# Relative Humidity¶

The relative humidity (RH) values in the POWER Archive are calculated from pressure (Pa in kPa), dry-bulb temperature (Ta in °C), and mixing ratio (e.g. specific humidity, q in kg/kg), parameters that are available from the re-gridded MERRA-2 and represents estimates at 2 m above the local surface averaged over the original source spatial resolution. The following is a summary of the expressions used to calculate RH. The units are indicated in square brackets.

## Equation¶

\begin{align} RH = (e_a/e_{sat}) × 100 \text{ } \text{ } \text{ } \text{ } [\%] \tag {Equation 1} \end{align}
\begin{align} Where: \\ e_a: & \text{ the water vapor pressure }\\ e_{sat}: & \text{ the saturation water vapor pressure at the ambient temperature $T_a$.}\\ \end{align}
\begin{align} \text{The 100 has been added to cast RH in terms of percent.} \end{align}

Since water vapor and dry air (a mixture of inert gases) can be treated as ideal gases, it can be shown that (Iribarne and Godson, pg 74, Eq. 76; note that the symbol, $r$, use in Eq. 76 for the mixing ratio has been replaced by $w$ and the factor of 10 has been added to convert the units to hPa.)

\begin{align} e_a = (10×P_a×w)/(ε + w) \text{ } \text{ } \text{ } \text{ } [hPa] \tag {Equation 2} \end{align}
\begin{align} Where: \\ w: & \text{ mixing ratio define as the ratio of mass of water to dry air }\\ \end{align}
\begin{align} ϵ = \frac{R'}{R_v} = \frac{287.05}{461.5} ≃ 0.622 \tag {Equation 3} \end{align}
\begin{align} Where: \\ R': & \text{ Dry vapor gas constant }\\ R_v: & \text{ Water vapor gas constant } \end{align}

There is no exact consensus for the gas constants past 3 significant digits, therefore the value of the ratio is kept to 3 significant digits. The mixing ratio is related to specific humidity by the relation (Jupp 2003, pg.37):

\begin{align} w = q / (1-q) \text{ } \text{ } \text{ } \text{ } [kg/kg] \tag {Equation 4} \end{align}

Combining Equation 2 and Equation 4 leads to the following expression for $e$ in terms of $q$:

\begin{align} e_a = q × 10 × P_a / [ϵ + q × (1- ϵ)] \text{ } \text{ } \text{ } \text{ } [hPa] \tag {Equation 5} \end{align}

An eighth-order polynomial fit (Flatau, et. al. 1992) to measurements of vapor pressure over ice and over water provides an expression to calculate the saturated water vapor pressure over ice and over water. The eight-order fit for $e_{wsat}$ is given by:

\begin{align} e_{wsat} & = A_{1w} + A_{2w}×(T_a) + …+ A_{(n-1)w}×(T_a)^n \tag {Equation 6} \\ e_{isat} & = A_{li} + A_{2i}×(T_a) + …+ A_{(n-1)i}×(T_a)^n \tag {Equation 7} \end{align}
\begin{align} Where: \\ e_{wsat}: & \text{ Saturated vapor pressure over water in [hPa = mb] }\\ e_{isat}: & \text{ Saturated vapor pressure over ice [hPa = mb] }\\ T_a: & \text{ Ambient dry bulb temperature in °C } \end{align}

## Coefficients¶

Coefficients for $e_{sat}$ over water and over ice and the temperature range over which the coefficient are applicable.

Over Water
(Valid: -85 °C to 70 °C)
Over Ice
(Valid: -90 °C to 0 °C)
A1w = 6.11583699 A1i = 6.09868993
A2w = 0.444606896 A2i = 0.499320233
A3w = 0.143177157E-1 A3i = 0.184672631E-1
A4w = 0.264224321E-3 A4i = 0.402737184E-3
A5w = 0.299291081E-5 A5i = 0.565392987E-5
A6w = 0.203154182E-7 A6i = 0.521693933E-7
A7w = 0.702620698E-10 A7i = 0.307839583E-9
A8w = 0.379534310E-13 A8i = 0.105758160E-11
A9w = -0.321582393E-15 A9i = 0.161444444E-14

Note

Only the relative humidity over water is calculated and provided in the POWER Archive consistent with the values reported by the National Weather Service.

## Validation¶

Scatter plot of the re-gridded MERRA-2 daily relative humidity vs. the daily station observations values from the NCEI GSOD files for every 3rd year from 1981 – 2014. The color bar along the right side of the scatter plot provides a measure of the distribution of the NCEI, MERRA-2 relative humidity pairs in bins of 1%. For example, the left column along the vertical color bar shows that each data point in dark blue represents the number ground site/MERRA-2 data pairs within 10% (i.e. <=1,255.5) of the maximum number of data pairs within a 1% bins in the plot (i.e. 12,555). Additionally, from the right side of the vertical color bar it can be seen that all the points shown in dark blue contain 16.83% of the total number of ground site, MERRA-2 data pairs (8,514,703). The remaining 83.2% of the data pairs are concentrated along the 1:1 grey line.