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Re: Refraction near the Horizon ? Ob servation vs. Calculation
From: Marcel Tschudin
Date: 2013 Apr 7, 01:53 +0300
From: Marcel Tschudin
Date: 2013 Apr 7, 01:53 +0300
Brad,
When collecting dip measurements the dip would not have to be adjusted to other atmospheric conditions. You just would write down the measured value and the environmental conditions during the measurements. Once a reasonably large data set exists one would then try to find out whether the environmental data would allow an improved estimation/calculation of the dip with e.g. a sophisticated calculation of k derived from some environmental data.
However, if for some reason you would have a need to convert the dip to different atmospheric conditions than those the k-value relates to , then you would have to proceed e.g. as follows:
1) You start EITHER by selecting a value for k which you found from analyzing measurements that it is the correct one to be used, OR by *assuming* it. With k=0.164 the resulting dips will agree well with those calculated with the general, simple formula as mentioned in Bowditch (and N.A.?). For this example we select k=0.164 and assume that this value relates to a dip at 10C and 1010 hPa.
2) Calculate with the selected earth radius Re=6400000 m a "refraction radius" Rr=Re/(1-k)
3) Calculate the geometrical and the refracted Rr, i.e. the geometrical Rr with k=0 and the refracted with k=0.164, thus
Rr_geom=6400000m
Rr_refr=7692308m
4) Calculate with the two Rr values the geometrical dip DIPgeom and the refracted dip DIPrefr for height of eye of e.g. HoE=4m.
You can do that EITHER with:
DIPgeom=arccos[Rr_geom/(Rr_geom+HoE)]=0.06405861 degrees
DIPrefr=arccos[Rr_refr/(Rr_refr+HoE)]=0.05843045 degrees
OR with:
DIPgeom=sqrt(2*HoE/Rr_geom)*180/pi=0.064058629 degrees
DIPrefr=sqrt(2*HoE/Rr_refr)*180/pi=0.05843046 degrees
5) Calculate the refraction: Refr = DIPgeom-DIPrefr = 0.00563 degrees
6) Scale refraction for different P and T. In your example you have P=1013.2 and T=279.7 then your Q has to be calculated (different to your version) as
Q = (P/1010)(283.15/T) = (1013.2/1010)(283.15/279.7) = 1.0155
Refr(T,P) = 0.00563 * 1.0155 = 0.00572 degrees
(Regarding Q: Note that refraction increases with higher pressures and *lower* temperatures.)
7) "Add" the scaled refraction to the geometrical dip
DIPrefr(P,T) = DIPgeom - Refr = 0.064059 - 0.00572 = 0.058339 degrees
which differs by only 0.3 seconds of arc from the dip at 10C and 1010 hPa ..
The example here assumes that the refraction part of the dip converts to different conditions (P,T) with the same relationship, Q, as "generally" used for scaling refractions to non-standard conditions. This appears actually to be not quite true, but that is an other story ...
Marcel
When collecting dip measurements the dip would not have to be adjusted to other atmospheric conditions. You just would write down the measured value and the environmental conditions during the measurements. Once a reasonably large data set exists one would then try to find out whether the environmental data would allow an improved estimation/calculation of the dip with e.g. a sophisticated calculation of k derived from some environmental data.
However, if for some reason you would have a need to convert the dip to different atmospheric conditions than those the k-value relates to , then you would have to proceed e.g. as follows:
1) You start EITHER by selecting a value for k which you found from analyzing measurements that it is the correct one to be used, OR by *assuming* it. With k=0.164 the resulting dips will agree well with those calculated with the general, simple formula as mentioned in Bowditch (and N.A.?). For this example we select k=0.164 and assume that this value relates to a dip at 10C and 1010 hPa.
2) Calculate with the selected earth radius Re=6400000 m a "refraction radius" Rr=Re/(1-k)
3) Calculate the geometrical and the refracted Rr, i.e. the geometrical Rr with k=0 and the refracted with k=0.164, thus
Rr_geom=6400000m
Rr_refr=7692308m
4) Calculate with the two Rr values the geometrical dip DIPgeom and the refracted dip DIPrefr for height of eye of e.g. HoE=4m.
You can do that EITHER with:
DIPgeom=arccos[Rr_geom/(Rr_geom+HoE)]=0.06405861 degrees
DIPrefr=arccos[Rr_refr/(Rr_refr+HoE)]=0.05843045 degrees
OR with:
DIPgeom=sqrt(2*HoE/Rr_geom)*180/pi=0.064058629 degrees
DIPrefr=sqrt(2*HoE/Rr_refr)*180/pi=0.05843046 degrees
5) Calculate the refraction: Refr = DIPgeom-DIPrefr = 0.00563 degrees
6) Scale refraction for different P and T. In your example you have P=1013.2 and T=279.7 then your Q has to be calculated (different to your version) as
Q = (P/1010)(283.15/T) = (1013.2/1010)(283.15/279.7) = 1.0155
Refr(T,P) = 0.00563 * 1.0155 = 0.00572 degrees
(Regarding Q: Note that refraction increases with higher pressures and *lower* temperatures.)
7) "Add" the scaled refraction to the geometrical dip
DIPrefr(P,T) = DIPgeom - Refr = 0.064059 - 0.00572 = 0.058339 degrees
which differs by only 0.3 seconds of arc from the dip at 10C and 1010 hPa ..
The example here assumes that the refraction part of the dip converts to different conditions (P,T) with the same relationship, Q, as "generally" used for scaling refractions to non-standard conditions. This appears actually to be not quite true, but that is an other story ...
Marcel
On Sat, Apr 6, 2013 at 10:20 PM, Brad Morris <bradley.r.morris@gmail.com> wrote:
Marcel
With all due respect ....
You indicated that my pressure and temperature procedure is wrong.
I asked for you to show the procedure you use, with a worked example.
You responded without a procedure. The procedure is critical for understanding.
Put your procedure & numbers on a piece of paper. Submit it to the group.
Brad
On Apr 6, 2013 2:50 PM, "Marcel Tschudin" <marcel.e.tschudin---com> wrote:Hi Brad, All your mentioned equations are approximations. It really depends on the accuracy one wants to achieve and whether this accuracy is really necessary or not. If one wants really to be exact, it is in this context actually not the radius of the earth which is of interest, it's rather the earth's radius of curvature at the location of the observer. Because the earth's shape is approximately an ellipsoid this radius of curvature depends on latitude and also on the azimuth, except at the poles. If you do not require this accuracy you can use any other less accurate mean earth radius which in the context of calculating dip are likely to be completely sufficient. Regarding the pressure and temperature dependency of refraction here some further explanations: The scaling factor (P/1010)/(T/283) adjusts the refactivity property of air, n, and as a consequence of it also the refraction resulting from it. However, the cause for light rays being bend in the atmosphere are temperature-GRADIENTS. The parameter k is introduced to describe in a simplified model the bended ray for terrestrial refraction and dip. From observations one obtained then typical values for it. Marcel
