When One Agency Publishes Two Heat Numbers, the Difference Is in the Inputs
Two numbers from the same agency
A National Weather Service forecast page can carry two temperature-like numbers for the same hour and the same location, and they will not agree. One is the heat index, the number that appears beside the air temperature in most forecast products. The other is Wet Bulb Globe Temperature, or WBGT, which several NWS field offices publish as a separate gridded product. When the two are compared on a hot afternoon, the natural assumption is that one of them is a correction of the other, or that a newer instrument has replaced an older one.
Neither is the case. The gap between them is not a disagreement about the weather. It is a difference in what each number was defined to take in. The heat index accepts two variables. WBGT accepts five. Everything else that separates them follows from that.
The heat index published by the NWS is not a reading from any instrument. It is the output of a multiple regression fitted to a table, and the equation is public. The NOAA Weather Prediction Center, Heat Index Equation page prints it in full:
HI = -42.379 + 2.04901523*T + 10.14333127*RH - .22475541*T*RH - .00683783*T*T - .05481717*RH*RH + .00122874*T*T*RH + .00085282*T*RH*RH - .00000199*T*T*RH*RH
Nine coefficients. Two variables: T, the air temperature in degrees Fahrenheit, and RH, the relative humidity in percent. There is no term for wind, no term for solar radiation, no term for cloud cover, and no term for time of day. Whatever the sky is doing, the same pair of numbers produces the same heat index.
Where the shade characterisation actually comes from
It is worth being precise about which document says what, because the two claims live on different pages. The equation page prints the regression and its conditions. It does not mention shade, sun, or solar radiation anywhere. The characterisation of the heat index as a shade number appears instead on the NWS pages that describe WBGT, where the two indices are contrasted directly. The NOAA National Weather Service, Wet Bulb Globe Temperature page defines WBGT as "a measure of the heat stress in direct sunlight, which takes into account: temperature, humidity, wind speed, sun angle and cloud cover (solar radiation)." It then draws the contrast in a single sentence: "This differs from the heat index, which takes into consideration temperature and humidity and is calculated for shady areas."
That sentence is doing a great deal of work. It is the agency stating, in its own words, the exposure condition its heat index number is built around. The equation itself contains no shade parameter that could be switched off; the shade condition is a property of the data the regression was fitted to, not a variable in the regression.
What follows from this is narrow and worth stating carefully. It does not follow that a specific number of degrees can be added to convert a shade heat index into a sun figure. No such conversion factor appears on any of the NWS pages examined here, and no figure is offered for one. What does follow is that the two numbers are outputs of two different definitions, and that comparing them as if they measured the same quantity is a category error.
The Chain
The published heat index is not produced by evaluating one equation. It is produced by a sequence with a branch and two conditional corrections in it. The NOAA Weather Prediction Center, Calculating the Heat Index page and the equation page together set out the order:
- A simpler formula runs first. The equation page states: "In practice, the simple formula is computed first and the result averaged with the temperature." That formula is
HI = 0.5 * {T + 61.0 + [(T-68.0)*1.2] + (RH*0.094)}. - The result is averaged with the air temperature. This averaged value is the quantity the next step tests.
- A threshold decides which equation governs. "If this heat index value is 80 degrees F or higher, the full regression equation along with any adjustment as described above is applied." Below that, the averaged value stands and the regression is never evaluated.
- The nine-coefficient regression runs. This is the equation printed above.
- Two conditional adjustments may fire. If relative humidity is under 13% and the temperature is between 80 and 112 °F, an amount is subtracted. If relative humidity is over 85% and the temperature is between 80 and 87 °F, an amount is added. Outside those two windows, neither correction applies.
- The result is published as a whole number of degrees.
The 80 °F threshold in step three is a hard switch between two different functions, and the two functions do not agree at the switch point. Evaluating both at the crossing shows the size of the seam. At 30% relative humidity the sequence changes equations near 80.4 °F, and the value drops by about 0.5 °F as it crosses. At 70% relative humidity the crossing sits near 79.5 °F and the value jumps upward by roughly 2.4 °F. The direction of the step is not fixed; it depends on the humidity at which the crossing happens.
