# Accuracy of direct ECI (Earth-Centered Inertial) to LLA (Longitude, Latitude, Altitude) conversion via sub-satellite point?

The process of converting between ECI and LLA is generally known to be complex, as it involves a intermediate conversion to ECEF (Earth-Centered, Earth-Fixed). This, in turn, involves the consideration of precession, nutation and polar motion to transform the ECI coordinate frame to an ECEF one.

However, I have seen in a few commonly used software libraries the use of a CelesTrak article's algorithm for a direct ECI to LLA operation, whereby they calculate the Longitude, Latitude and Altitude of a satellite using values from an ECI frame (without going through ECEF) by instead calculating the LLA of the sub-satellite point - that being, the point on the Earth's surface directly below the satellite. It also involves the usage of the Julian Date in order to determine the rotation of the Earth beneath the satellite.

How accurate is the use of this method in calculating the LLA of a satellite's position from its ECI values? This includes both how it compares to the ECEF method and the legitimacy of this usage in general (it looks like it came from a book made in 1996, before the developments of the IAU SOFA library.) Moreover, would it be valid to use the CelesTrak method if its only use was to plot a satellite's position above the map of an Earth?

## 2 Answers

The transformation in the related question and the transformation in this question make the same low fidelity approximation, which is that precession, nutation, and polar motion can be ignored. This is probably fine for a low fidelity approximation of a satellite's orbit (e.g., two line elements). For applications where satellite position determined from two line elements is too coarse, this low fidelity ECI to latitude / longitude / altitude almost certainly is also too coarse.

The accuracy of any mathematical operation is determined by the accuracy of the data entered into the operation. Error propagation is a well-developed field in statistics. I recommend getting a copy of Bevington's "Data Reduction and Error Analysis" as that is straightforward and full of useful examples. (I"m assuming you know at least introductory Calculus)

Without knowing the error-bars for all the source data used in the various methods you reference, (as well as how the data are processed) there's no way to say in advance which is the preferred approach.