What I actually would need is an estimation of all the water content
in the atmosphere from surface data. This parameter is of importance
for the diurnal temperature range (DTR) at a location. It's actually
for this parameter for which I try to find an estimation with a fitted
function using only surface data.

Finally I found also some time for checking the suggested estimation
function. Here now the estimated precipitable water vapour values
which were compared to the values reported from weather balloon
soundings of the same day. The selected locations provide some extreme
values. The daily mean value was used where more than one value was
measured at the same day. Note: I'm aware that the suggested (fitted)
function was derived of monthly mean values and that it is therefore
not really correct to use it on daily base. However, as the results
show, the function delivers even half way reasonable estimates for
daily values. The values shown for W correspond to mm/m^2

Location Day W(sounding) W(estimation)
Manaus 01.05.2006 61.01 45.19
Manaus 02.05.2006 62.42 46.76
Manaus 03.05.2006 58.09 45.19
Manaus 04.05.2006 63.19 45.19
Manaus 05.05.2006 60.04 42.21
Kuching 01.12.2006 63.16 48.38
Kuching 02.12.2006 64.13 48.38
Kuching 03.12.2006 61.64 48.38
Kuching 04.12.2006 62.19 48.38
Kuching 05.12.2006 59.61 48.38
San Diego 21.10.2007 13.71 18.00
San Diego 22.10.2007 1.01 7.67
San Diego 23.10.2007 0.00 10.79
San Diego 24.10.2007 3.90 13.24
San Diego 25.10.2007 4.36 18.00
Halley 11.07.2007 1.93 2.03
Halley 12.07.2007 2.37 3.06
Halley 13.07.2007 1.98 2.03
Halley 14.07.2007 0.86 1.03
Halley 17.07.2007 1.32 0.68

These results suggest that the estimation generally underestimates
high W values. Low W values seem to be well represented in case of low
temperature (Antarctica). However, low W values in the context of high
temperatures and very dry air are overestimated.