The person inside the equation
The regression was not fitted to raw observations. It was fitted to a table produced by a biometeorological model, and that model required a specific person doing a specific thing. The NWS technical attachment that introduced the equation, NOAA National Weather Service, Technical Attachment SR 90-23, The Heat Index "Equation", sets this out plainly: "In order to arrive at an equation which uses more conventional independent variables, a multiple regression analysis was performed on the data from Steadman's table." It then lists the parameters "from Steadman, 1979", with assumed magnitudes in parentheses.
Several of those assumptions are physical descriptions of a hypothetical human being. Dimensions of a human: 5 feet 7 inches tall, 147 pounds. Significant diameter of a human, based on the body's volume and density: 15.3 cm. Effective radiation area of skin, a ratio depending on skin surface area: 0.80. Core temperature: 98.6 °F. Core vapor pressure: 5.65 kPa. Ambient vapor pressure of the atmosphere: 1.6 kPa.
Others describe what that person is wearing and doing. Clothing cover: "Long trousers and short-sleeved shirt is assumed," at 84% coverage. Activity, which determines metabolic output: 180 watts per square metre of skin area, specified for "the model person walking outdoors at a speed of 3.1 mph." Effective wind speed, the vector sum of the body's movement and an average wind speed: 5 knots. Ventilation rate, the heat lost through exhaling: 2 to 12 percent depending on humidity.
This is the structural point. Wind is in there. Clothing is in there. Body size is in there. None of them are inputs. They were fixed at single values before the regression was fitted, and they are now constants folded into those nine coefficients. A forecast heat index cannot respond to a change in any of them, because after the fitting step there is nothing left to respond with.
The same document also states the precision of the fit: "Because this equation is obtained by multiple regression analysis, the heat index value (HI) has an error of ±1.3°F." That figure is a useful yardstick for the rest of this discussion.
What ten points of humidity are worth
The NWS publishes the equation but not a table of what it does. Running it over a grid of inputs shows a property that the printed coefficients do not make obvious: the response to humidity is strongly non-linear, and the non-linearity grows with temperature. The values below were computed from the published regression using unrounded inputs, with rounding applied only for display.
| Air temperature | RH 20% → 30% | RH 40% → 50% | RH 60% → 70% | RH 70% → 80% |
|---|---|---|---|---|
| 85 °F | +0.8 °F | +2.1 °F | +3.5 °F | +4.1 °F |
| 90 °F | +1.6 °F | +3.9 °F | +6.2 °F | +7.4 °F |
| 95 °F | +2.9 °F | +6.2 °F | +9.5 °F | +11.2 °F |
| 100 °F | +4.8 °F | +9.1 °F | +13.3 °F | +15.4 °F |
| 105 °F | +7.3 °F | +12.4 °F | +17.5 °F | +20.1 °F |
Read across a single row and the same ten-point change in relative humidity is worth several times more at high humidity than at low. At 95 °F, moving from 20% to 30% relative humidity raises the heat index by 2.9 °F; moving from 70% to 80% raises it by 11.2 °F, roughly 3.8 times as much. Read down a column and the same ten-point change grows with temperature: the 70-to-80 step is worth 4.1 °F at 85 °F and 20.1 °F at 105 °F.
This is why the heat index and the air temperature do not track each other. At 85 °F and 20% relative humidity the index sits at 82.0 °F, which is nearly three degrees below the air temperature. At 95 °F and 70% it reaches 122.6 °F, more than twenty-seven degrees above it. The distance between the two numbers is not a fixed offset, and it does not even keep a fixed sign; it is a surface.
The chart below plots that surface along four horizontal slices. Each curve holds the air temperature fixed and sweeps relative humidity from 20% to 85%. The four curves start within about fifteen degrees of one another on the left and end far apart on the right: at 85% relative humidity the four values are 99.1 °F, 117.5 °F, 140.0 °F and 166.7 °F, spanning more than sixty-seven degrees for a fifteen-degree spread in air temperature. The fanning is the non-linearity in the table above, drawn.
