Water Vapor - References

I used a number of references to research this area and create the programs. This page contains additional notes and comments about those.

When dealing with equations and data, typos are a common problem. In the following, I have pointed out several.

Holger Vömel

Saturation vapor pressure formulations, Holger Vömel (CIRES, University of Colorado, Boulder).

This is my primary reference for 25 (of 30+) equations. In order to reduce the number of typos, I used the provided Fortran source code and made only simple modifications to convert it to javascript.

I originally used a 2011 version of this which had 25 formulations (19 on the html page).
The current 2016 version
- Fixes Wexler 1996 over water sign error in 2011 version
- Adds Wexler 1997 over ice
- Adds Hardy (1998) over water and over ice - this updates Wexler from ITS68 to ITS90
- Adds McIDAS over water and over ice
The earliest version in the wayback machine is dated 2003 and lists only 14 formulations on the html page.

The associated code contains several formulations not included in the html page.

When a value over ice is requested, the associated code replaces the result with HylandWexler (water) for any temperature above 0°C. I have elected not to do that in my code.

Wexler

In the 2011 version, the Wexler over water formulation contains 2 errors, both fixed in a 2014 version.

The reference is for Wexler's over ice formulation

The sign of one of the coefficients is wrong

- 4.4412543E-13 * T⁴ // this should be plus, not minus

The 2011 code does not actually contain Wexler's over ice formulation, just a reference to it and an incorrect over water formulation.

Sonntag Typo

This was fairly difficult to find - basically, one of the Sonntag over ice constants is different in 4 separate papers and the primary Sonntag article is behind a paywall.

  Nielsen      Murphy      Vömel       Alduchov 96
 29.32707     24.7219    24.721994      24.72219     over ice coefficient

Nielsen is the odd one because he computes the saturation pressure in Pa and the others use hPa - a scale factor difference of 100. Therefore, the first thing to do was to convert the Nielsen value by subtracting ln(100).

  Nielsen
 29.32707 - ln(100) = 24.7218998 => 24.7219

Since this is basically the same as MurphyKoop, I assumed that this is correct.

But where did the extra Vömel digits come form?

Assuming that he copied the over water function and just typed new coefficients (something many of us do), he might not have noticed that the over ice value has only 4 digits after the decimal and that the over water value has more.

     Vömel
   16.635794  over water
   24.721994  over ice

Since Nielsen and MurphyKoop have basically the same coefficient, and since I can explain the Vömel value as a typo, I think that it is very likely that Alduchov also contains a typo.

So, How big an error is this?

Sonntag fix - change is not significant
 °C     old      new
-30   0.38003  0.37999
-10   2.59917  2.59893

Murphy and Koop

Review of the vapour pressures of ice and supercooled water for atmospheric applications by D. M. Murphy and T. Koop, 29 Dec 2006

This is my primary source for the following

The vapour pressure of both ice and liquid water at the triple point is
P_t = 611.657 ± 0.01 Pa at temperature T_t = 273.16 K
(Guildner et al. 1976, Vapor pressure of water at its triple point. J. Res. Natl. Bur. Stand., 80A, 505–521)

Appendix A contains a number of equations that provide pressure in Pa - since the Vömel equations provide pressure in hPa, there is a difference of 100 (2 in the log's below).

There are 3 Goff and Gratch equations - 1946, 1957, 1965.

The 1946 equation differs in the final term from Vömel - the decimal point is moved 2 places (as explained above) and the number of significant digits is different.
The 1957 equation over ice has several differences
The 1965 equation is not in Vömel or my program

Algorithm	Murphy and Koop	Vömel
Goff and Gratch 1946 over water	log (101325)	log (1013.246)	5.005716612414 3.005714897949
Goff 1957 over water WMO_Goff	log(611.14) 2.78614071	0.78614	Close enough
Goff 1957 over ice WMO_Goff	log(611.14) 3.566506	0.78614 3.56654	2 constants are different

Appendix C contains a table with the expected saturation vapor pressure values both over ice and over water - the intended purpose is to check computer code. Those values (specified with 4 to 6 significant digits) agree with the values computed by my software when rounded to the same precision. Note that the values in the table are specified in Pa and my software produces results in hPa (mbar) - the difference being 100 (10^2).

The following formulations are in Appendix A, including a few not found in other sources.

Goff and Gratch (1946)
Goff (1957)
Goff (1965) also referenced as Goff (1963)
Hyland and Wexler (1983)
Jancso et al. (1970) data fit
Jancso et al. (1970) thermodynamic derivation
Koop et al. (2000)
Marti and Mauersberger (1993)
Mauersberger and Krankowsky (2003)
Sonntag (1990)
Wright (1997), US Meteorological Handbook
Wagner et al. (1994)
Wexler (1976) - IPTS-68

The saturation vapor pressure over ice depends on the crystallization - hexagonal, cubic (below 200K), amorphous (below 160K). (Once, when cleaning a home freezer, I saw what appeared to be cubic ice on some old ice cream - about -20°F (244K). Before that, I had no idea such a thing was possible.)

