Ellipsoid of accuracy

Figure 6.9. Sketch of the ellipsoid of accuracy (EOA) used in ISAT to control the local interpolation error. The initial EOA is usually conservative, and thus the EOA is grown in subsequent iterations.

Now we arrive at an important result, namely, with an accuracy of the square of the flattening this equation characterizes the ellipsoid of rotation. From the last equation we have... 

This may have been the situation 20 years ago. Today it is pessimistic to a totally unwarranted degree. When based on carefully collected data and careful analysis, it frequently appears that the derived thermal parameters make good sense, in that the ellipsoids correlate well in orientation and magnitude with the chemical structure. Furthermore, there are cases in which x-ray and neutron data yield virtually identical thermal parameters. Some degree of caution in quantitative assessment of accuracy is nevertheless advisable. The reason is that the thermal parameters are sensitive to any systematic errors correlated with scattering angle, and such errors are quite difficult to assess. 

The formal uncertainties provide a statistical estimate of precision, but give no information about accuracy. Obviously, the precision of the measurement increases with the number of data, resulting in a smaller error ellipsoid. However, systematic measurement and model errors may introduce bias, which cannot be accounted for by the error ellipsoid. The accuracy of the measurement is the deviation from its true value and can only be measured when ground truth information is available. 

According to his deduction the common finding of ellipsoidal deformation of the reflections is indicative for affine deformation. Moreover, he arrives at an equation that permits to determine with high accuracy the microscopical draw ratio, Xd, of the structural entities from the ellipticity of the deformed Debye sphere. This value can be compared to the macroscopical draw ratio. Even the intensity distribution along the ellipsoidal ridge is predicted for a bcc-lattice of spheres, and deviations of experimental data are discussed. 

Integral c(x) can be taken with the adequate accuracy by saddle-point technique [1,5]. Change of (13) introduces an essential difference between w(s) and w(x) the last determines the probability w(x)dx of fact that the SAR W trajectory at given values m, N and cr will finished in the elementary volume dx = ] [ dxi lying on the surface of the ellipsoid... 

Figure 11.3 is a plot of G(p) as a fimction of p for both prolate and oblate ellipsoids. For simplicity and reasonable accuracy over all ranges of the volume fraction, the factor hi ) is given by Happel and Brenner... 

Lasentec Partec 100 was found to give reasonably accurate data for the chord length distribution of sherical particles but the accuracy deteriorated progressively as the shape became more ellipsoidal [123],... 

The vio for all particles that fall on a curve such as curve 1 in Figure 1 is the p in the first-dimension gel determined for particles that form the portion of curve 1 that is parallel to the dotted line (i.e., the smaller particles). Accuracy as high as 2% in measuring uo s can be achieved with previously-described (22) procedures of control for voltage gradient and temperature. In case R is constant and po varies, all bands are found on one size line. If discrete bands are not resolved, a single ellipsoidal band with its longer axis coincident with the size line is formed (see refs. 19, 23). 

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