
    "control": "Evaluation porosity (%)",
    "columns": [
      "Evaluation porosity (%)",
      "Dataset A fitted k (mD)",
      "Dataset B fitted k (mD)"
    ],
    "rows": [
      [
        10,
        0.9120108393559097,
        1.096478196143185
      ],
      [
        15,
        9.54992586021435,
        10.471285480508996
      ],
      [
        20,
        100,
        100
      ],
      [
        25,
        1047.1285480508984,
        954.9925860214369
      ],
      [
        30,
        10964.781961431829,
        9120.108393559096
      ]
    ],
    "baseline": 2,
    "rowNotes": [
      "Evaluation porosity (%) = 10. Dataset A fitted k (mD): 0.912010839; Dataset B fitted k (mD): 1.0964782.",
      "Evaluation porosity (%) = 15. Dataset A fitted k (mD): 9.54992586; Dataset B fitted k (mD): 10.4712855.",
      "Evaluation porosity (%) = 20. Dataset A fitted k (mD): 100; Dataset B fitted k (mD): 100.",
      "Evaluation porosity (%) = 25. Dataset A fitted k (mD): 1047.12855; Dataset B fitted k (mD): 954.992586.",
      "Evaluation porosity (%) = 30. Dataset A fitted k (mD): 10964.782; Dataset B fitted k (mD): 9120.10839."
    ],
    "provenance": "Original synthetic teaching calculation. Equations and fixed controls are stated here; these are not field measurements.",
    "questions": [
      {
        "question": "Reproduce the baseline Dataset A fitted k (mD) without rounding intermediate values.",
        "answer": "Fitted slopes A = 0.204, B = 0.196; both predict 100 mD at 20% porosity, with identical log-space residual RMSE but different predictions elsewhere. Dataset A fitted k (mD) = 100. Equal residual RMSE does not imply identical fitted parameters or predictions. These deterministic perturbations are not a probability model."
      },
      {
        "question": "Compare the first and last recorded scenarios. Which input changes?",
        "answer": "Evaluation porosity (%) changes from 10 to 30. Other controls remain as stated in the premise. Dataset A fitted k (mD) changes from 0.912010839 to 10964.782."
      },
      {
        "question": "What would be needed to apply this example to a different physical system?",
        "answer": "Replace the synthetic controls with compatible documented inputs and check the relation's assumptions. Both use unweighted log-space least squares. The residual pattern is shared, while opposite linear perturbations shift the fitted slopes. Equal residual RMSE does not imply identical fitted parameters or predictions. These deterministic perturbations are not a probability model."
      }
    ]
  },
  "recorded": {
    "A": {
      "a": -2.08,
      "b": 0.204,
      "observations": [
        [
          10,
          0.72443596007499
        ],
        [
          15,
          10.715193052376065
        ],
        [
          20,
          125.89254117941675
        ],
        [
          25,
          1174.897554939529
        ],
        [
          30,
          8709.635899560806
        ]
      ]
    },
    "B": {
      "a": -1.92,
      "b": 0.196,
      "observations": [
        [
          10,
          0.8709635899560806
        ],
        [
          15,
          11.748975549395297
        ],
        [
          20,
          125.89254117941675
        ],
        [
          25,
          1071.519305237606
        ],
        [
          30,
          7244.359600749898
        ]
      ]
    },
    "noiseA": [
      -0.14,
      0.03,
      0.1,
      0.07,
      -0.06
    ],
    "noiseB": [
      -0.06,
      0.07,
      0.1,
      0.03,
      -0.14
    ]
  }
}