SoilHydraulicProperties

{
  "reference": {
    "notes": "source_ref: UrbanCanopy.jl@9200edd4f15609f20339d3a7d104de8064544310:src/hydrology.jl:322\nUpstream `SoilHydraulicProperties` is an MTK `@component` whose three inputs (pct_sand, pct_clay, θ_i) are `@parameters` and whose five outputs (k_sat, θ_sat, B, ψ_sat, ψ) are time-indexed `@variables` with five algebraic equations (no derivatives). Identical structure preserved here.\n\nUnicode parameter / variable names θ_i, θ_sat, ψ_sat, ψ are preserved from upstream — sibling files in this directory (soil_water_flux.esm, water_table_depth.esm, interface_hydraulic_conductivity.esm, soil_water_content_calc.esm) already established that unicode identifiers are the rig convention for new urban_canopy migrations and the canonical Python runner accepts them.\n\nThe upstream component declares three `@constants` (k_coeff, ψ_coeff, ψ_min); they are encoded here as parameters with the upstream default values, mirroring the convention used by sibling files (groundwater_drainage.esm, surface_runoff_infiltration.esm) for `@constants`-to-parameter promotion. Default values: k_coeff = 7.0556e-6 m/s (paper's 0.0070556 mm/s converted to m/s, Eq. 5.70), ψ_coeff = -0.010 m (paper's -10.0 mm converted to m, Eq. 5.74), ψ_min = -1.0e5 m (paper's -1e8 mm converted to m). The mm-to-m conversions are applied at the @constants declaration site upstream; this file inherits the SI values so its outputs are SI-consistent without further conversion.\n\nEq. 5.73 (matric potential) carries two clamps on the water-content ratio (θ_i / θ_sat) to keep ψ finite: an inner `max(θ_i, 0.01 * θ_sat)` floor on the numerator keeps the ratio ≥ 0.01 (prevents division-by-zero and excessive negative exponents as the soil approaches dryness), and an outer `max(..., ψ_min)` floor on ψ itself keeps it ≥ -1e5 m even when the inner clamp's value still produces an absurdly negative matric potential (e.g. at very high B for clay-rich soils). Both clamps are preserved verbatim from upstream Eq. 5.73 — they are physical regularizers, not numerical hacks, and the canonical Python runner reproduces the upstream Julia results to machine precision when both clamps fire.\n\nObservability classification: this component is functionally observed-only — all five output variables are given by single algebraic equations in terms of the parameters (the `(t)` time index has no role since there is no derivative). Test scenarios use `time_span: [0, 1]` and sample at t=0 to mirror the convention used by the other observed-only ESM files in this directory (soil_water_flux.esm, water_table_depth.esm, groundwater_drainage.esm, interface_hydraulic_conductivity.esm).\n\nDefault parameter values: pct_sand = 50 %, pct_clay = 20 %, θ_i = 0.3 m^3/m^3 — a typical CLMU loam at partial saturation (θ_i ≈ 0.7 × θ_sat). The qualitative-test scenarios in the upstream `SoilHydraulicProperties - Qualitative Behavior` testitem (UrbanCanopy.jl test/hydrology_test.jl:970) use the same pct_clay = 20, θ_i = 0.3 baseline.\n\nTolerance rationale (model-level rel=1e-9): observed-only algebraic system (no ODE integration); the five equations involve O(10) IEEE-754 float64 operations each, with k_sat and ψ_sat carrying one `^` (10.0^x) call (≈1 ULP of relative error) and ψ carrying one `^` plus two `max` calls — worst-case roundoff is ~10*ULP ≈ 1e-15. 1e-9 leaves comfortable headroom and is well below any plausible physical-discrimination scale on soil hydraulic properties (k_sat ∈ [1e-7, 1e-4] m/s across realistic Cosby sand fractions; θ_sat ∈ [0.4, 0.49]; B ∈ [3, 11]; ψ_sat ∈ [-0.8, -0.07] m). Tests that pin upstream test:466's `≈ rtol=1e-6` assertions use the looser `rel: 1e-6` tolerance from the upstream to confirm cross-runner consistency."
