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Backus, G, Rieutord M.  2017.  Completeness of inertial modes of an incompressible inviscid fluid in a corotating ellipsoid. Physical Review E. 95   10.1103/PhysRevE.95.053116   AbstractWebsite

Inertial modes are the eigenmodes of contained rotating fluids restored by the Coriolis force. When the fluid is incompressible, inviscid, and contained in a rigid container, these modes satisfy Poincares equation that has the peculiarity of being hyperbolic with boundary conditions. Inertial modes are, therefore, solutions of an ill-posed boundary-value problem. In this paper, we investigate the mathematical side of this problem. We first show that the Poincare problem can be formulated in the Hilbert space of square-integrable functions, with no hypothesis on the continuity or the differentiability of velocity fields. We observe that with this formulation, the Poincare operator is bounded and self-adjoint, and as such, its spectrum is the union of the point spectrum (the set of eigenvalues) and the continuous spectrum only. When the fluid volume is an ellipsoid, we show that the inertial modes form a complete base of polynomial velocity fields for the square-integrable velocity fields defined over the ellipsoid and meeting the boundary conditions. If the ellipsoid is axisymmetric, then the base can be identified with the set of Poincare modes, first obtained by Bryan [Philos. Trans. R. Soc. London 180, 187 (1889)], and completed with the geostrophic modes.

Backus, G, Parker RL, Constable C.  2005.  Foundations of geomagnetism. :xiv,369p.., Cambridge ; New York: Cambridge University PressWebsite
Walker, AD, Backus GE.  1997.  A six-parameter statistical model of the Earth's magnetic field. Geophysical Journal International. 130:693-700.   10.1111/j.1365-246X.1997.tb01863.x   AbstractWebsite

A six-parameter statistical model of the non-dipole geomagnetic field is fitted to 2597 harmonic coefficients determined by Cain, Holter & Sandee (1990) from MAGSAT data. The model includes sources in the core, sources in the crust, and instrument errors. External fields are included with instrument errors. The core and instrument statistics are invariant under rotation about the centre of the Earth, and one of the six parameters describes the deviation of the crustal statistics from rotational invariance. The model treats the harmonic coefficients as independent random samples drawn from a Gaussian distribution. The statistical model of the core held has a correlation length of about 500 km at the core-mantle boundary, too long to be attributed to a white noise source just below the boundary layers at the top of the core. The estimate of instrument errors obtained from the statistical model is in good agreement with an independent estimate based on tests of the instruments (Langel, Ousley & Berbert 1982).

Walker, AD, Backus GE.  1996.  On the difference between the average values of B-r(2) in the Atlantic and Pacific hemispheres. Geophysical Research Letters. 23:1965-1968.   10.1029/96gl01854   AbstractWebsite

Recent work on reversals of the earth's magnetic field [Clement, 1991; Constable, 1992; Laj et al., 1991; Runcorn, 1992] and work on low rates of change of the radial magnetic field, B-r, in the Pacific hemisphere (see Jacobs [1994] page 8, for a summary) suggest that the statistics of B-r differ between the Pacific and Atlantic hemispheres. The question of statistical invariance of the magnetic field is interesting in its own right [Courtillot et al, 1992], and we study this hypothesis by investigating the statistical significance of the difference between the average value of B-r(2) in the Atlantic and Pacific hemispheres at three different radii. The differences between the average value of B-r(2) in the Atlantic hemisphere and the average value of B-r(2) in the Pacific hemisphere at all three radii are incompatibly large with two statistical models of B-r. The differences are incompatibly large at two of three radii when a third statistical model is used. The results suggest that further work on the statistical properties of B-r is warranted and provide another test for normality of spherical harmonic coefficients.

Walker, AD, Backus GE.  1996.  Is the non-dipole magnetic field random? Geophysical Journal International. 124:315-319.   10.1111/j.1365-246X.1996.tb06373.x   AbstractWebsite

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 B-r produced by the core, but the results apply to any scalar held on the core-mantle boundary (CMB) which determines B outside the CMB.

Backus, GE.  1996.  Trimming and procrastination as inversion techniques. Physics of the Earth and Planetary Interiors. 98:101-142.   10.1016/s0031-9201(96)03183-4   AbstractWebsite

By examining the processes of truncating and approximating the model space (trimming it), and by committing to neither the objectivist nor the subjectivist interpretation of probability (procrastinating), we construct a formal scheme for solving linear and non-linear geophysical inverse problems. The necessary prior information about the correct model x(E) can be either a collection of inequalities or a probability measure describing where x(E) was likely to be in the model space X before the data vector y(0) was measured. The results of the inversion are (1) a vector z(0) that estimates some numerical properties z(E) of x(E); (2) an estimate of the error delta z = z(0) - z(E). As y(0) is finite dimensional, so is z(0), and hence in principle inversion cannot describe all of x(E). The error delta z is studied under successively more specialized assumptions about the inverse problem, culminating in a complete analysis of the linear inverse problem with a prior quadratic bound on x(E). Our formalism appears to encompass and provide error estimates for many of the inversion schemes current in geomagnetism, and would be equally applicable in geodesy and seismology if adequate prior information were available there. As an idealized example we study the magnetic field at the core-mantle boundary, using satellite measurements of field elements at sites assumed to be almost uniformly distributed on a single spherical surface. Magnetospheric currents are neglected and the crustal field is idealized as a random process with rotationally invariant statistics. We find that an appropriate data compression diagonalizes the variance matrix of the crustal signal and permits an analytic trimming of the idealized problem.

