WetBulbTracker — Scientific Methodology

A complete specification of every heat-stress metric, algorithm, physical constant, and adjustment used by WetBulbTracker.com.

Last updated: June 2026 · Engine version: sprint-2 (full Liljegren WBGT)

This document is intended for scientists, clinicians, safety officers, and engineers who need to audit or reproduce the numbers shown in the app. Every formula below is implemented verbatim in the open metrics engine; source-file references are given for each section so the code and this document can be cross-checked line by line.

1. Scope and conventions

• Internal units. Every quantity is computed and stored in SI / metric units: temperature in degrees Celsius (°C), wind in metres per second (m/s), pressure in hectopascals (hPa), shortwave irradiance in watts per square metre (W/m²). Imperial conversion happens only at the display layer, so the calculation engine has a single source of truth.
• Display conversion. °F = °C · 9/5 + 32; mph = (m/s) · 2.236936; km/h = (m/s) · 3.6. The default display unit is chosen from the location's country (°F for the United States, °C elsewhere) and can be overridden.
• Determinism. The metrics engine is a set of pure functions: given the same raw inputs it always returns the same outputs. It performs no I/O and has no dependency on the UI framework.
• Reference temperature for thermodynamics. Energy-balance solvers work in Kelvin (K = °C + 273.15) and pascals (Pa = hPa · 100) internally, converting back to °C for the public result.

Source: src/lib/units.ts, src/lib/metrics/index.ts.

2. Data sources and pipeline

Raw weather fields requested (current conditions): temperature_2m, relative_humidity_2m, apparent_temperature, is_day, wind_speed_10m, surface_pressure, shortwave_radiation, dew_point_2m, uv_index, cloud_cover, precipitation. Wind is requested in m/s. Open-Meteo's underlying model is primarily ECMWF/national NWP blends at roughly 1–11 km resolution.

Freshness. Current conditions are cached for 15 minutes server-side; the forecast and historical archive for 1 hour and 24 hours respectively. A failed upstream request is retried once without cache before erroring.

Solar instant. For "current conditions" the solar geometry (§4) is evaluated at the present UTC instant; the sun moves slowly enough that the ≤15-minute data lag is immaterial. For the forecast strip and map time scrubber, each hour is evaluated at its own true UTC instant.

Source: src/lib/openmeteo.ts, src/app/api/conditions/route.ts.

2.1 Input variables

3. Metric overview

WBGT is the primary metric because it is the only one that incorporates solar radiation and is the recognised standard for occupational, athletic, and military heat-stress management.

4. Solar geometry

The Liljegren WBGT model needs two solar quantities: the cosine of the solar zenith angle and the fraction of incoming shortwave that arrives as a direct beam. Source: src/lib/solar.ts.

4.1 Cosine of the solar zenith angle (NOAA equations)

Let doy be the day of year (Jan 1 = 1) and h the UTC time in fractional hours. Define the fractional-year angle γ (radians):

γ = (2π / 365) · (doy − 1 + (h − 12) / 24)

Equation of time (minutes):

EqTime = 229.18 · ( 0.000075
                   + 0.001868·cos γ  − 0.032077·sin γ
                   − 0.014615·cos 2γ − 0.040849·sin 2γ )

Solar declination δ (radians):

δ = 0.006918 − 0.399912·cos γ  + 0.070257·sin γ
              − 0.006758·cos 2γ + 0.000907·sin 2γ
              − 0.002697·cos 3γ + 0.001480·sin 3γ

True solar time and hour angle (lng east-positive degrees):

TST       = h·60 + EqTime + 4·lng           (minutes)
HourAngle = (TST / 4 − 180) · π/180         (radians)

Cosine of the zenith angle (φ = latitude):

cos Z = sin φ · sin δ + cos φ · cos δ · cos(HourAngle)      (clamped to [−1, 1])

cos Z > 0 means the sun is above the horizon.

4.2 Direct-beam fraction (Liljegren clearness index)

Top-of-atmosphere irradiance on a horizontal surface, including the Earth-Sun distance correction (S₀ = 1367 W/m²):

TOA = S₀ · (1 + 0.033·cos(2π(doy − 1)/365)) · cos Z

Clearness index and direct fraction:

Kt   = min(1, S / TOA)
fdir = exp(3 − 1.34·Kt − 1.65/Kt)        clamped to [0, 0.9]

fdir = 0 whenever the sun is below the horizon (cos Z ≤ 0) or S ≤ 0.

5. Primary metric — WBGT (Liljegren et al., 2008)

WBGT combines three temperatures:

WBGT (outdoor, in sun) = 0.7·Tnwb + 0.2·Tg + 0.1·Ta
WBGT (shade / indoor)  = 0.7·Tnwb + 0.3·Ta

where Tnwb is the natural wet-bulb temperature, Tg the black-globe temperature, and Ta the dry-bulb (air) temperature. The app uses the outdoor (in sun) form, solving Tnwb and Tg from two coupled radiative-convective energy balances by iteration.