  },
  "tolerance": {
    "rel": 1e-09
  },
  "variables": {
    "pct_sand": {
      "type": "parameter",
      "units": "1",
      "default": 50.0,
      "description": "Percent sand content of the soil layer (dimensionless mass percent, 0-100)"
    },
    "pct_clay": {
      "type": "parameter",
      "units": "1",
      "default": 20.0,
      "description": "Percent clay content of the soil layer (dimensionless mass percent, 0-100)"
    },
    "θ_i": {
      "type": "parameter",
      "units": "m^3/m^3",
      "default": 0.3,
      "description": "Total volumetric soil water content of the layer (Oleson Eq. 5.73 input)"
    },
    "k_coeff": {
      "type": "parameter",
      "units": "m/s",
      "default": 7.0556e-06,
      "description": "Coefficient for saturated hydraulic conductivity (Oleson Eq. 5.70, converted from 0.0070556 mm/s to 7.0556e-6 m/s)"
    },
    "ψ_coeff": {
      "type": "parameter",
      "units": "m",
      "default": -0.01,
      "description": "Coefficient for saturated matric potential (Oleson Eq. 5.74, converted from -10.0 mm to -0.010 m)"
    },
    "ψ_min": {
      "type": "parameter",
      "units": "m",
      "default": -100000.0,
      "description": "Minimum (most-negative) soil matric potential (Oleson Eq. 5.73 outer clamp, converted from -1e8 mm to -1e5 m)"
    },
    "k_sat": {
      "type": "unknown",
      "units": "m/s",
      "description": "Saturated hydraulic conductivity (Oleson Eq. 5.70)"
    },
    "θ_sat": {
      "type": "unknown",
      "units": "m^3/m^3",
      "description": "Volumetric water content at saturation / porosity (Oleson Eq. 5.71)"
    },
    "B": {
      "type": "unknown",
      "units": "1",
      "description": "Clapp-Hornberger exponent (Oleson Eq. 5.72, dimensionless)"
    },
    "ψ_sat": {
      "type": "unknown",
      "units": "m",
      "description": "Saturated soil matric potential (Oleson Eq. 5.74)"
    },
    "ψ": {
      "type": "unknown",
      "units": "m",
      "description": "Soil matric potential (Oleson Eq. 5.73, clamped: inner θ_i/θ_sat ratio ≥ 0.01, outer ψ ≥ ψ_min)"
    }
  },
  "equations": [
    {
      "lhs": "k_sat",
      "rhs": {
        "op": "*",
        "args": [
          "k_coeff",
          {
            "op": "^",
            "args": [
              10.0,
              {
                "op": "+",
                "args": [
                  -0.884,
                  {
                    "op": "*",
                    "args": [
                      0.0153,
                      "pct_sand"
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "θ_sat",
      "rhs": {
        "op": "-",
        "args": [
          0.489,
          {
            "op": "*",
            "args": [
              0.00126,
              "pct_sand"
            ]
          }
        ]
      }
    },
    {
      "lhs": "B",
      "rhs": {
        "op": "+",
        "args": [
          2.91,
          {
            "op": "*",
            "args": [
              0.159,
              "pct_clay"
            ]
          }
        ]
      }
    },
    {
      "lhs": "ψ_sat",
      "rhs": {
        "op": "*",
        "args": [
          "ψ_coeff",
          {
            "op": "^",
            "args": [
              10.0,
              {
                "op": "-",
                "args": [
                  1.88,
                  {
                    "op": "*",
                    "args": [
                      0.0131,
                      "pct_sand"
                    ]
                  }
                ]
              }
            ]
          }
        ]
      }
    },
    {
      "lhs": "ψ",
      "rhs": {
        "op": "max",
        "args": [
          {
            "op": "*",
            "args": [
              "ψ_sat",
              {
                "op": "^",
                "args": [
                  {
                    "op": "/",
                    "args": [
                      {
                        "op": "max",
                        "args": [
                          "θ_i",
                          {
                            "op": "*",
                            "args": [
                              0.01,
                              "θ_sat"
                            ]
                          }
                        ]
                      },
                      "θ_sat"
                    ]
                  },
                  {
                    "op": "*",
                    "args": [
                      -1,
                      "B"
                    ]
