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Satellite Laser Ranging (SLR) and Global Navigation Satellite System (GNSS) both measure the ellipsoidal heights. Which measurement of ellipsoidal height is more accurate?

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The ellipsoid is a fiction, a very useful fiction, but a fiction nonetheless.

Satellite laser ranging measures the distance to the surface with high precision. No ellipsoid is needed. Those measurements can be cast to latitude, longitude, and ellipsoid heights with knowledge of the satellite's location. This knowledge is inherently imperfect. For example, two line elements (TLEs) are not very good. No sane operator of a satellite that uses an altimeter of some sort will use TLEs. Laser ranging, along with measurements from related altimetry devices, are extremely precise, perhaps to the centimeter level.

A GPS receiver estimates its Earth-fixed Cartesian location based on signals received from four or more GPS satellites. Those measurements can be cast to latitude, longitude, and ellipsoid heights. The data received from a GPS satellite are inherently imperfect, and the greatest uncertainty is typically in the vertical direction. GPS ellipsoidal height measurements are often off by ten meters or more.

However, suppose one places a GPS receiver at a location whose ellipsoidal height is extremely well known by other techniques. Another nearby GPS receiver will see the same GPS satellites as those seen by the reference receiver. Relative GPS techniques make the vertical (and horizontal) errors cancel out, or nearly so. Relative GPS can be competitive with satellite altimetry.

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The International Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) Service uses GNSS, SLR, and high-precision Doppler measurement with earth science satellites in low earth orbits to track the 3D positions and velocities with respect to the earth's center of gravity for their 60ish ground stations around the world to a precision of a fraction of a millimeter per year.

When you measure things that precisely, you discover some of the questions we're used to asking aren't anywhere near as simple to answer as we once thought, because the more accurate you want to be, the more complicated your model of the world has to become.

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