This is a faithful port of the PyWBGT Cython implementation (github.com/QINQINKONG/PyWBGT), itself a port of Liljegren's original C code. Constants and equations are unchanged; the only deviation is the root-finder (robust bisection here versus Brent's method in the reference — both converge to the same root). Source: src/lib/metrics/liljegren.ts.

5.1 Physical constants

Derived: RATIO = CP·MAIR/MH2O, RAIR = RGAS/MAIR, Pr = CP / (CP + 1.25·RAIR) (Prandtl number).

5.2 Thermophysical helper functions

Air dynamic viscosity μ(T) [kg/(m·s)], with Ω = 1.2945 − T/1141.17647:

μ(T) = 2.6693×10⁻⁶ · √(28.97·T) / (13.082689 · Ω)

Thermal conductivity: k(T) = (CP + 1.25·RAIR) · μ(T).

Saturation vapour pressure (Pa), Buck-type with a pressure-enhancement factor; separate branches over water (T > 273.15 K) and ice:

water:  es = 611.21 · exp(17.502·(T−273.15)/(T−32.18)) · (1.0007 + 3.46×10⁻⁶·Ps_hPa)
ice:    es = 611.15 · exp(22.452·(T−273.15)/(T− 0.60)) · (1.0003 + 4.18×10⁻⁶·Ps_hPa)

Atmospheric emissivity, with vapour pressure e in hPa:

e      = RH·0.01 · (es·0.01)
εatm   = 0.575 · e^0.143

Water-vapour diffusivity in air:

D(T, Ps) = 2.471773765×10⁻⁵ · (T·0.00342105637)^2.334 / (Ps/101325)

Latent heat of vaporisation: λ(T) = 1665134.5 + 2370·T.

Convective heat-transfer coefficients (ρ = Ps/(RAIR·T)):

Sphere   (globe):  Re = v·ρ·Dglobe/μ;  Nu = 2 + 0.6·Re^0.5·Pr^0.3333;  h = Nu·k/Dglobe
Cylinder (wick):   Re = v·ρ·Dwick /μ;  Nu = 0.281·Re^0.6·Pr^0.44;       h = Nu·k/Dwick

5.3 Black-globe energy balance

Direct-beam geometry term (0 when fdir = 0 or cos Z ≤ 0):

directTerm = (0.5/cos Z − 1) · fdir

Radiative constant:

C0 = 0.5·(1 + εatm)·Ta⁴
   + (1 / (2·εg·σ)) · S·(1 − αg)·(1 + directTerm + αsfc)

Globe temperature Tg (K) is the root of, with h the sphere coefficient evaluated at the film temperature ½(Ta + Tg):

C0 − (1/(εg·σ))·h·(Tg − Ta) − Tg⁴ = 0          bracket [Ta−50, Ta+90]

5.4 Natural wet-bulb energy balance

eair       = RH·0.01 · es(Ta)
directTerm = ( tan(arccos(min(1, cos Z)))/π + Dwick/(4·Lwick) ) · fdir
D4         = εwick·0.5·σ·Ta⁴·(εatm + 1)
           + (1 − αwick)·S·( (1 + Dwick/(4·Lwick))·(1 − fdir) + directTerm + αsfc )

Natural wet-bulb temperature Tnwb (K) is the root of, with Tref = ½(Ta+Tw), Sc = μ(Tref)/(ρ·D(Tref)) the Schmidt number, h the cylinder coefficient at Tref, and Fatm = D4 − εwick·σ·Tw⁴:

Ta − (λ(Tref)/RATIO)·((es(Tw) − eair)/(Ps − es(Tw)))·(Pr/Sc)^0.56 + Fatm/h − Tw = 0

bracket [Ta − (100 − RH)/5 − 50, min(Ta + 70, 340)].

5.5 Adjustments and solver details

• Wind height correction. Inputs are 10 m winds; the model expects ~2 m. Converted with the power law v₂ = v₁₀·(2/10)^0.2 and floored at 0.13 m/s, as in the reference (a still-air floor prevents division blow-up).
• Humidity clamp. RH is clamped to [1, 100] %.
• Irradiance / geometry guards. S is floored at 0; fdir is forced to 0 when the sun is below the horizon.
• Root finder. Bisection over the brackets above, tolerance 1×10⁻⁴ K, up to 80 iterations. If either balance fails to bracket a root (no sign change), the engine falls back to the simplified estimate (§5.7) rather than returning a bad number.

