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  • 1
    Monograph available for loan
    Monograph available for loan
    Cambridge : Cambridge Univ. Press
    Call number: M 97.0112 ; M 97.0044
    Type of Medium: Monograph available for loan
    Pages: xiv, 369 S.
    ISBN: 0521410061
    Classification: A.2.3.
    Language: English
    Location: Upper compact magazine
    Location: Upper compact magazine
    Branch Library: GFZ Library
    Branch Library: GFZ Library
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let $$\hat n$$ be the outward unit normal to S, ▽ S the surface gradient on S, I S the metric tensor on S, gij the four covariant components of I S (i,j = 1, 2), h ij the four covariant components of - $$\hat n$$ xI S , and D i covariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ϕ and ψ on S, unique to within additive constants, such that $$u = \nabla _s \varphi - \hat n \times \nabla _s \psi $$ ; the covariant components of u are $$u_i = D_i \varphi + h_i^j D_j \psi $$ . This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are $$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$ where $$E_{ij} (M,N) = ( - \nabla _s ^2 M)g_{ij} + 2D_i D_j M + (h_i ^k D_j + h_j ^k D_i )D_k N$$ It is shown that H and L are uniquely determined by F but that M and N are not. The set of complex scalar fields ℳ′ = M′+iN′ such that E ij (M′, N′)=0 is shown to constitute a four-dimensional complex linear space $$\mathfrak{W}$$ . The scalars M and N which help to generate a given F are uniquely determined by F and the condition that, for every ℳ′ in $$\mathfrak{W}$$ , $$\mathop \smallint \limits_s (M - iN)\mathcal{M}\prime dS = 0$$ The real linear space of second-order tangent tensor fields on S which have simultaneously the form E(M, 0) and the form E(0, N) is shown to have dimension zero on a sphere, dimension four on a non-spherical, intrinsically axisymmetric spheroid (a spheroid whose isometries form a compact, one parameter group), and dimension six on a spheroid which is not intrinsically axisymmetric. Applications of the representation theorem to tensor problems in spherical coordinates are briefly discussed.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    [s.l.] : Nature Publishing Group
    Nature 201 (1964), S. 591-592 
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] The purpose of this communication is to suggest some measurements in the South Atlantic which would test simultaneously Vine and Matthews's hypothesis and the hypothesis that South America and Africa have drifted apart as flotsam on a convection current in the mantle which rises under the ...
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    [s.l.] : Nature Publishing Group
    Nature 377 (1995), S. 198-199 
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] IN 1600, W. Gilbert1 absolved the pole star Polaris of responsibility for the Earth's magnetic field, B. He found that a uni-formly magnetized sphere produced the same latitude dependence of the angle between the field and the local vertical as was being reported by the navigators of ...
    Type of Medium: Electronic Resource
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  • 5
    Publication Date: 2013-08-31
    Description: In the uniqueness part of a geophysical inverse problem, the observer wants to predict all likely values of P unknown numerical properties z=(z sub 1,...,z sub p) of the earth from measurement of D other numerical properties y (sup 0) = (y (sub 1) (sup 0), ..., y (sub D (sup 0)), using full or partial knowledge of the statistical distribution of the random errors in y (sup 0). The data space Y containing y(sup 0) is D-dimensional, so when the model space X is infinite-dimensional the linear uniqueness problem usually is insoluble without prior information about the correct earth model x. If that information is a quadratic bound on x, Bayesian inference (BI) and stochastic inversion (SI) inject spurious structure into x, implied by neither the data nor the quadratic bound. Confidence set inference (CSI) provides an alternative inversion technique free of this objection. Confidence set inference is illustrated in the problem of estimating the geomagnetic field B at the core-mantle boundary (CMB) from components of B measured on or above the earth's surface.
    Keywords: GEOPHYSICS
    Type: NAS 1.26:184839 , NASA-CR-184839
    Format: application/pdf
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  • 6
    Publication Date: 2013-08-31
    Description: Let R be the real numbers, R(n) the linear space of all real n-tuples, and R(infinity) the linear space of all infinite real sequences x = (x sub 1, x sub 2,...). Let P sub n :R(infinity) approaches R(n) be the projection operator with P sub n (x) = (x sub 1,...,x sub n). Let p(infinity) be a probability measure on the smallest sigma-ring of subsets of R(infinity) which includes all of the cylinder sets P sub n(-1) (B sub n), where B sub n is an arbitrary Borel subset of R(n). Let p sub n be the marginal distribution of p(infinity) on R(n), so p sub n(B sub n) = p(infinity)(P sub n to the -1(B sub n)) for each B sub n. A measure on R(n) is isotropic if it is invariant under all orthogonal transformations of R(n). All members of the set of all isotropic probability distributions on R(n) are described. The result calls into question both stochastic inversion and Bayesian inference, as currently used in many geophysical inverse problems.
