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Hello toves,
quote from text below..
"At this level of accuracy, what is meant by the "position" of
a station starts to get lost in the noise of the planet's shape."
There are (of course) programs that will try to tell you to the nearest
inch what the distance is between any two points on the globe, and throw in the
azimuth for good measure.
The following text, admittedly a bit longish, I copied from an
amateur radio book. It is about programs giving "distance and bearing" between
two points. Just replace "station" by "location" and you're allright.
I did not include the actual (Basic) programs. If anyone wants them, let me
know. They are from 1985. But it is really the reasoning I thought was
interesting.The text is by John Morris, GM4ANB for illuminati. We meet him just
after he has explained a basic, sphere-based program.
-- snip --
Latitude correction So far it has been assumed that the Earth is a perfect sphere. Of course it is not, being somewhat flattened at the poles and bulging at the equator. It is convenient to assume that the Earth is ellipsoidal; that is a cross section through the equator is circular, but a cross section through a line of longitude is elliptical. This assumption introduces two corrections. The most obvious one is that the distances must now be measured along the boundary of a more or less eccentric ellipse. Surprisingly, this is a relatively small correction. A rather larger error arises because latitude is not quite what most people think it is. Fig 5.2 shows a cross section of the Earth through a line of longitude. The eccentricity is much exaggerated. For a given point P, on the surface of the Earth, the angle ECP is called the central angle. It is this angle which all of the preceding distance and bearing formulae assume is the latitude. In fact the latitude is calculated from the normal to the surface of the Earth at P, giving angle ETP. This arises from the way latitude was historically measured; by the positions of the stars relative to the horizon. The distance a is called the semi-major axis of the ellipse, and b is the semi-minor axis. For a given latitude the central angle can be calculated as follows: Tan(central angle) = (b/a).(b/a).tan(latitude) For the Earth the value of a is about 6378km, and b is about 6356km. The average of these, 6367km, can be used as an average Earth radius, as was done in Programs 5.1 and 5.2. Program 5.3 shows the lines which should be changed or added in Program 5.2 so that central angles are used instead of latitudes. Line 30 sets up a few constants giving the size and shape of the Earth. The new subroutine, in lines 2000-2100, is very similar to the original in Program 5.2, but instead of the latitudes uses the central angles, Nl and N2, in the formulae. The latter are calculated in line 2000 using the above formula and the assumed shape of the Earth. A typical run after the lines of Program 5.3 has been included is shown in Fig 5.3. The results are similar to those from Program 5.2, but not identical. In this version of the subroutine the maximum distance error is reduced from 0.5% to about 0.2%. The residual error arises because the distance calculation itself still assumes a spherical Earth, even though the correct central angles have been used. The azimuth error is extremely small. The bearing from one point to another does not depend on the shape of the Earth, only on the central angles. Once these are known no further correction for the shape of the Earth is needed. Ellipticity correction Program 5.4 shows a replacement subroutine for those in Programs 5.2 and 5.3 which takes account of the ellipticity of the Earth when calculating distance. As may be seen, getting the last few tenths of a percent of accuracy is a messy business. The mathematics behind this routine are rather long, but the steps involved can be summarised as follows: Lines 2000 to 2070 find the central angle and azimuth, as previously. The rest of the routine finds the distance. Lines 2080 to 2170 find the great circle on which the two stations lie. This great circle is in fact an ellipse, its ellipticity depending on its inclination to the equator. If the great circle intersects the equator at a large angle (such as a line of longitude) the ellipticity will be the same as that of the Earth. If the angle is small the ellipticity will be small; in the case of the equator itself it is zero. AL is this angle of intersection. XI and (XI+CA) are the angular distances round the great circle from the equator of the two stations. EP is the square of the eccentricity of the ellipse on which both stations lie. Once the ellipse of interest and the positions of the two stations on it have been identified lines 2180 to 2200 perform the actual distance calculation. The formula in line 2200 is an approximation to an infinite series in EP. As EP will, for the Earth, always be small (its maximum is about 0.007) terms containing second or higher order powers can safely be ignored. The subroutine is messy enough already, without making it worse by adding spurious accuracy! A typical run using this subroutine substituted for that in Program 5.2 (note that line 30 from Program 5.3 is also needed) is shown in Fig 5.4 If the Earth were perfectly ellipsoidal and the major and minor axis measured exactly right then the maximum error in the distance given by Program 5.4 would be 0.000,08% - or about 150m for an antipodeal contact. In fact neither of these conditions is satisfied, as annoying imperfections in the shape of the Earth (continents and such like) and uncertainty of the the Earth's dimensions limit the accuracy to about 0.1%. There is no way to greatly improve this accuracy, even if the shape of the Earth is known in detail, because the whole meaning of "great circle distance" becomes a little bit fuzzy when you look this closely. For example, if both stations are on top of 2,000m high mountains should the effective earth radius be increased by 2km (0.03%) or should the local sea level value be used? And then is it mean sea level, or that at the time of the contact? At this level of accuracy, what is meant by the "position" of a station starts to get lost in the noise of the planet's shape. The moral is that, at least in amateur radio, there is no agreed definition of exactly how the distance between two stations should be measured. Everyone blandly says "along the great circle", but as we have seen this is inadequate for great accuracy. It is not really a problem. Let's face it, even the 0.5 or 0.2% accuracies given by earlier programs will usually be sufficient, and the programs are much, much simpler. The 0.1% of Program 5.4 should be more than adequate for just about all amateur radio purposes, especially as the wave that propagates precisely along a great circle has yet to be radiated. -- snip -- Enjoy your ponderings! Rudolf Attachment converted: Macintosh HD:lat.tif (TIFF/JVWR) (000B6387) |