5.6 Validation status

The Liljegren method is the validated reference model and is reported to be more accurate than handheld WBGT meters (Duke Nicholas Institute, 2024). Full numerical validation against NWS WBGT output (target ±1.5 °C) is the sprint-2 deliverable. Physical sanity checks pass: dry desert heat (e.g. Phoenix) reads moderate WBGT, humid tropical nights (e.g. Miami) read elevated WBGT, and in direct sun Tnwb < WBGT < Tg, with Tg rising sharply under high irradiance and low wind.

5.7 Simplified fallback (only when geometry/pressure are unavailable)

When cos Z, fdir, or surface pressure are missing, WBGT uses a closed-form estimate: a Stull wet bulb (§7) for Tnwb and a parameterised globe term:

Tg ≈ Ta + min(25, 0.024·S / √max(0.5, v))
WBGT = 0.7·Tnwb_Stull + 0.2·Tg + 0.1·Ta

This is directionally correct but not validated to the ±1.5 °C target; in normal operation (with full solar geometry) the Liljegren path is always used.

6. Heat Index (NWS Rothfusz regression)

The familiar US "feels like in the shade" number. Defined in °F; the engine converts in and out so the rest of the pipeline stays in °C. Source: src/lib/metrics/heatindex.ts.

A Steadman blend is always computed first (T in °F, R = RH %):

HI = 0.5·(T + 61 + (T − 68)·1.2 + R·0.094)

If the average (HI + T)/2 ≥ 80 °F, the full Rothfusz regression replaces it:

HI = −42.379
   + 2.04901523·T   + 10.14333127·R
   − 0.22475541·T·R − 0.00683783·T²
   − 0.05481717·R²  + 0.00122874·T²·R
   + 0.00085282·T·R² − 0.00000199·T²·R²

with the two standard corrections:

if R < 13 and 80 ≤ T ≤ 112:   HI −= ((13 − R)/4)·√((17 − |T − 95|)/17)
if R > 85 and 80 ≤ T ≤  87:   HI += ((R − 85)/10)·((87 − T)/5)

The result is converted back to °C.

7. Wet-bulb temperature (Stull, 2011)

A fast psychrometric (thermodynamic) wet-bulb approximation, valid near sea-level pressure for roughly T ∈ [−20, 50] °C and RH ∈ [5, 99] %, accurate to about ±0.3 °C. RH is clamped to [1, 100] %. Source: src/lib/metrics/wetbulb.ts.

Tw = T·arctan(0.151977·√(RH + 8.313659))
   + arctan(T + RH) − arctan(RH − 1.676331)
   + 0.00391838·RH^1.5·arctan(0.023101·RH)
   − 4.686035

Note: this thermodynamic wet bulb does not include radiation or wind, so in direct sun it slightly underestimates the *natural* wet bulb. The app surfaces it as its own "Wet Bulb" metric and uses it as the Tnwb term only in the simplified WBGT fallback (§5.7); the primary WBGT uses the full Liljegren natural wet-bulb solver.

8. Apparent Temperature / "Real Feel" (BoM, Steadman)

The Australian Bureau of Meteorology apparent-temperature formulation. Unlike Heat Index it includes the cooling effect of wind, making it the better everyday "real feel". Water-vapour pressure e in hPa, wind v in m/s. Source: src/lib/metrics/apparent.ts.

e   = (RH/100) · 6.105 · exp(17.27·Ta / (237.7 + Ta))
AT  = Ta + 0.33·e − 0.70·v − 4.00

When Open-Meteo supplies a native apparent_temperature, that value is used in preference to this formula; otherwise the formula above is computed.

9. Dew point (Magnus-Tetens)

Coefficients from Alduchov & Eskridge (1996); accurate to about ±0.4 °C for 0 < T < 60 °C. RH clamped to [1, 100] %. a = 17.625, b = 243.04. Source: src/lib/metrics/dewpoint.ts.

γ  = ln(RH/100) + (a·Ta)/(b + Ta)
Td = (b·γ) / (a − γ)

The provider's native dew_point_2m is preferred when present.

10. Supporting context fields

These are displayed for context and are not inputs to the heat-stress calculations:

• UV Index (uv_index) — bucketed Low / Moderate / High / Very High / Extreme at 3 / 6 / 8 / 11.
• Air Quality (us_aqi) — US AQI, bucketed Good / Moderate / Sensitive / Unhealthy / Very Bad / Hazardous at 50 / 100 / 150 / 200 / 300.
• Cloud cover (cloud_cover, %), surface pressure (surface_pressure, hPa), precipitation (precipitation, mm; shown in inches when imperial).

11. Classification scales

All thresholds are stored in °C (engine units); °F equivalents are given here for convenience. Bands share a common green → yellow → orange → red → purple → black severity gradient so that colour means the same thing across metrics. Source: src/lib/metrics/scales.ts.

