These forecast products display the probability of the daily minimum, maximum, and mean heat index exceeding specific thresholds.  Color filled displays of probabilities and expected heat index values are based on Weather Prediction Center (WPC) or Model Output Statistics (MOS) maximum and minimum temperature and dew point temperature forecast grids.   The MOS is used for projection day 3; the WPC forecasts are used for projection days 4 through 7. These contribute the "human" component to the heat index forecast. The probabilities have the normal distribution (bell curve) about the WPC or MOS forecast heat indexes with spatially varying standard deviations representing forecast uncertainty derived from an ensemble, which includes the NCEP Global Forecast System (GFS) high resolution model and the NCEP Global Ensemble Forecast System (GEFS).   The maximum and minimum heat index values are averaged to get daily mean heat index values for each forecast projection day (3 through 7).  The calculations are done at regularly spaced grid points approximately 20 km apart across the continental United States (CONUS).   Contours of probability are drawn with color fill denoting the forecast chance of the heat index exceeding specified thresholds.

  1. Introduction

So called heat waves, periods of sustained high temperature and high humidity, have long been recognized as a significant weather hazard.  Attempts to quantify the combined effect of high temperature and humidity on the human body date back many years.  Culminating work on the subject was done by Steadman (Journal of Applied Meteorology, 1979, Volume 18, Issue 7, pages 861-873) and reported in his paper:  "The Assessment of Sultriness.  Part I:  A Temperature-Humidity Index Based on Human Physiology and Clothing Science."  Steadman summarizes his results as a table of numbers.  Others have used Steadman's table of discrete values to derive regression equations for the apparent temperature (or heat index as it has come to be called) as a continuous function of temperature and relative humidity. It is important to note that these regression equations cannot be applied for temperature and relative humidity values outside the range of data studied by Steadman.   While high values of the heat index can pose a health risk to anyone engaged in outdoor activity over a short period of time, a greater general danger to public health exists when the heat index remains high for an extended period of time.  These occasions are marked by high daily average values of the heat index.  For this reason, the focus of this forecast product is on the daily mean heat index for each forecast day, but the extrema are included to enhance the value of this guidance.
  2.  Calculating the Heat Index

WPC forecasters have skill in modifying the Environmental Modeling Center's (EMC) GFS Medium Range Forecast (MRF) Model based Model Output Statistics (MOS)   guidance generated by the Meteorological Development Laboratory (MDL) for maximum and minimum temperatures.  In addition to the temperature, the heat index equation given below requires a relative humidity value.  The WPC forecasters do not currently modify the MOS dew point temperature forecasts; therefore, derived dew point temperatures valid at 12 UTC and the following 00 UTC are matched with the WPC minimum and maximum temperatures, respectively, to compute the relative humidity values needed to calculate the forecast heat index values.   The dew point temperatures are obtained in the manner described in the documentation of the WPC 5-km resolution grid data products. Here, the 00 UTC value is that for the next forecast day.  Over the CONUS, 00 UTC falls in the early evening and thus corresponds roughly to the usual time of maximum temperature.   The ensemble is described in more detail below. The WPC 5-km grid products are used to generate the heat index guidance for projection days 4 through 7. MOS temperature and dewpoint temperature forecasts gridded at 5-km resolution are used for projection day 3.

The computation of the heat index is a refinement of a result obtained by multiple regression analysis carried out by Lans P. Rothfusz and described in a 1990 National Weather Service (NWS) Technical Attachment (SR 90-23).  The regression equation of Rothfusz is

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
where T is temperature in degrees F and RH is relative humidity in percent.  HI is the heat index expressed as an apparent temperature in degrees F.  If the RH is less than 13% and the temperature is between 80 and 112 degrees F, then the following adjustment is subtracted from HI:
ADJUSTMENT = [(13-RH)/4]*SQRT{[17-ABS(T-95.)]/17}
where ABS and SQRT are the absolute value and square root functions, respectively.  On the other hand, if the RH is greater than 85% and the temperature is between 80 and 87 degrees F, then the following adjustment is added to HI:
ADJUSTMENT = [(RH-85)/10] * [(87-T)/5]
The Rothfusz regression is not appropriate when conditions of temperature and humidity warrant a heat index value below about 80 degrees F. In those cases, a simpler formula (not shown) is applied to calculate values consistent with Steadman's results. This regression is not valid for extreme temperature and relative humidity conditions beyond the range of data considered by Steadman. The allowed values for temperature (at and above 80 degrees F) and relative humidity are displayed in the NWS Heat Index Table.

