Error ellipsoids

The network was to be homogeneous and isotropic, viz. the error ellipsoids ought, under free net adjustment with minimal trace of the covariance matrix of pylon coordinates, to be all the same size and circular in shape. 

Fig. 6.14. 3-D-error ellipsoid for visualization of the spatial localization errors. Errors Ox, and <3x can be foiuid as the projections of the ellipsoid in the direction of X, y and z, respectively.

It is an important fact that for a homogeneous material the shape of the error ellipsoid is determined by geometrical relationships between source location and sensor distribution only, as summarized in the matrix G of the partial derivatives (Eq. 6.16 and 6.17). The data uncertainty is assumed to obey Gaussian statistics and acts as a scaling factor for C. 

Thus it is possible to visualize the localization results and to interpret them according to their location imcertainties by defining a maximal spatial error as the length of the major axis of the error ellipsoid. 

Fig. 6.16. Localization errors resulting from an erroneous arrival time (same arrangement as Fig. 6.15, arrival time at sensor 1 altered by 5 rs) calculated and depicted for an array of 40 by 40 points. Top left actual localization errors as vector Top right, actual localization errors as density function (equidistance 0.01 m) Bottom left mean arrival time residuals as density function (equidistance 0.25 ps) Bottom right major axis of the error ellipsoid.

Using the same theoretical example as mentioned above. Fig. 6.16 illustrates the effect of a systematically erroneous arrival time on the source localization. For an array of 40 by 40 AE-sources the theoretical arrival times at the four sensors were calculated. To introduce an error, 5 ps were added to the arrival times of Sensor 1. The iterative localization algorithm then yields AE-source locations that minimize the travel time residuals and thus distribute the error in arrival times over all sensors. Fig. 6.16 top depicts the difference between the actual and the calculated AE-source location, on the left side as error vectors and on the right side as a density function of the error value. Fig. 6.16 bottom left shows a density function of the minimized travel time residuals (mean value over all sensors) and bottom right the major axis of the error ellipsoid. In most cases the size and orientation of calculated location uncertainties (bottom right) corresponds well to the actual error vector (top left). 

Fig. 6.23. Nonlinear localization results compared to linearized localization results. The 68%-error ellipsoid estimated by the standard linearized localization method is plotted. See also Fig. 6.20 for the experimental setup.

The results of the nonlinear localization (confidence volumes and maximum likelihood source coordinates, indicated by gray scales and white stars) are compared to the point source solution (white circles). The 68% error ellipsoid estimated by the standard linearized localization method is plotted. In both cases, a constant propagation velocity was assumed. For the well observed event El no differences between both solutions are evident. The event E2 occurred at the edge of the sensor network and therefore has a greater localization error. There are slight deviations between the error ellipsoid and the shape of Ihe PDF, because the nonlinear relationship between source coordinates and travel times is taken into account. For this reason, the PDF can have a more irregular shape in the case of a 3-D velocity model. 

Fig. 6.25. Nonlinear localization results considering the air-filled duct (confidence levels in gray scales and maximum likelihood solution as stars) compared to hnea-rized results for a plain concrete model (black circles as point sonrce with error ellipsoid).

Systematic location errors could occur due to high deformation of the rock specimen. To minimize the travel-time residuals, systematic location errors associated with picking errors and the velocity variations due to microcracking were removed by the application of the joint-hypocenter determination (JHD) method (Frohlich 1979). Using the JHD method, "station corrections" can be determined that account for consistent inaccuracies of the wave velocity along the travel path especially near sensor positions. To delineate structures inside a clouded AE event distribution the collapsing method, which was first reported by Jones and Steward [1997], can be applied. This method describes how the location of an AE event can be moved within its error ellipsoid in order that the distribution of movements for every event of a cloud approximates that of normally distributed location uncertainties. This does not make the location uncertainties in the dataset smaller but it highlights structures already inherent within the unfocussed dataset. 

In practical applications the real position of the source is not known. Nevertheless, the accuracy of localization can be estimated, when a signal is recorded by more than four sensors, by calculating the error of the source coordinates. These can be described as error ellipses in plane and as error ellipsoids in space, respectively (cf. chapter 6.3.5, Fig. 6.14). The accuracy of localization results can be used for judging the plausibility of mechanical processes (cf. chapter 6.4, Fig. 6.17). 

For this purpose, we reanalyze all the available static EOS data for Th, as shown in table 1, with a set of three different EOS forms, and compare the effect of the different EOS forms with the effects resulting from different data sets. As EOS forms we use the Birch equation (Birch 1978) in second order, BE2, and two recently proposed forms (Holzapfel 1990,1991) in second-order form, H02 and HI2, which are related to the Thomas-Fermi theory and are distinguished by the fact that H12 is bound to approach the Fermi-gas limit at infinite compression. A close inspection of table 1 shows very clearly that most of the data are fitted almost equally well by any of these forms, without any significant difference in the fitted parameters Kq and K g or in the minimized standard deviation of the pressure, Tp. In contrast to many other publications, table 1 presents the unrestricted standard deviations of Kq and K, which correspond to the extreme values of the error ellipsoids presented in fig. 11, and not just to the width of the error ellipsoids along and K at the center points, which are usually given as (restricted) statistical errors. Thus, it becomes obvious that... 

Fig. II. Illustration of the correlation in the errors oI Kq and K lor Th at ambient conditions according to the different fits presented in table I. The keys for references are given in table I. The error ellipsoids mark the level of one standard deviation in the least-squares fitting. The vertical and horizontal bars represent the commonly given restricted errors. The heavily drawn ellipsoid represents the result of the best fitting form, H12 in table 1, for all the data together, including the data of Vohra and Akella (1991) with weight ten, but with Kg free.

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. 

The model covariance matrix defines a fourdimensional error ellipsoid, whose projections provide the two-dimensional epicenter error ellipse and the one-dimensional estimates of depth and origin time uncertainties. The formal uncertainties represent the two-dimensional confidence interval for the epicenter and the one-dimensional confidence intervals for depth and origin time. These uncertainties are typically scaled to a specified percentile level (90 % or 95 %). In other words, the error ellipse scaled to the 90 % confidence level encompasses the region that contains the true location with 90 % probability. The surface of an error ellipsoid scaled to the pth percentile confidence level is defined as the hypocenters (m) satisfying... 

The interpretation of the error ellipsoid is a bit tricky. The error ellipsoid described above is called absolute, and one may expect that errors (and thus also accuracy) thus measured refer to the coordinate system in which the positions are determined. They do not They actually refer to the point (points) given to the network adjustment (cf, Section II.D) for fixing the position of the adjusted point configuration. This point (points) is sometimes called the datum for the adjustment, and we can say that the absolute confidence regions are really relative with respect to the adjustment datum. As such, they have a natural tendency to grow in size with the growing distance of the point of interest from the adjustment datum. This behavior curtails somewhat the usefulness of these measures. 

Hence, in some applications, relative error ellipsoids (confidence regions) are sought. These measure errors (accuracy) of one position. A, with respect to another posi-... 

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