Nielsen, et al. 2004

Uncertainty in the generation of humidity Jan Nielsen, Jeremy Lovell-Smith, Martin de Groot, Stephanie Bell

A paper presented at the EUROMET 2004 Workshop on Uncertainty in Humidity Measurements. 22th February 2004, EUROMET THERM Technical Committee Meeting.

As the name suggests - this paper is primarily focused on uncertainty in the calibration of humidity sensors.

This discusses the Wexler formulation and provides a table of coefficients. In particular, the Sonntag (1990) coefficients fit the Wexler formulation to ITS-90.

The following table compares the coefficients in this paper to those in the Vömel paper. The first conversion is easy - Nielsen provides results in Pa, Vömel uses hPa - a difference of 100. However, the comparison is difficult because they don't appear to use the same versions of each algorithm. For instance, Nielsen provides coefficients for Sonntag (1990) and Vömel uses Sonntag (1994). Over both water and ice, these two have identical coefficients except for the scale factor. As shown in the table below, the coefficients are (nearly) equivalent to the scale factor of 100 .. except that one of them does not have enough significant figures. Unfortunately, since I do not have access to the original Sonntag papers - they are about $33 each to read! - I don't know which source is wrong.

The 4 Wexler papers are a bigger problem.

The reference dates don't match
Over water, the sign of one of the minor coefficients is different

I only compared the following algorithms (coefficients in Table 12, p 24)

Algorithm	Nielsen	Vömel 2011
Over water	Wexler (1976)	Wexler (1977)	The coefficients are identical except for the sign of the T^4 term +/- 4.4412543E-13 * T_K^4 =0.003597416 when T_K = 300 Vömel fixed this in 2014 and now they are identical
Over ice	Wexler (1977)	HylandWexler (1983)	These have identical coefficients Note: these are not the same as the Wexler 77 coefficients over ice found in all the actual Wexler reference
Over water	Sonntag (1990)	Sonntag (1994)	scale factor - 21.2409642 vs 16.635794 exp(21.2409642 - 16.635794) = 100 The other 4 coefficients are identical
Over ice	Sonntag (1990)	Sonntag (1994)	scale factor - 29.32707 vs 24.721994 exp(29.32707 - 24.721994) = 99.99058184 The other 4 coefficients are identical

The error in Sonntag over ice is actually pretty big when you consider that someone manually made the conversion between Pa and hPa - or, perhaps, the difference is in the original source. Who, knows?

ln(100)	4.605170186	Computed via Excel
21.2409642 - 16.635794	4.605170200	Very close
29.32707 - 24.721994	4.605076000	This is the problem

Table 12 also has ITS-90 coefficients for Hardy 1989, Wagner and Pruss 1993, Wagner, et al. 1994.

GISS Model-E

NASA provides source code for their climate simulation model - GISS Model-E - used for IPCC AR4 and AR5. I spent several months trying to get it to work, but it contained too many problems - mostly divide by zero errors.

At any rate, I used the opportunity to look at the science behind their model. The two saturation vapor pressure formulations (one equation, but with different parameters over water and over ice) are implemented in the QSAT function in UTILDBL.f. I was surprised to see that it had a different form than any of the Vömel equations. I eventually learned that they are using the highly inaccurate August equation, but with different parameters. It turns out that over the range of atmospheric temperatures, their formulation is probably accurate enough.

About 6% error at -40°C (-40°F)
About 4% error at  40°C (104°F)
 < 0.06% error at   6°C ( 44°F)

6.108 mBar @ 0.01°C

(Apparently, accuracy does not matter for climate models.)

JPL Paper

The two saturation vapor pressure formulations used in NASA's climate simulation model - GISS Model-E - produce 6.108 mbar at 0.01°C (the triple point). Since that differs from the standard given by Murphy and Koop, I tried to determine where it came from. As a result, I came across a paper from JPL prepared for the Air Force with funding from NASA.

Atmospheric Noise Temperature Induced by Clouds and Other Weather Phenomena at SHF Band (1-45 GHz)
Christian Ho, Stephen Slobin, and Kelly Gritton, August 11, 2005

On page 18, it says

6.108 mbar at 0°C

Since there is no decimal point, that could be at the triple point - or not.

I discuss this reference in detail. On page 18, they provide a number of saturation vapor pressures, but provide no reference on where they come from (or how they were computed). Since the pressures are provided with 2 decimal places, I assume that the temperatures given are exact and not rounded from something close. While their values don't exactly match any of the formulas in my software, they almost match WMO_Goff.

Table 5 provides

Water Vapor	Attenuation scale height = 2.0 km
Oxygen	Attenuation scale height = 5.4 km
Cloud	base = 1 km top = 3 km 2 km thick

Pressure scale heights are typically computed at a specified temperature. From the text, it appears that the attenuation scale heights are measured (computed from data) for a real atmosphere.

