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Figure 2. The observed values of the signal u(t) (crosses) and of its time derivative u'(t) (dots). Two solutions have then been worked out. One comes from col- location, as applied to the whole set of w’(¢;) and to the five known

Figure 2 The observed values of the signal u(t) (crosses) and of its time derivative u'(t) (dots). Two solutions have then been worked out. One comes from col- location, as applied to the whole set of w’(¢;) and to the five known

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Figure 1. Tomography of D in the straight ray approximation.
Figure 1. Tomography of D in the straight ray approximation.
Figure 3. The predicted values of the signal u(t) using collocation (black) and general kriging (grey). Crosses are the original observed values.
Figure 3. The predicted values of the signal u(t) using collocation (black) and general kriging (grey). Crosses are the original observed values.
Figure 4. The prediction error of the signal u(t) using collocation (black) and general kriging (grey). A decrease of 20 per cent in rm.s. (~30 per cent in variance) is certainly not exceptional, though significant. Naturally, this is due to the choice we made to simulate a u(t) with a not very high average; a larger average would have improved the outperformance of general kriging versus collocation.
Figure 4. The prediction error of the signal u(t) using collocation (black) and general kriging (grey). A decrease of 20 per cent in rm.s. (~30 per cent in variance) is certainly not exceptional, though significant. Naturally, this is due to the choice we made to simulate a u(t) with a not very high average; a larger average would have improved the outperformance of general kriging versus collocation.
Figure 5. The domain D, its projection Ds on the sphere S and the obser- vation points { P;}. (4) The applicability of eq. (54) relies on the possibility of esti- mating one basic covariance function. Typically this is the gravity anomaly covariance based on a model like (53) and adapted to local data. This means that we implicitly assume that u(P) should have zero average on Dg (cf. Fig. 5). Wide practical experience says that this happens, to a reasonable degree of approximation, for L(w), i.e. for the residual gravity anomalies, in particular if the degree of T is sufficiently high (e.g. 180) and the diameter of Ds is at least some degrees. The same does not happen to the average of u: even in areas of dozens of degrees, the geoid (corresponding to u/y, y ~ 10° mgal) can have an average of more than 10 cm, which is well above the threshold of the noise. Nor will a small number of observed values of u(P;) help in determining the average of u ina reliable way, as we have already seen in Example 1 and will see in the next examples. So we are in a situation where GK theory can be advantageously applied.
Figure 5. The domain D, its projection Ds on the sphere S and the obser- vation points { P;}. (4) The applicability of eq. (54) relies on the possibility of esti- mating one basic covariance function. Typically this is the gravity anomaly covariance based on a model like (53) and adapted to local data. This means that we implicitly assume that u(P) should have zero average on Dg (cf. Fig. 5). Wide practical experience says that this happens, to a reasonable degree of approximation, for L(w), i.e. for the residual gravity anomalies, in particular if the degree of T is sufficiently high (e.g. 180) and the diameter of Ds is at least some degrees. The same does not happen to the average of u: even in areas of dozens of degrees, the geoid (corresponding to u/y, y ~ 10° mgal) can have an average of more than 10 cm, which is well above the threshold of the noise. Nor will a small number of observed values of u(P;) help in determining the average of u ina reliable way, as we have already seen in Example 1 and will see in the next examples. So we are in a situation where GK theory can be advantageously applied.
Figure 7. Gridded values of geoid undulations from the GPM98A model. Crosses are the observed values. Figure 6. Gridded values of gravity anomalies from the GPM98A model.
Figure 7. Gridded values of geoid undulations from the GPM98A model. Crosses are the observed values. Figure 6. Gridded values of gravity anomalies from the GPM98A model.
In particular when both points P, Q are on S, rp = rg = R, (55) reduces basically to (49), i.e.
In particular when both points P, Q are on S, rp = rg = R, (55) reduces basically to (49), i.e.
Figure 8. Gravity anomaly covariance function. Dots represent the empir- ical values.
Figure 8. Gravity anomaly covariance function. Dots represent the empir- ical values.
Figure 9. Geoid undulation covariance function.
Figure 9. Geoid undulation covariance function.
Figure 10. Gravity anomaly variogram.
Figure 10. Gravity anomaly variogram.
Figure 11. Geoid undulation variogram.
Figure 11. Geoid undulation variogram.
Figure 12. Geoid undulation prediction error by using collocation.
Figure 12. Geoid undulation prediction error by using collocation.
Figure 13. Geoid undulation prediction error by using general kriging.
Figure 13. Geoid undulation prediction error by using general kriging.
Figure 14. Histogram of the geoid undulation prediction error by using collocation. Border effects have been removed.
Figure 14. Histogram of the geoid undulation prediction error by using collocation. Border effects have been removed.
Figure 15. Histogram of the geoid undulation prediction error by using general kriging. Border effects have been removed. Figure 16. Differences between collocation and general kriging prediction error.
Figure 15. Histogram of the geoid undulation prediction error by using general kriging. Border effects have been removed. Figure 16. Differences between collocation and general kriging prediction error.
Figure 18. Gridded values of geoid undulations on a 2° x 2° area. Crosses are the observed values.
Figure 18. Gridded values of geoid undulations on a 2° x 2° area. Crosses are the observed values.
Figure 17. Gridded values of gravity anomalies on a 2° x 2° area.
Figure 17. Gridded values of gravity anomalies on a 2° x 2° area.
Figure 20. Histogram of the general kriging prediction error in the 2° x 2° area. Border effects have been removed. Figure 19. Histogram of the collocation prediction error in the 2° x 2° area. Border effects have been removed.
Figure 20. Histogram of the general kriging prediction error in the 2° x 2° area. Border effects have been removed. Figure 19. Histogram of the collocation prediction error in the 2° x 2° area. Border effects have been removed.
Figure 21. Prediction error histogram in the 10° x 10° area by least-squares collocation with parameters. Border effects have been removed.
Figure 21. Prediction error histogram in the 10° x 10° area by least-squares collocation with parameters. Border effects have been removed.
Figure 22. Prediction error histogram in the 2° x 2° area by least-squares collocation with parameters. Border effects have been removed.
Figure 22. Prediction error histogram in the 2° x 2° area by least-squares collocation with parameters. Border effects have been removed.
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MathematicsGeodesyGeologyGeophysicsGeomatic EngineeringRandom FieldCovariance function
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