Refractivity formula

[Huw Lewis]

We need to review the use of the 'Rueger' refractivity formula in ROPP.

Since the release of ROPP-2 we have discovered that the Rueger formula is in fact not officially adopted by IAU and not in widespread use within the community. Evidence from Sean also shows detrimental impact in experiements (see email below).

For ROPP-3, we should re-adopt the 'traditional' refractivity formulation used in ROPP-1 (perhaps with compile-time option --with-rueger for validation against ROPP-2).

Some notes on the refractivity coefficients
---------------------------------------------

I'm just writing a few notes that might stimulate discussion
tomorrow. At a recent telecon, Kent asked if ECMWF planned to
use the Rueger refractivity coefficients. I have performed
some experiments with the Rueger coefficients and I am
not convinced about making the change.

To recap in operations at ECMWF (and the Met Office) we use

N = 77.6 P/T + 3.37E5/T**2

(p = dry partial pressure, e = water vapour pressure).

Rueger suggests

N = 77.689 P/T - 6.3938 e/T + 3.75463E5 e/T**2

The Rueger values are now used in ROPP. I have most
trouble with the change in the dry term (k1) from 77.6 to 77.689.
That is a 0.115% increase, which Huw has shown maps into
a 0.115% increase in the simulated bending angles.
I thought k1=77.6? value was pretty well known (Eg paper
from Bevis 1994).

Rueger makes the point that the value must be increased because
of the increasing CO2, but from his calculations
the change in CO2 only accounts for a just a 0.004 of the
0.089 increase.

I have run the experiments for 2 months (Dec 08, Jan 09) and
using Rueger cools the troposphere by ~-0.1 K.
It is slightly degrading the fit to radiosonde
height measurements. I do not feel able to make this change
for operations. We have a parallel suite coming up, but
I am not going to propose the change.


Non-ideal gas effects.
-----------------------

One reason why Rueger might be too big is because I think
his coefficients essentially contain non-ideal effects.
It appears the experimentalists measure N
for a range of P and T and then derive "k1" assuming

N = k1 P/T

k1 = N T/p

(Assuming DRY air for clarity)
However, if you take into account non-ideal effects the
equation becomes

N = (k1 P/T) Z

where Z in the inverse compressibility. At T=273.15K and
P = 1013.25 hPa, Z = 1.000588 which is small.
But if you plug it into Ruegers formula k1 is reduced from
k1 = 77.689 to k1 = 77.643, so the increase has reduced by
about a half, to 0.055%. (Z-1) falls with pressure. I think
Rueger's k1 value is really (k1*Z), with Z evaluated at the
surface, so his k1 is not appropriate higher up as (Z-1)
gets smaller.

Data selection
---------------

We also have some issues about the data he used for his
best available coefficients. EG, Mike noticed that he
used a k1 value attributed to Liebe (1977) of k1 = 77.6764,
but Liebe has used k1 = 77.64 since 1985.

I think it is worth noting that the only person who uses Rueger in 
operations is Josep (Paul Poli does not!) and he has found that he
has to include non-ideal gas effects in his hydrostatic
equation to reduce biases.

(I think we may have got lucky here in the past. We ignore
the compressibilty when calculating N, but also ignore it
when calculating geopotential height and I think these two
errors might cancel out)

Use of Rueger formula
-----------------------

I have no doubt that Rueger know's this area much better than
me, but his formula has not been published and is not the
IAG standard. In GPSRO, only Josep has adopted it and he seems to
have bias problems. At the moment, I think if someone asked why
do you use Rueger in ROPP, it would be difficult to point to
a paper that explains why. We'd also have to explain why we
don't use the Rueger values operationally at the met office
and ecmwf.

Visiting Scientist Activity?
--------------------------

Mike and I have been looking at the Rueger papers and others. Its
confusing to be honest. I am wondering whether this might be
a good area for a VS activity. I (with Mike) could write up
a short report on what we know now. A short VS activity to really
critically examine all the papers might be very useful for
the community. I do *not* have a person in mind to do the work.


Summary
-------

I've coded up Rueger and run experiments. The results are not
great so I've gone back to the papers to understand why k1
is so big. I think the non-ideal gas/compressibilty is
important.

My 2d operator code in the ROPP branch currently uses
Rueger. I am not sure if it should.


Thanks

Sean.

[Huw Lewis]

Sean has completed a report reviewing the history of refractivity coefficients. This will be attached here when finalised (and written up as a GSR document). The use of Rueger coefficients had a detrimental impact on forecast skill. It has therefore been decided to revert to using the Smith and Weintraub coefficients in all ROPP routines, until such time that we have better understanding.

The ability of using '3 coefficients' remains in the ROPP forward model code and TL, AD for future flexibility. At present, the third coefficient is set to zero.

Ticket closed as fixed.