11.1 WBGT — "Heat Stress" (primary)

11.2 Wet-bulb temperature

The ~35 °C wet-bulb survivability limit is the temperature above which a healthy person can no longer shed metabolic heat by sweating, even at rest in shade.

11.3 Heat Index

11.4 Real Feel (apparent temperature)

11.5 Dew point (comfort)

11.6 Unified advisory

The card shows one persistent heat-danger advisory: the most severe guidance across the three monotonic heat-stress metrics (WBGT, Heat Index, Wet Bulb), independent of which metric the user is viewing. Severity is ranked by band index and mapped to the WBGT advisory ladder:

Dew point (a comfort scale) and Real Feel (which has cold bands) are excluded from the unified advisory.

12. "This day in history"

Places today's heat in climatological context using ERA5 reanalysis. Source: src/app/api/history/route.ts, src/lib/openmeteo.ts.

• Variable. Daily maximum apparent temperature (apparent_temperature_max), a humidity-aware, precomputed daily aggregate — chosen so decades of context come from a single cheap request rather than recomputing hourly WBGT.
• Window. A ±3-day window around today's calendar day, across the most recent 35 complete years.
• Per-year peak. For each year, the maximum daily feels-like high within the window. Window days are grouped by *climatological season-year* so a window that straddles 1 January stays in one group (correct for Southern-Hemisphere summers).
• Outputs. Percentile rank of today's forecast feels-like high among the per-year peaks; the climatological normal (mean of the peaks); the all-time record (value and year); and the full per-year series for the sparkline.
• Today's value. Today's forecast apparent_temperature_max.

If fewer than five valid years are available, the panel hides itself.

13. Map visualisation (interpolation)

The heat map is a continuous field built from point samples. For a viewport the app samples a grid of points (current conditions, or a forecast hour from the time scrubber); at low zoom it uses an always-on coarse global grid (36 × 18 ≈ 648 points, CDN-cached hourly). Each sampled point is run through the same metrics engine, then the grid is bilinearly interpolated per pixel and coloured along the metric's continuous gradient; ocean / missing nodes fade to transparent so the field flows seamlessly. This is a visualisation step only and does not alter the per-point metric values. Source: src/components/HeatMap.tsx, src/app/api/grid/route.ts, src/app/api/global/route.ts.

14. Limitations and assumptions

• Inputs are NWP model output, not in-situ observations; local microclimate (pavement, shade, buildings) is not captured.
• The Liljegren surface albedo (0.45) and the assumption of a standing person in the open are fixed; true WBGT varies with ground cover and posture.
• Shortwave radiation is global horizontal; the direct/diffuse split is inferred from a clearness-index parameterisation, not measured.
• Stull wet bulb assumes near-sea-level pressure.
• Historical context uses daily apparent-temperature maxima (not WBGT) for tractability, and ERA5 has a coarser grid than the live forecast.
• The simplified WBGT fallback (§5.7) is not validated and is used only when solar geometry or pressure are unavailable.

15. References

1. Liljegren, J. C., Carhart, R. A., Lawday, P., Tschopp, S., & Sharp, R. (2008). *Modeling the Wet Bulb Globe Temperature Using Standard Meteorological Measurements.* Journal of Occupational and Environmental Hygiene, 5(10), 645–655.
2. Stull, R. (2011). *Wet-Bulb Temperature from Relative Humidity and Air Temperature.* Journal of Applied Meteorology and Climatology, 50(11), 2267–2269.
3. Rothfusz, L. P. (1990). *The Heat Index Equation.* NWS Southern Region Technical Attachment SR/SSD 90-23.
4. Steadman, R. G. (1984). *A Universal Scale of Apparent Temperature.* Journal of Climate and Applied Meteorology, 23(12), 1674–1687. (BoM apparent temperature.)
5. Alduchov, O. A., & Eskridge, R. E. (1996). *Improved Magnus Form Approximation of Saturation Vapor Pressure.* Journal of Applied Meteorology, 35(4), 601–609.
6. NOAA Global Monitoring Laboratory. *Solar Position Calculator equations* (general solar geometry).
7. Kong, Q., & Huber, M. *PyWBGT* — reference implementation of the Liljegren WBGT model. github.com/QINQINKONG/PyWBGT
8. Hersbach, H., et al. (2020). *The ERA5 global reanalysis.* Quarterly Journal of the Royal Meteorological Society, 146(730), 1999–2049.
9. Open-Meteo. *Open-Meteo Weather, Air-Quality, and Historical Reanalysis APIs.* open-meteo.com

*Generated for WetBulbTracker.com. The metrics engine is a pure, framework-free module; every formula above is implemented in src/lib/metrics/ and src/lib/solar.ts and can be audited directly.*