Standardize RH calcs on WMO8

Base all RH calculations on WMO8 standard using vapor partial pressure
ratio. This includes calculations previously "directly" calculating from
 ratio of mixing ratios. Corrections cited. See #2951 for more info.

src/metpy/calc/thermo.py

@@ -55,7 +55,7 @@ def relative_humidity_from_dewpoint(temperature, dewpoint):
 
     Notes
     -----
-    .. math:: rh = \frac{e(T_d)}{e_s(T)}
+    .. math:: RH = \frac{e(T_d)}{e_s(T)}
 
     .. versionchanged:: 1.0
        Renamed ``dewpt`` parameter to ``dewpoint``
@@ -2026,28 +2026,33 @@ def mixing_ratio_from_relative_humidity(pressure, temperature, relative_humidity
     >>> T = 28.1 * units.degC
     >>> rh = .65
     >>> mixing_ratio_from_relative_humidity(p, T, rh).to('g/kg')
-    <Quantity(15.9828362, 'gram / kilogram')>
+    <Quantity(15.7646969, 'gram / kilogram')>
 
     See Also
     --------
     relative_humidity_from_mixing_ratio, saturation_mixing_ratio
 
     Notes
     -----
-    Formula adapted from [Hobbs1977]_ pg. 74.
+    Employs [WMO8]_ eq. 4.A.16 as derived from WMO relative humidity definition based on
+    vapor partial pressures (eq. 4.A.15).
 
-    .. math:: w = (rh)(w_s)
+    .. math:: RH = \frac{w}{\epsilon + w} \frac{\epsilon + w_s}{w_s}
+    .. math:: \therefore w = \frac{\epsilon * w_s * RH}{\epsilon + w_s (1 - RH)}
 
     * :math:`w` is mixing ratio
-    * :math:`rh` is relative humidity as a unitless ratio
     * :math:`w_s` is the saturation mixing ratio
+    * :math:`\epsilon` is the molecular weight ratio of vapor to dry air
+    * :math:`RH` is relative humidity as a unitless ratio
+
 
     .. versionchanged:: 1.0
        Changed signature from ``(relative_humidity, temperature, pressure)``
 
     """
-    return (relative_humidity
-            * saturation_mixing_ratio(pressure, temperature)).to('dimensionless')
+    w_s = saturation_mixing_ratio(pressure, temperature)
+    return (mpconsts.nounit.epsilon * w_s * relative_humidity
+            / (mpconsts.nounit.epsilon + w_s * (1 - relative_humidity))).to('dimensionless')
 
 
 @exporter.export
@@ -2081,27 +2086,31 @@ def relative_humidity_from_mixing_ratio(pressure, temperature, mixing_ratio):
     >>> from metpy.units import units
     >>> relative_humidity_from_mixing_ratio(1013.25 * units.hPa,
     ...                                     30 * units.degC, 18/1000).to('percent')
-    <Quantity(66.1763544, 'percent')>
+    <Quantity(67.1277085, 'percent')>
 
     See Also
     --------
     mixing_ratio_from_relative_humidity, saturation_mixing_ratio
 
     Notes
     -----
-    Formula based on that from [Hobbs1977]_ pg. 74.
+    Employs [WMO8]_ eq. 4.A.16 as derived from WMO relative humidity definition based on
+    vapor partial pressures (eq. 4.A.15).
 
-    .. math:: rh = \frac{w}{w_s}
+    .. math:: RH = \frac{w}{\epsilon + w} \frac{\epsilon + w_s}{w_s}
 
-    * :math:`rh` is relative humidity as a unitless ratio
     * :math:`w` is mixing ratio
     * :math:`w_s` is the saturation mixing ratio
+    * :math:`\epsilon` is the molecular weight ratio of vapor to dry air
+    * :math:`RH` is relative humidity as a unitless ratio
 
     .. versionchanged:: 1.0
        Changed signature from ``(mixing_ratio, temperature, pressure)``
 
     """
-    return mixing_ratio / saturation_mixing_ratio(pressure, temperature)
+    w_s = saturation_mixing_ratio(pressure, temperature)
+    return (mixing_ratio / (mpconsts.nounit.epsilon + mixing_ratio)
+            * (mpconsts.nounit.epsilon + w_s) / w_s)
 
 
 @exporter.export
@@ -2217,28 +2226,33 @@ def relative_humidity_from_specific_humidity(pressure, temperature, specific_hum
     >>> from metpy.units import units
     >>> relative_humidity_from_specific_humidity(1013.25 * units.hPa,
     ...                                          30 * units.degC, 18/1000).to('percent')
-    <Quantity(67.3893629, 'percent')>
+    <Quantity(68.3229304, 'percent')>
 
     See Also
     --------
     relative_humidity_from_mixing_ratio
 
     Notes
     -----
-    Formula based on that from [Hobbs1977]_ pg. 74. and [Salby1996]_ pg. 118.
+    Employs [WMO8]_ eq. 4.A.16 as derived from WMO relative humidity definition based on
+    vapor partial pressures (eq. 4.A.15).
 
-    .. math:: RH = \frac{q}{(1-q)w_s}
+    .. math:: RH = \frac{w}{\epsilon + w} \frac{\epsilon + w_s}{w_s}
 
-    * :math:`RH` is relative humidity as a unitless ratio
-    * :math:`q` is specific humidity
+    given :math: w = \frac{q}{1-q}
+
+    * :math:`w` is mixing ratio
     * :math:`w_s` is the saturation mixing ratio
+    * :math:`q` is the specific humidity
+    * :math:`\epsilon` is the molecular weight ratio of vapor to dry air
+    * :math:`RH` is relative humidity as a unitless ratio
 
     .. versionchanged:: 1.0
        Changed signature from ``(specific_humidity, temperature, pressure)``
 
     """
-    return (mixing_ratio_from_specific_humidity(specific_humidity)
-            / saturation_mixing_ratio(pressure, temperature))
+    return relative_humidity_from_mixing_ratio(
+        pressure, temperature, mixing_ratio_from_specific_humidity(specific_humidity))
 
 
 @exporter.export