[GIS] Precision of conversion from WSG84 to ECEF and ECEF to ENU

coordinate systemprecision

To convert from a global to a local coordinate system, I apply the following transformations.

WSG84 to ECEG:

$WSG to ECEF$

with

$ECEF to ENU$

and ECEF to ENU:

$ECEF to ENU$

Let's assume that the semi-major axes a and e are known with high precision (e.g. a = 6378137.0 and e²=6.69437999014e-3), and the reference position in the second transformation is correct.

What is the precision I can expect from this transformation within a short distance from the reference position (let's say less than a mile)?

Best Answer

Both of these equations ("Geodetic/Wgs84 to ECEF" and "ECEF to ENU") are exact equations for the transformations and they do not as such introduce any error to the position.

The limiting factors for the accuracy are basically the accuracy of the ellipsoid parameters and the numerical accuracy of the actual implementation (float point numbers have limited accuracy).

What I would personally do to check out the accuracy is to try out some conversions (e.g. with Wikipedia Coordinate Conversion equations)

Geodetic -> ENU-> Geodetic and
ENU -> Geodetic -> ENU

and see how well the numbers are the same after one "full loop". My guess is that with almost any impementation the accuracy is highly limited by the actual measurement of the location (be it, in Geodetic coordinates, for example).

Related Solutions

[GIS] Trasformation from ECEF to ENU

The easiest way to check that this is right is to multiply out the matrices to get

     [     -sin(lambda)            cos(lambda)          0     ]
R =  [ -sin(phi)*cos(lambda)  -sin(phi)*sin(lambda)  cos(phi) ]
     [  cos(phi)*cos(lambda)   cos(phi)*sin(lambda)  sin(phi) ]

The three rows are the unit vectors E, N, and U in ECEF coordinates; so when you form R.r_ecef you are computing the components of r_enu by resolving r_ecef into its ENU coordinates.

Incidently, you are missing the translation component in this conversion. The origin of the ENU system is normally on the surface of the ellipsoid while the origin of ECEF is the center of the earth.

See, for example, Other Earth-based coordinate systems and Conversion calculations at Geodetic datum.

[GIS] How much error should occur when converting WGS84 to ECEF coordinates

The implementation here actually appears correct. The problem exists in what is being checked, and what is expected. The distance checked is the longitude while the expected distance is the latitude.

We can perform two quick checks to satisfy that the our conversions are correct:

The magnitude of the ECR vector should be ~637.814 km at the equator.

For both of the given vectors I found their magnitude to be 637.814 km.

The longitudinal change at the equator should be about 111.321 km.

The straight line distance between the values given is approximately 111.318 km. This is actually only ~3m off from what should be expected. The difference is because I did not take into account the curvature of the earth.

Now as to why is the difference (-0.971 km, 111.313 km)? We must remember that ENU is a tangent plane and X is merely a distance to the plane. Also since 1 km along the plane is not 1 km along the surface of the earth, we cannot expect to see X slope away at the given rate.

If none of that helps, then this answer may help Longitude Length ...in miles

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[GIS] Trasformation from ECEF to ENU

[GIS] How much error should occur when converting WGS84 to ECEF coordinates

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