Backus, G, Parker RL, Constable C.  1996.  Foundations of geomagnetism. :xiv,369p.., Cambridge England ; New York: Cambridge University Press AbstractWebsite

The main magnetic field of the Earth is a complex phenomenon. To understand its origins in the fluid of the Earth's core, and how it changes in time requires a variety of mathematical and physical tools. This book presents the foundations of geomagnetism, in detail and developed from first principles. The book is based on George Backus' courses for graduate students at the University of California, San Diego. The material is mathematically rigorous, but is logically developed and has consistent notation, making it accessible to a broad range of readers. The book starts with an overview of the phenomena of interest in geomagnetism, and then goes on to deal with the phenomena in detail, building the necessary techniques in a thorough and consistent manner. Students and researchers will find this book to be an invaluable resource in the appreciation of the mathematical and physical foundations of geomagnetism.

Ander, ME, Zumberge MA, Lautzenhiser T, Parker RL, Aiken CLV, Gorman MR, Nieto MM, Cooper APR, Ferguson JF, Fisher E, McMechan GA, Sasagawa G, Stevenson JM, Backus G, Chave AD, Greer J, Hammer P, Hansen BL, Hildebrand JA, Kelty JR, Sidles C, Wirtz J.  1989.  Test of Newtons Inverse-Square Law in the Greenland Ice Cap. Physical Review Letters. 62:985-988.   10.1103/PhysRevLett.62.985   Website
Backus, GE.  1989.  Confidence Set Inference with a Prior Quadratic Bound. Geophysical Journal-Oxford. 97:119-150.   10.1111/j.1365-246X.1989.tb00489.x   Website
Backus, GE.  1988.  The Field Lines of an Axisymmetric Magnetic-Field. Geophysical Journal-Oxford. 93:413-417.   10.1111/j.1365-246X.1988.tb03869.x   Website
Backus, GE.  1988.  Bayesian-Inference in Geomagnetism. Geophysical Journal-Oxford. 92:125-142.   10.1111/j.1365-246X.1988.tb01127.x   Website
Backus, G.  1987.  Isotropic Probability-Measures in Infinite-Dimensional Spaces. Proceedings of the National Academy of Sciences of the United States of America. 84:8755-8757.   10.1073/pnas.84.24.8755   Website
Backus, GE, Estes RH, Chinn D, Langel RA.  1987.  Comparing the Jerk with Other Global-Models of the Geomagnetic-Field from 1960 to 1978. Journal of Geophysical Research-Solid Earth and Planets. 92:3615-3622.   10.1029/JB092iB05p03615   Website
Backus, GE, Lemouel JL.  1986.  The Region on the Core Mantle Boundary Where a Geostrophic Velocity-Field Can Be Determined from Frozen-Flux Magnetic Data. Geophysical Journal of the Royal Astronomical Society. 85:617-628.   10.1111/j.1365-246X.1986.tb04536.x   Website
Backus, G.  1986.  Poloidal and Toroidal Fields in Geomagnetic-Field Modeling. Reviews of Geophysics. 24:75-109.   10.1029/RG024i001p00075   Website
Backus, G, Hough S.  1985.  Some Models of the Geomagnetic-Field in Western-Europe from 1960 to 1980. Physics of the Earth and Planetary Interiors. 39:243-254.   10.1016/0031-9201(85)90137-2   Website
Voorhies, CV, Backus GE.  1985.  Steady Flows at the Top of the Core from Geomagnetic-Field Models - the Steady Motions Theorem. Geophysical and Astrophysical Fluid Dynamics. 32:163-173.   10.1080/03091928508208783   Website
Joyce, T, Backus R, Baker K, Blackwelder P, Brown O, Cowles T, Evans R, Fryxell G, Mountain D, Olson D, Schlitz R, Schmitt R, Smith P, Smith R, Wiebe P.  1984.  Rapid Evolution of a Gulf-Stream Warm-Core Ring. Nature. 308:837-840.   10.1038/308837a0   Website
Backus, GE.  1983.  Application of Mantle Filter Theory to the Magnetic Jerk of 1969. Geophysical Journal of the Royal Astronomical Society. 74:713-746.Website
Backus, GE.  1982.  The Electric-Field Produced in the Mantle by the Dynamo in the Core. Physics of the Earth and Planetary Interiors. 28:191-214.   10.1016/0031-9201(82)90002-4   Website
Shure, L, Parker RL, Backus GE.  1982.  Harmonic Splines for Geomagnetic Modeling. Physics of the Earth and Planetary Interiors. 28:215-229.   10.1016/0031-9201(82)90003-6   Website
Backus, G, Park J, Garbasz D.  1981.  On the Relative Importance of the Driving Forces of Plate Motion. Geophysical Journal of the Royal Astronomical Society. 67:415-435.   10.1111/j.1365-246X.1981.tb02758.x   Website
Backus, GE.  1979.  L2 Form of Bernsteins Inequality. Proceedings of the National Academy of Sciences of the United States of America. 76:3061-3064.   10.1073/pnas.76.7.3061   Website