Reduced point mass or multipole base functions. - Niels Bohr Institutet

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$Turbulence course, 6. week problem 1 . The fractal dimension and ...$

As pointed out by Arabelos (1980) we can not simply put to zero the first N terms.Here a solution was found, i.e. that the first terms were not put to zero, but putequal to the so-called error-degree variances, σ 2 e,i contingently scaled by a factor αso as to reflect if the model was better (α≤1) or worse (α≥1) in an area.To get a little more insight into this, let us interpret the point mass potential as areproducing kernel in a Hilbert space, where the functions are harmonic down to aBjerhammar-sphere with radius R B inside the Earth and r Q < R B . Using a Kelvintransformation we obtain a point D outside the sphereR B2r P r D= r Qr P(7)So that L=1 − 2 R B 2cos(ψ) + R B 2 2 2Rr P r D r P r . Then K(P,D)= B/L is the so-calledD r P r DKrarup kernel, Krarup (1969). When interpreted as a covariance function, it hasunitless degree-variances equal to 1, i.e. it is not well suited to represent theanomalous potential, since the degree-variances of T tends to zero like n -3 * q, q
∞i=2A kF k r P , r Q, ψ = r i+1Q P(i − k) r i (cosψ)P(9)Each of these may be represented by a closed expression. The first termscorresponding to the maximal degree of the global model used, should besubstituted by the (scaled potential) error-degree-variances.Ncov N r P , r D, ψ = aσ iei=2 R B 2 i+1 Pr D r i (cosψ)P(10)5. Reduced point mass and excentric multipole base functions - examples.The interesting functions are those which represent the geoid (height anomaly) andthe first and second derivative with respect to r P . All quantities are anomalousquantities with respect to EGM96 complete to degree N=24. We consider geoidand gravity disturbance at height zero and the second order radial derivative at 250km altitude. For the latter we will put the mass-point at depth 242.5 km,corresponding to data (r P ) at 250 km. The Bjerhammar sphere will be put at 1.46km depth.In the first example (Table 1) we use as observation one (anomalous) radial gravitygradient observation equal to 0.15 Eötvöes, at height 250 km (M=1 in eq. (3)). Thelatitude and the longitude are set to zero for the mass-point, but the latitude for thepoint P increases in steps from zero. Table 2 shows the same function, now withthe quantities evaluated at zero altitude.In Table 3 we have corresponding values obtained using the covariance functioneq. (8), (10) and again with the same gravity gradient observation as used in Table1 and 2.We have here calculated values with spherical distance 1.0 degree and so that thevalue for the second order radial derivative is 0.15 Eötvös.
Page 1 and 2: Reduced point mass or multipole bas
Page 3: 2. Higher order derivatives,For der
Page 7 and 8: Table 2. Closed and Reduced Point m
Page 9: frequencies which the global model

Reduced point mass or multipole base functions. - Niels Bohr Institutet

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