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I have read the answers for both of these questions -Finding a point along a line a certain distance from another point and Find a point along line on Earth and I am trying to convert the transformations given in this question that I asked here - Trigonometric formula derivation into their spherical trigonometric versions. All longitudes and latitudes have been defined with respect to the WGS84 datum

$P(\lambda^`,\phi^`) $ lies on a geodesic curve from $O(\lambda_0,\phi_0) $ and I want the coordinates of $R(\lambda,\phi) $ which presumably lies along the geodesic distance between P and O. So O is what is called the nadir point of the satellite on the earth's surface. P is the coordinates of a ray at a slant angle $\theta$ from the location of the satellite above the earth's surface.

I have also been the height and from that I am presuming the arc length PR can be deduced by the following equation $$ h * tan(\theta) $$ and this arc lies along the geodesic OP.

I am wanting to find out the coordinates of $R(\lambda,\phi) $ where R is the coordinates of the actual cloud top position but I have been given the coordinates of P which is the apparent cloud top position. Hence a parallax correction needs to be applied.

Satellite parallax

I believe the Matlab/Octave function reckon does what I want but I want to do this in my own programming language and hence I cannot use that API call. I would like to do this from first principles.

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  • $\begingroup$ See also physics.stackexchange.com/questions/25917/… (though there is no answer there right now). $\endgroup$ – davidlowryduda♦ Apr 26 '18 at 11:24
  • $\begingroup$ @mixedmath - yes I flagged that for the moderators to look at. $\endgroup$ – gansub Apr 26 '18 at 11:44
  • $\begingroup$ Yes, that's how I became aware of it. You'd asked whether these should be merged or closed as duplicates (neither of which are possible since they're on different sites). But what my comment does is let others who visit this question know that there is another place which might have an answer, and thus reduce duplication of effort. $\endgroup$ – davidlowryduda♦ Apr 26 '18 at 13:07
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You could try the following approach:

First, calculate the geocentric xyz cartesian coordinates of the satellite and point P with the spherical coordinate system formulas. Then you can find the intersection between the satellite line of sight and a sphere with a radius = Earth radius + height of cloud top, using these formulas. Then convert the xyz coordinates of the intersection back to lat lon with the cartesian-to-spherical conversion. This will give you the lat lon of your cloud top.

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  • $\begingroup$ the spherical approximation for an inherent ellipsoid is a fair one to make ? $\endgroup$ – gansub Apr 30 '18 at 23:05
  • $\begingroup$ I think so, for the intended useage with relatively small heights. The difference is only about 0.5% at maximum. 0.5% of a few kilometers is not much. Using an ellipsoid would considerably complexify matters for very little practical gain.Unless you do need better accuracy? $\endgroup$ – FSimardGIS Apr 30 '18 at 23:36
  • $\begingroup$ It would be good to run some tests or determine how much is the error. The cloud top can reach up to 20 kms in the tropics. $\endgroup$ – gansub May 1 '18 at 0:26
  • $\begingroup$ I tested a few points with the ellipsoid vs sphere, with a satellite at 500 km and clouds at 20km, within 17° for the nadir angle, and the errors in the evaluation of the Lat,Lon of Point R are within 0.1", so +- 2-3 m. I think it is accurate enough for most purposes. If the resolution of your space radar image datasets is coarser than that, it will not cause any problems. $\endgroup$ – FSimardGIS May 1 '18 at 2:57
  • $\begingroup$ you da man ! I will try this out and let you know. $\endgroup$ – gansub May 1 '18 at 3:04

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