                  }
                ]
              }
            ]
          },
          "ψ_min"
        ]
      }
    }
  ],
  "tests": [
    {
      "id": "eq_verify_saturated",
      "description": "Reproduces upstream Equation Verification fixture (UrbanCanopy.jl test/hydrology_test.jl:466) with pct_sand = 50, pct_clay = 20, θ_i = θ_sat. Anchors all five outputs against the Cosby/Clapp-Hornberger closed-form values for a mid-range loam: k_sat = 0.0070556e-3 * 10^(-0.884 + 0.0153 * 50) ≈ 5.365e-6 m/s; θ_sat = 0.489 - 0.00126*50 = 0.426 m^3/m^3; B = 2.91 + 0.159*20 = 6.09; ψ_sat = -0.010 * 10^(1.88 - 0.0131*50) ≈ -0.16788 m. At saturation θ_i = θ_sat the ratio is exactly 1 so ψ = ψ_sat — a key invariant of Eq. 5.73. Tolerances loosened to rel = 1e-6 here to match the upstream Julia test's pinned tolerance and confirm cross-runner consistency (the Python runner reproduces the Julia closed-form to ≈1 ULP, far tighter than 1e-6).",
      "parameter_overrides": {
        "pct_sand": 50.0,
        "pct_clay": 20.0,
        "θ_i": 0.426
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 5.364558079579147e-06,
          "tolerance": {
            "rel": 1e-06
          }
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.426,
          "tolerance": {
            "rel": 1e-06
          }
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 6.09,
          "tolerance": {
            "rel": 1e-06
          }
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.16788040181225597,
          "tolerance": {
            "rel": 1e-06
          }
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -0.16788040181225597,
          "tolerance": {
            "rel": 0.0001
          }
        }
      ]
    },
    {
      "id": "typical_partial_saturation",
      "description": "Realistic CLMU operating point: pct_sand = 50, pct_clay = 20, θ_i = 0.3 m^3/m^3 (partial saturation, θ_i / θ_sat ≈ 0.7042). k_sat, θ_sat, B, ψ_sat are identical to the eq_verify_saturated case (they depend only on pct_sand and pct_clay), but ψ now reflects the partial-saturation matric potential: ψ = ψ_sat * (0.3/0.426)^(-6.09) ≈ -0.16788 * 8.4768 ≈ -1.42048 m. Neither clamp fires. Anchors the non-equilibrium branch of Eq. 5.73 — the dominant operating regime for the urban-canopy soil column.",
      "parameter_overrides": {
        "pct_sand": 50.0,
        "pct_clay": 20.0,
        "θ_i": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 5.364558079579147e-06
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.426
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 6.09
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.16788040181225597
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -1.420482923905238
        }
      ]
    },
    {
      "id": "sandy_low_clay_dry",
      "description": "Sandy soil with low clay at moderate-to-dry water content: pct_sand = 70, pct_clay = 10, θ_i = 0.15 m^3/m^3. Tests Eq. 5.70-5.74 in the sandy / less-clayey regime: k_sat increases with sand (≈1.085e-5 m/s, ~2× the default loam), θ_sat decreases (0.4008 m^3/m^3), B decreases with clay (4.5), ψ_sat is less negative for sandy soils (≈-0.0918 m). With θ_i / θ_sat ≈ 0.374 and B = 4.5, the (-B) exponent drives ψ to ≈ -7.65 m — moderately dry but well above the ψ_min clamp.",
      "parameter_overrides": {
        "pct_sand": 70.0,
        "pct_clay": 10.0,
        "θ_i": 0.15
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 1.0852603880122091e-05
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.4008
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 4.5
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.09183325964835806
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -7.651807612176636
        }
      ]
    },
    {
      "id": "clayey_partial",
      "description": "Clay-rich soil at partial saturation: pct_sand = 10, pct_clay = 50, θ_i = 0.3 m^3/m^3. Tests Eq. 5.70-5.74 in the clayey regime — low sand suppresses k_sat (≈1.31e-6 m/s, ~4× lower than the default loam), high clay increases B sharply (10.86 vs 6.09), and ψ_sat is significantly more negative (-0.561 m vs -0.168 m for the default loam). The large B = 10.86 amplifies the (θ_i / θ_sat)^(-B) factor: ψ ≈ -85.16 m — high-clay soils retain water more tightly. Still well above the ψ_min = -1e5 m clamp.",
      "parameter_overrides": {
        "pct_sand": 10.0,
        "pct_clay": 50.0,
        "θ_i": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 1.3107925113344989e-06
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.4764