The two conditional corrections, measured
The equation page adds two corrections that fire only inside narrow windows, and neither is large. Computing them across their full permitted ranges puts numbers on them.
The low-humidity correction applies when relative humidity is below 13% and the temperature lies between 80 and 112 °F, and it is subtracted. Its formula makes the deduction largest where the square-root term is largest, at exactly 95 °F. At 95 °F and 5% relative humidity the deduction is 2.0 °F, taking the raw regression value of 90.2 °F down to 88.2 °F. At the same humidity but 110 °F it is only 0.69 °F, and at 112 °F the window closes and the correction stops entirely.
The high-humidity correction applies when relative humidity is above 85% and the temperature lies between 80 and 87 °F, and it is added. It is largest at the corner of its window: at 80 °F and 100% relative humidity it adds 2.1 °F, taking 87.2 °F to 89.3 °F. At 86 °F and 90% it adds 0.1 °F. At 87 °F the window closes.
Both corrections are therefore bounded by roughly the same magnitude as the ±1.3 °F fitting error stated in the technical attachment. They are refinements at the edge of the equation's resolution, not adjustments that move a number into a different range. They also introduce small discontinuities: an air mass drying from 13.1% to 12.9% relative humidity at 95 °F crosses into the correction window, and the published value steps by a few hundredths of a degree at the boundary and grows from there as humidity falls further.
Why the second index exists at all
WBGT is not a corrected heat index. It is a different instrument concept with a different input list. The NWS description names five: temperature, humidity, wind speed, sun angle, and cloud cover as a proxy for solar radiation. Three of those five have no representation in the heat index regression as variables at all. Sun angle and cloud cover are absent entirely. Wind exists only as the fixed 5-knot constant buried in Steadman's parameter list.
That difference in input count is the whole explanation for why the two numbers separate and by how much they separate on any given afternoon. A day with high sun and light wind and a day with heavy overcast and a stiff breeze can share a temperature and a relative humidity. The heat index cannot distinguish them, because it was never given the variables that differ. WBGT can, because it was.
Neither number is wrong. Each is the correct output of its own definition. The failure mode is a reading failure rather than a measurement failure: two products from the same agency get treated as competing estimates of one quantity, when they are exact answers to two different questions.
Where This Doesn't Apply
This frame breaks down in several places, and the documents say so themselves.
First, below the branch point. When the averaged value from step two falls under 80 °F, the nine-coefficient regression is never evaluated, and nothing described above about its coefficients, its adjustments, or its ±1.3 °F fitting error applies to the number that gets published. The WPC page is explicit: "The Rothfusz regression is not appropriate when conditions of temperature and humidity warrant a heat index value below about 80 degrees F." Below that line, a much simpler linear expression is the whole story.
Second, at the top of the range. The regression is an extrapolation outside the data it was fitted to, and NOAA marks the limit without giving a numeric cutoff: "The Rothfusz regression is not valid for extreme temperature and relative humidity conditions beyond the range of data considered by Steadman." The upper right of the chart above sits in that direction. A value of 166.7 °F at 100 °F air temperature and 85% relative humidity is what the arithmetic returns; the agency's own statement is the reason not to treat it as a physical claim.
Third, the shade characterisation is a sentence on a WBGT page, not a parameter in the equation. It describes the conditions the underlying table was built for. It does not license any arithmetic conversion between a shade figure and a sunlit one, and no NWS page examined here supplies a factor for that.
Fourth, the assumed person. A regression fitted to a table generated for a 5-foot-7, 147-pound figure in long trousers and a short-sleeved shirt walking at 3.1 mph is a statement about that figure. Every departure from those parameters is outside the model's stated scope, and the equation offers no mechanism to represent the departure.
Finally, WBGT products are not uniformly available. WBGT appears on individual NWS field office pages rather than as a single national forecast field of the same standing as the heat index, so the comparison drawn here is available in some places and not others. Where only one of the two numbers is published, the definitional gap does not disappear. It simply becomes invisible.
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