# How Variance of Latitude/Longitude Relates to Variance of Distance on Earth Surface.

#### Hunter

##### New Member
I have been looking at using the position of the sun to estimate an observer's lat/lon. This relies on only three measurements, the sun's azimuth, elevation, and the time of observation. I wanted to get an idea of how accurately these three measurements must be known in order to determine the observer's position on the earth's surface. To do this I calculated the partial derivatives of lat and lon with respect to azimuth, elevation, and time measurements. From this I think I figured out an expression for the variance of lat and longitude as a linear combination of the variance of the azimuth, elevation, and time measurements (pretty much used the expression cov(f) = J * cov(x) * Jtranspose). Now however I want to find the variance of distance error as a function of the variance of my measurements. I know that an error in the east-west direction is equal to the error in the longitude * the radius of the earth * the cosine of the latitude and that the error in the north-south direction is equal to the error in the latitude * the radius of the earth. Also, the calculated latitude is used to calculate the longitude so that errors in the latitude are correlated with errors in the longitude. Any help on how to find the variance of the distance error would be appreciated, I have kind of gotten confused about the whole thing. Thanks!

Hunter