In least square adjustments, what do error ellipses represent?

In least squares adjustments, error ellipses are graphical representations that provide insight into the uncertainty associated with adjusted coordinates. When dealing with survey data, it is important to account for the various sources of error that can impact the measurements taken. The least squares method helps to mathematically optimize the values of these measurements to derive the most probable coordinates.

The error ellipses visualize the precision of the adjusted coordinates by illustrating the covariance between the coordinates' estimated errors. The major and minor axes of the ellipses indicate the magnitude and direction of the uncertainties. A tighter ellipse indicates higher confidence in the adjusted coordinates, while a more elongated ellipse suggests greater uncertainty and a wider range of potential values.

In contrast, unadjusted coordinates, measurements, or adjusted angles do not encapsulate the complexities of error propagation or the relationships among various parameters. Error ellipses are specifically associated with adjusted coordinates as they reflect the result of applying statistical techniques like least squares to minimize error, hence providing a clearer understanding of the reliability of the adjusted data.