The equation for HI above with the appropriate adjustment is applied to compute a maximum heat index using the WPC or MOS forecast maximum temperature and the 00 UTC dew point temperature at each forecast grid point location for each forecast projection day.  Similarly, a minimum heat index is computed using the minimum temperature and corresponding 12 UTC derived dew point temperature.  The forecast daily mean heat index for the projection day is the average of these two values, the maximum heat index and the minimum heat index.
  3.  The Ensemble

For each forecast day, the mean and spread of the heat index is computed from the combined GFS and GEFS ensemble at 6-hour intervals, beginning at valid time 06 UTC and ending at 00 UTC. The ensemble mean for the minimum heat index is the minimum of the 06 and 12 UTC values. The ensemble spread for the minimum heat index is the maximum of the spreads computed at 06 and 12 UTC. The ensemble mean for the maximum heat index is the maximum of the ensemble means at 18 and 00 UTC. The ensemble spread for the maximum heat index is the maximum of spreads computed at 18 and 00 UTC. The mean and spread for the mean heat index are the respective averages of those obtained for the minimum and maximum. These calculations are done at each point on a grid having 20-km resolution.

The variance of the ensemble about the human forecast from the WPC or the MOS is computed by squaring the ensemble spread to obtain a variance and adding to it the square of the difference between the human forecast and the ensemble mean. The square root of this variance of the ensemble about the human forecast is the spread about the human forecast. It is this spread value that is taken as the standard deviation of a normal distribution to compute the probabilities (see next section below).

Prior to the warm season of 2007, the spread was computed from ensemble MOS point data objectively analyzed to the grid. This spread was under dispersive due to the tendency of MOS to gravitate toward climatology, especially at longer projection times. An arbitrary adjustment was done to enhance that ensemble MOS spread. The approach described here, using the ensemble spread directly, gives a better estimate in a much more efficient computational procedure that does not involve an arbitrary enhancement of spread.
  4.  Calculating Probabilities

The probability forecast at any given point intends to answer the following question:  Based on the ensemble estimate of forecast uncertainty, what is the chance that the observed heat index parameter will be higher than H given that the WPC or MOS forecast heat index parameter is F = HI, computed as described above?  It is assumed that the forecast error at each point at each forecast day has a normal distribution about F.  This assumption allows a rather straight forward calculation of the probability that a given heat index threshold, H, will be exceeded, given a forecast value of F.  If F stands for the forecast value and O stands for the observed value, then, under the assumption, (O-F) has a normal distribution.  Furthermore,  Z = (O-F) / S has the standard normal distribution, where S is the forecast uncertainty standard deviation.  To compute the probability that a threshold will be exceeded for a given forecast F, the question posed above becomes what is the probability that (O-F) will be greater than or equal to (H-F)?  This is the probability that the standard normally distributed z is greater than or equal to Z = (H-F) / S.  The old fashioned way to get the probability would be to compute Z and then use a table of the values of the standard normal distribution to get the probability.  In practice, numerical integration is used to obtain the probability.  Here is an example to help clarify this calculation:  Suppose that for the day 5 forecast at Richmond, VA, the forecast uncertainty standard deviation is 3 degrees F.  If the WPC or MOS forecast heat index parameter (either the minimum, maximum, or mean) for projection day 5 is 87 degrees F, what is the probability of the heat index parameter exceeding 90 degrees F?  The answer, 16%, is given in the table below and was obtained the old fashioned way using tabulated standard normal distribution probabilities.

S (std dev)
F (forecast)
H (threshold)
These forecast probabilities are based on ensemble statistics and have not been calibrated by an extended period of verification.  The result in the example above may imply that the heat index will exceed 90 degrees F 16 out of 100 times when the forecast value is 85, but an actual verification may show that 90 is exceeded more or less frequently.
  5.  Displaying Probability Forecasts

Contours are drawn for the 10%, 40%, and 70% probabilities, with shading between the lines and for all areas of greater than 70% probability.  These shaded regions, 10 to 40%, 40 to 70%, and greater than 70%, correspond roughly to low, moderate, and high chance of the observed heat index exceeding the given threshold.   Tabulated forecast values of the probabilities and the heat index parameters presented as text are interpolated linearly from the gridded computational domain to the display locations.
  6.  Summary

Heat index probability forecasts are generated by combining GFS, GEFS, and WPC or MOS forecasts of dew point and maximum/minimum temperatures to compute forecast daily minimum, maximum, and mean heat indexes, which become the basis for computing the probability of these heat index parameters exceeding specific thresholds using an ensemble estimate of the inherent forecast uncertainty standard deviation and the assumption of normally distributed error.  This is an evolving product, and future work is suggested.  The probabilities could be compared to observed frequencies and calibrated statistically, or, using more advance ensemble analysis techniques, other methods of computing the probabilities could be developed. The heat index and probability forecasts are archived for verification.