    Keywords: STATISTICS AND PROBABILITY
    Type: NAS 1.26:181555 , NASA-CR-181555
    Format: application/pdf
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  • 7
    Publication Date: 2013-08-31
    Description: In linear inversion of a finite-dimensional data vector y to estimate a finite-dimensional prediction vector z, prior information about X sub E is essential if y is to supply useful limits for z. The one exception occurs when all the prediction functionals are linear combinations of the data functionals. Two forms of prior information are compared: a soft bound on X sub E is a probability distribution p sub x on X which describeds the observer's opinion about where X sub E is likely to be in X; a hard bound on X sub E is an inequality Q sub x(X sub E, X sub E) is equal to or less than 1, where Q sub x is a positive definite quadratic form on X. A hard bound Q sub x can be softened to many different probability distributions p sub x, but all these p sub x's carry much new information about X sub E which is absent from Q sub x, and some information which contradicts Q sub x. Both stochastic inversion (SI) and Bayesian inference (BI) estimate z from y and a soft prior bound p sub x. If that probability distribution was obtained by softening a hard prior bound Q sub x, rather than by objective statistical inference independent of y, then p sub x contains so much unsupported new information absent from Q sub x that conclusions about z obtained with SI or BI would seen to be suspect.
    Keywords: GEOPHYSICS
    Type: NAS 1.26:181557 , NASA-CR-181557
    Format: application/pdf
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  • 8
    Publication Date: 2013-08-31
    Description: The objective of the NASA Geodynamics program for magnetic field measurements is to study the physical state, processes and evolution of the Earth and its environment via interpretation of measurements of the near Earth magnetic field in conjunction with other geophysical data. The fields measured derive from sources in the core, the lithosphere, the ionosphere, and the magnetosphere. Panel recommendations include initiation of multi-decade long continuous scalar and vector measurements of the Earth's magnetic field by launching a five year satellite mission to measure the field to about 1 nT accuracy, improvement of our resolution of the lithographic component of the field by developing a low altitude satellite mission, and support of theoretical studies and continuing analysis of data to better understand the source physics and improve the modeling capabilities for different source regions.
    Keywords: GEOPHYSICS
    Type: National Aeronautics and Space Administration, Solid Earth Science in the 1990s. Volume 2: Panel Reports; 35 p
    Format: application/pdf
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  • 9
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: In the uniqueness part of a geophysical inverse problem, the observer wants to predict all likely values of P unknown numerical properties z = (z1,…, zp) of the earth from measurement of D other numerical properties Y(0)= (y1(0),…, yD(0)), using full or partial knowledge of the statistical distribution of the random errors in Y(0). the data space Y containing Y(0) is D-dimensional, so when the model space X is infinite-dimensional the linear uniqueness problem usually is insoluble without prior information about the correct earth model x. If that information is a quadratic bound on x (e.g. energy or dissipation rate), Bayesian inference (BI) and stochastic inversion (SI) inject spurious structure into x, implied by neither the data nor the quadratic bound. Confidence set inference (CSI) provides an alternative inversion technique free of this objection. the first step in CSI is to estimate unmodelled systematic errors in Y(0) and z. the second step is to choose any finite-dimensional subspace XN of X, and to use the prior quadratic bound to estimate the truncation error when the full data function F:X→ Y in the forward problem is approximated by restricting it to XN to give a finite-dimensional function FN:XN→ Y. Step three calculates the eigenstructure (singular value decomposition) of FN. Step 4 uses this eigenstructure to find for each positive ρ≤ 1 a Neyman subset Kz(ρ) of the P-dimensional prediction space Z such that either the correct value of the prediction vector z is a member of the confidence set Kz(ρ) or an event has occurred whose probability was no more than ρ. In contrast to SI and BI, CSI offers no incentive for considering any value of P except 1. CSI is illustrated in the problem of estimating the geomagnetic field B at the core-mantle boundary (CMB) from components of B measured on or above the earth's surface. Neither the heat flow nor the energy bound is strong enough to permit estimation of Br at single points on the CMB, but the heat flow bound permits estimation of uniform averages of Br over discs on the CMB, and both bounds permit weighted disc-averages with continuous weighting kernels. Both bounds also permit estimation of low-degree Gauss coefficients at the CMB. the heat flow bound resolves them up to degree 8 if the crustal field at satellite altitudes must be treated as a systematic error, but can resolve to degree 11 under the most favorable statistical treatment of the crust. These two limits produce circles of confusion on the CMB with diameters of 25° and 19° respectively.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: Statistical modelling of the Earth's magnetic field B has a long history (see e.g. McDonald 1957; Gubbins 1982; McLeod 1986; Constable & Parker 1988). In particular, the spherical harmonic coefficients of scalar fields derived from B can be treated as Gaussian random variables (Constable & Parker 1988). In this paper, we give examples of highly organized fields whose spherical harmonic coefficients pass tests for independent Gaussian random variables. The fact that coefficients at some depth may be usefully summarized as independent samples from a normal distribution need not imply that there really is some physical, random process at that depth. In fact, the field can be extremely structured and still be regarded for some purposes as random. In this paper we examined the radial magnetic field Br produced by the core, but the results apply to any scalar field on the core–mantle boundary (CMB) which determines B outside the CMB.
    Type of Medium: Electronic Resource
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