## Circular error probable

In contrast to the DOP metric, the CEP metric is volumetric and non-normalized. The CEP defines the radius of the smallest circle, centered at the estimate, that has a 50% probability of containing the true value. A linear approximation of the CEP can be derived from the estimate covariance.

where aL and as are the major and minor axes of the uncertainty ellipse as shown in Fig. 7. The major and minor axes are the eigenvalues of the two-dimensional covariance matrix. The difference between the true CEP and the approximation of equation (22) is less than 1.5% when the uncertainty ellipse has a low aspect ratio (0.5aL <as< 0^0.5), otherwise a quadratic approximation should be used (Nelson, 1988).

The CEP metric is volumetric because it uses the principle directions rather than the Cartesian directions, but is not normalized and therefore is a function of both the geometry and the range uncertainty.

Fig. 7. Illustration of the covariance metrics. Geometrically the 2D covariance can be represented with an ellipse. The diagonal terms of the fully populated 2x2 matrix are ffj and Oy. The square root of the two eigenvalues are one half major (aL) and minor (as) axes of the ellipse. (Figure is adapted from (Kaplan, 1996).)

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