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 10.86
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.5610479760324704
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -85.1601483413643
        }
      ]
    },
    {
      "id": "inner_clamp_fires_only",
      "description": "Very dry soil where the inner `max(θ_i, 0.01 * θ_sat)` clamp on the ratio fires, but the outer ψ_min clamp does NOT — isolates the inner-clamp branch of Eq. 5.73. pct_sand = 80, pct_clay = 0 (extreme: low B = 2.91), θ_i = 0.001 m^3/m^3. 0.01 * θ_sat = 0.01 * 0.3882 = 0.003882 > θ_i = 0.001, so the ratio is pinned at 0.01. (0.01)^(-2.91) = 100^2.91 = 10^5.82 ≈ 6.607e5. ψ = -0.0679 * 6.607e5 ≈ -44874.54 m, which is more negative than the typical operating range but still > ψ_min = -1e5, so the outer clamp does not engage. Verifies that the inner clamp protects against extreme dryness without prematurely triggering the outer clamp.",
      "parameter_overrides": {
        "pct_sand": 80.0,
        "pct_clay": 0.0,
        "θ_i": 0.001
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 1.5435970913938464e-05
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.3882
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 2.91
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.06792036326171844
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -44874.53899331323
        }
      ]
    },
    {
      "id": "both_clamps_fire",
      "description": "Extreme dry state where BOTH clamps in Eq. 5.73 fire: pct_sand = 50, pct_clay = 20, θ_i = 0.001 m^3/m^3. 0.01 * θ_sat = 0.00426 > θ_i = 0.001 so the inner clamp pins the ratio at 0.01. (0.01)^(-6.09) = 100^6.09 = 10^12.18 ≈ 1.514e12. ψ_sat * ratio^(-B) ≈ -0.168 * 1.514e12 ≈ -2.541e11 m — far below ψ_min = -1e5. The outer clamp engages and pins ψ at exactly ψ_min = -1.0e5 m. Verifies the outer-clamp branch and confirms that the IEEE-754 result hits ψ_min exactly (abs = 1e-15 floor).",
      "parameter_overrides": {
        "pct_sand": 50.0,
        "pct_clay": 20.0,
        "θ_i": 0.001
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 5.364558079579147e-06
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.426
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 6.09
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.16788040181225597
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -100000.0,
          "tolerance": {
            "abs": 1e-10
          }
        }
      ]
    },
    {
      "id": "qual_low_sand_ksat",
      "description": "Lower-sand monotonicity-check anchor for the upstream `SoilHydraulicProperties - Qualitative Behavior` testitem (UrbanCanopy.jl test/hydrology_test.jl:982): higher sand content should produce higher k_sat (Eq. 5.70's 10^(0.0153 * pct_sand) factor is monotone-increasing in pct_sand). At pct_sand = 20, pct_clay = 20, θ_i = 0.3: k_sat ≈ 1.864e-6 m/s, θ_sat = 0.4638 m^3/m^3 (higher porosity for clay-rich soils, consistent with Eq. 5.71's negative sand-dependence), ψ_sat ≈ -0.415 m (more negative, also consistent with Eq. 5.74). Pairs with qual_high_sand_ksat to verify the k_sat / θ_sat / ψ_sat sand-dependence directions.",
      "parameter_overrides": {
        "pct_sand": 20.0,
        "pct_clay": 20.0,
        "θ_i": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 1.8643779228160723e-06
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.4638
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 6.09
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.41495404263436286
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -5.892312769696793
        }
      ]
    },
    {
      "id": "qual_high_sand_ksat",
      "description": "Higher-sand monotonicity-check companion to qual_low_sand_ksat: at pct_sand = 80 (other parameters held: pct_clay = 20, θ_i = 0.3), k_sat ≈ 1.544e-5 m/s — about 8.3× larger than the pct_sand = 20 value, confirming Eq. 5.70's strong sand-dependence (decade per ~65 % sand). θ_sat drops to 0.3882 m^3/m^3 (lower porosity in sandy soils) and ψ_sat rises to ≈ -0.0679 m (less negative — sandy soils are less retentive). B = 6.09 is unchanged since it depends only on clay. ψ ≈ -0.326 m — less negative than the loam at the same θ_i / θ_sat because the higher θ_sat-relative ratio (0.3 / 0.3882 ≈ 0.773) keeps the (-B)-power factor closer to 1.",
      "parameter_overrides": {
        "pct_sand": 80.0,
        "pct_clay": 20.0,
        "θ_i": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "assertions": [
        {
          "variable": "k_sat",
          "time": 0.0,
          "expected": 1.5435970913938464e-05