A surface water vapor density (absolute humidity - AH) of about 7.7 g/m3 "results from a surface temperature of 15 C and a relative humidity of about 58%".

Alduchov and Eskridge

There are 2, almost but not quite, identical papers

Improved Magnus Form Approximation of Saturation Vapor Pressure

The abstract and some of the text is different, but they are basically the same paper .. with the same name. I have no idea why the typed version has a later date than the journal article.

Table I in the typed (1997) version (not in the 1996 paper) appears to be wrong. I created a calculator to perform the computations and got significantly different answers.

1. Introduction (paragraph 3)

The most precise formulation of vapor pressure over a plane surface of water was given by Wexler (1976).
The relative errors of Tetens' (1930) formula and one due to Buck (1981) (...)
are shown in Table I.


Table I. Relative errors(%) of Buck's (BU81) and Tetens' (TE30) approximations.

T °C	Table I		My Values
T °C	BU81	TE30	Buck 81	Tetens 67
14.3	0.456	0.394	0.0418	0.0689
8.6	0.380	0.328	0.0229	0.0745
5.5	0.291	0.262	0.0126	0.0891
-4.5	0.208	0.318	0.0025	0.2237
-21.2	0.373	0.988	0.2367	0.9758
-31.7	0.847	2.007	0.7410	2.0366
-43.9	2.044	4.130	1.9783	4.2099

Name in Table I	My Values	Name in Calculator
All values are compared to Wexler 1976 over water
BU81	Buck 81	Buck_original (1981)
TE30	Tetens 67	MagnusTetens (1967)
TE30 is Tetens 1930 which is identical to MagnusTetens (1967)

Given the fact that the equations are apparently identical (yes, I checked that), my computations and what they show in the table should be identical - not even close.

To use the calculator,

Click on the Wexler (1976 Over Water) field - the background will turn yellow. All other values will be compared to this.
Set the temperature
Read the values from the Buck_original (1981) and MagnusTetens (1967) "Percent %" fields

(The dates are shown in the status bar when the mouse is placed over the formulation name.)

Typo

There is a probable typo in one of the Sonntag 1990 over ice coefficients. To verify that, I had to subtract ln(100) from the Nielsen value (which appears to have its own problem).

  Nielsen
 29.32707 - ln(100) = 24.7218998 => 24.7219

 Murphy      Vömel       Alduchov 96
24.7219    24.721994      24.72219     over ice coefficient

As already pointed out, the Nielsen and Vömel values don't match .. but they are close.

By inspection, it appears to me that both Vömel and Alduchov contain a typo.

Wexler

There are many sources for the Wexler formulations - the following all agree.

Vapor Pressure Formulation for Water in Range 0 to 100 °C Arnold Wexler, 1976
Vapor Pressure Formulation for Ice Arnold Wexler, 1977
Uncertainty in the generation of humidity Jan Nielsen, Jeremy Lovell-Smith, Martin de Groot, Stephanie Bell
Improved Magnus Form Approximation of Saturation Vapor Pressure Alduchov (Ru) and Eskridge (US), 1997
Review of the vapour pressures of ice and supercooled water for atmospheric applications by D. M. Murphy and T. Koop, 29 Dec 2006, Appendix A

Table error - In Wexler, 1976, I think there is a typo in Table 2 - all the values in the first column agree with my calculator except

 °C      Table 2     Calculator
 80     47375.85     47374.847

Seinfeld and Pandis

One of the first saturation vapor pressure equations I found was in

Atmospheric Chemistry and Physics by Seinfeld and Pandis (2006)

My programs were originally written to see how accurate (or consistent) their algorithm was. I kept it in the program because it has a different form from those in Vömel. It is obviously calibrated for a boiling point of 100°C (373.15 K) at 1 atm (1013.25 mbar).

This formulation is identical to the 1971 Richards formulation with the exception that Richards used 373.16 K (vs 373.15 K) as the calibration point. Lowe77 eq7

Temperature

The problem of temperature is quite involved - I have a separate page discussing it.

Basically, there are several definitions of the metric temperature scale and various algorithms (formulations) use one standard or another.

Of course, this makes it very difficult to analyze atmospheric data over a large interval - the standards changed, so the measured values changed. However, it is hard to find the correct metadata to make the necessary adjustments.

The International Temperature Scales - 1878 to 2000

The International Temperature Scale of 1990 (ITS-90)

International Practical Temperature Scale of 1968 (IPTS-68)

Guide to the Realization of the ITS-90

Wikipedia

I really hate using Wikipedia as a reference. However, it would not be fair to refuse to credit a source I use so much, and it provides good definitions of many concepts. Occasionally, it is the only reference for some item - I assume that someone copied something from another source but forgot to provide a reference.

Most of the information on the Antoine and August equations comes from here. Not because there aren't other sources, but just because those sources are not available online.

Author: Robert Clemenzi