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 Subject: About satellite geometry Category: Science > Earth Sciences Asked by: timnan-ga List Price: \$50.00 Posted: 27 Sep 2002 02:14 PDT Expires: 27 Oct 2002 01:14 PST Question ID: 69640
 ```For using satellites data, when I know the location (in Lon, Lat) and the altitude of a satellite (from two-line elements), how do I calculate the lon-lat value of a viewing point (on earth) with a specified nadir angle and azimuth angle?``` Request for Question Clarification by livioflores-ga on 27 Sep 2002 09:29 PDT ```Can you clarify me about the altitude data, you know the altitude in meters (for example) of the satellite from the location point (like the altitude of a plane over the Earth surface) or you know the altitude in degrees?``` Clarification of Question by timnan-ga on 27 Sep 2002 21:55 PDT ```Dear livioflores: for example: the NOAA polor orbiting satellites is 800km above the earth. timnan``` Request for Question Clarification by livioflores-ga on 28 Sep 2002 00:27 PDT ```I can solve this using basic geometry, polar coordinates and trigonometry in order to obtain a very aproximate lat-lon value, without correction for the geodesic form of the Earth, considering the Earth surface as a plane and suposing, in the calculations, the satellite position at a given time, with no prediction to a later moment. I can give you a page with very complex calculations where you can see the position and prediction calculations did for the NOAA Polar satellite. Do you consider this a good answer?``` Request for Question Clarification by livioflores-ga on 28 Sep 2002 00:37 PDT ```Clarification of my request: When I say "considering the Earth surface as a plane", I am considering it locally, because a small amount of Earth surface is envolved. If this kind of answer donŽt satisfy you please let me know the degree complexity that you require. Thank you.``` Clarification of Question by timnan-ga on 28 Sep 2002 05:14 PDT ```Yes, your answer already meet my requirement. Answer without geodesic correction (regarding earth as a ellipsoid) is good enough for me, coz I am now dealing with the oceanic data, and trying to do some first-order correction. timnan-ga``` Clarification of Question by timnan-ga on 28 Sep 2002 06:09 PDT ```Also, if there is any reference, please list me one. Thanks timnan-ga```
 ```Well dear timnan, I will try to get an answer: First of all some definitions: -Azimut angle (Aa): The azimuth angle of a satellite is the angle on the horizon that begins to be measure from the North cardinal point towards to the east to the vertical of the object. -Nadir Point (Np): The point on the earth directly below the satellite at any given time during its orbit. -Nadir Angle (Na): The angle between the Satellite-Nadir point line (S-Np) and Satellite-Visualization point line (S-Vp). -Visualization Point (Vp): the point on the Earth where the observer is. Now I can start the calculations: Considering the local Earth surface as a plane, we have a right triangle formed by the Satellite, the Nadir point and the Visualization point. And we know the Nadir angle and the lenght of the adjacent side of this angle, such is the altitude of the satellite (H). If D is the distance between the Nadir point and the Visualization point: tan(Na) = D/H, then D = H x tan(Na) Actually we know the Azimuth angle and the distance D from the Visualization point to the Nadir point. If the Np is taken as origin of coordinates, we have the polar coordinates of the Vp. We need to convert it to a (x,y) pair coordinate system, where x express the distance from origin in the East-West direction (EW) and y the same but in the North-South direction (NS). It is not a hard task, but we must consider four situations: 0 =< Aa < 90 cos(Aa) = NS/D, then NS = D x cos(Aa) to the South. sen(Aa) = EW/D, then EW = D x sen(Aa) to the West. 90 =< Aa < 180 cos(180-Aa) = NS/D, then NS = D x cos(180-Aa) to the North. sen(180-Aa) = EW/D, then EW = D x sen(180-Aa) to the West. 180 =< Aa < 270 cos(Aa-180) = NS/D, then NS = D x cos(Aa-180) to the North. sen(Aa-180) = EW/D, then EW = D x sen(Aa-180) to the East. 270 =< Aa < 360 cos(360-Aa) = NS/D, then NS = D x cos(360-Aa) to the South. sen(360-Aa) = EW/D, then EW = D x sen(360-Aa) to the East. If your calculator not support the calculation of trigonometric functions with the angle expressed in degrees, you must divide the angles by 57.2958 in order to convert it to radians. Now we need to convert the distances in each direction to degrees. We know the lat-lon values for the Nadir point. If R is the radius of the Earth and O is its center (considering the Earth as a sphere), to specify the latitude of some point P on the surface, we must draw the radius OP to that point. Then the elevation angle of that point above or below the equator is its latitude. For each latitude angle we can draw a circle of radius r(lat) = R x cos(lat). Then the circunference of that circle is: 2 x pi x R x cos(lat); and a degree is a 360th part of this circunference (for this given latitude only, in this case the latitude of the Nadir point), then we have: 1șEW = 2 x pi x R x cos(lat)/360 = 3.1415 x 6378.137 x cos(lat)/180 [km] If we consider the semi-circle that describes any meridian, we can divide it in 180 parts that each one is equal to one degree variation in latitude. 1șNW = 3.1415 x 6378.137 /180 [km] We can convert the EW and NS distances to EW and NS degrees by a simple division, for example: EW degrees = EW/1șEW Now you only need to add or substract this calculated degrees to the lat-lon values of the Nadir point. You will found some interesting (but very complicated) calculations and related info at: "National Oceanic and Atmospheric Administration (NOAA)" website http://www2.ncdc.noaa.gov/docs/intro.htm "NOAA KLM USER'S GUIDE" page http://www2.ncdc.noaa.gov/docs/klm/ "APPENDIX I.2: Calculating the Earth's Coordinates" article http://www2.ncdc.noaa.gov/docs/klm/html/i/app-i2.htm Some ideas I got from: "Latitude and Longitude" from NASA website http://www-istp.gsfc.nasa.gov/stargaze/Slatlong.htm I hope this satisfy you as an answer, if you need some clarification and/or more info, please feel free to post a request for it. Best Regards livioflores-ga```
 timnan-ga rated this answer: `Thanks, this helps a lot.`