        },
        {
          "variable": "θ_sat",
          "time": 0.0,
          "expected": 0.3882
        },
        {
          "variable": "B",
          "time": 0.0,
          "expected": 6.09
        },
        {
          "variable": "ψ_sat",
          "time": 0.0,
          "expected": -0.06792036326171844
        },
        {
          "variable": "ψ",
          "time": 0.0,
          "expected": -0.3263470887762197
        }
      ]
    }
  ],
  "analyses": [
    {
      "id": "k_sat_vs_pct_sand",
      "description": "Saturated hydraulic conductivity k_sat vs. percent sand content (Eq. 5.70). Sweeping pct_sand from 5 % (clay-rich) to 95 % (nearly-pure sand) traces the canonical Cosby-style log-linear relationship — k_sat increases by ~1.5 decades over the realistic 5-95 % sand range. The y-axis is logarithmic because Eq. 5.70 places pct_sand inside a 10^(0.0153 * pct_sand) factor.",
      "initial_state": {},
      "parameters": {
        "pct_clay": 20.0,
        "θ_i": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "pct_sand",
            "range": {
              "start": 5.0,
              "stop": 95.0,
              "count": 19,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "k_sat_curve",
          "type": "line",
          "description": "k_sat vs pct_sand: log-linear increase from Eq. 5.70's 10^(0.0153 * pct_sand) factor.",
          "x": {
            "variable": "pct_sand",
            "label": "pct_sand (%)"
          },
          "y": {
            "variable": "k_sat",
            "label": "k_sat (m/s)"
          }
        }
      ]
    },
    {
      "id": "psi_retention_curve",
      "description": "Soil-water retention curve: matric potential ψ vs. volumetric water content θ_i for a typical Cosby loam (pct_sand = 50, pct_clay = 20). Sweeps θ_i from 0.05 (very dry) to 0.426 (saturated, θ_sat). Reproduces the canonical Clapp-Hornberger retention curve of Eq. 5.73 — ψ saturates at ψ_sat ≈ -0.168 m near saturation and decreases rapidly toward the inner-clamp regime as θ_i → 0. The y-axis spans many decades so a log-scale view is most informative.",
      "initial_state": {},
      "parameters": {
        "pct_sand": 50.0,
        "pct_clay": 20.0
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "θ_i",
            "range": {
              "start": 0.05,
              "stop": 0.426,
              "count": 30,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "psi_curve",
          "type": "line",
          "description": "ψ vs θ_i (Cosby loam, pct_sand=50, pct_clay=20) — the Clapp-Hornberger retention curve of Eq. 5.73.",
          "x": {
            "variable": "θ_i",
            "label": "θ_i (m^3/m^3)"
          },
          "y": {
            "variable": "ψ",
            "label": "ψ (m, more negative = drier)"
          }
        }
      ]
    },
    {
      "id": "psi_vs_sand_and_clay",
      "description": "Joint sweep over pct_sand and pct_clay at fixed θ_i = 0.3 m^3/m^3. Surfaces the iso-ψ contours of the soil hydraulic system: ψ depends on all three properties (ψ_sat from sand, B from clay, θ_sat from sand). Clay-rich / low-sand corners exhibit the most negative ψ (high B amplifies the (θ_i / θ_sat)^(-B) factor); high-sand / low-clay corners stay close to ψ_sat (small B, ratio close to 1). The default operating point (sand=50, clay=20) sits in the moderately-negative band.",
      "initial_state": {},
      "parameters": {
        "θ_i": 0.3
      },
      "time_span": {
        "start": 0.0,
        "end": 1.0
      },
      "parameter_sweep": {
        "type": "cartesian",
        "dimensions": [
          {
            "parameter": "pct_sand",
            "range": {
              "start": 10.0,
              "stop": 90.0,
              "count": 9,
              "scale": "linear"
            }
          },
          {
            "parameter": "pct_clay",
            "range": {
              "start": 5.0,
              "stop": 50.0,
              "count": 10,
              "scale": "linear"
            }
          }
        ]
      },
      "plots": [
        {
          "id": "psi_heatmap",
          "type": "heatmap",
          "description": "ψ as a function of sand and clay percentages at θ_i = 0.3 m^3/m^3.",
          "x": {
            "variable": "pct_sand",
            "label": "pct_sand (%)"
          },
          "y": {
            "variable": "pct_clay",
            "label": "pct_clay (%)"
          },
          "value": {
            "variable": "ψ",
            "reduce": "final"
          }
        }
      ]
    }
  ]
}