Isotropic probability measures in infinite dimensional spaces: Inverse problems/prior information/stochastic inversion

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

Completeness of Inertial Modes of an Incompressible Non-Viscous Fluid in a Corotating Ellipsoid

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 Poincar\'e's 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 Poincar\'e 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 Poincar\'e 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 Poincar\'e modes, first obtained by Bryan (1889), and completed with the geostrophic modes.Comment: 19 pages, 1 figure, to appear in Physical Review

Comparing hard and soft prior bounds in geophysical inverse problems

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

Contributions from geomagnetic inverse theory to the study of hydromagnetic conditions near the core-mantle boundary

The Final Report on contributions from geomagnetic inverse theory to the study of hydromagnetic conditions near the core-mantle boundary (CMB) is presented. The original proposal was to study five questions concerning what the surface and satellite magnetic data imply about hydromagnetic and electromagnetic conditions near the CMB. The five questions are: (1) what do the surface and satellite data imply about the geomagnetic field B near the surface of the earth; (2) how does one extrapolate B down through the conducting mantle to the CMB; (3) if B on the CMB is visible, how accurately does it satisfy the frozen-flux approximation; (4) if frozen flux is a good approximation on the CMB, what can be inferred about the fluid velocity v in the upper core; and (5) if v at the CMB is visible, does it suggest any dynamical properties of the core, such as vertical advection, Alfven-inertial waves, link instabilities, or mantle effects. A summary of the research is provided

Confidence set inference with a prior quadratic bound

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(0)=(y sub 1(0),...,y sub D(0)) 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. 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 B(r) at single points on the CMB, but the heat flow bound permits estimation of uniform averages of B(r) over discs on the CMB, and both bounds permit weighted disc-averages with continous 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 deg and 19 deg respectively

Hydromagnetic conditions near the core-mantle boundary

The main results of the grant were (1) finishing the manuscript of a proof of completeness of the Poincare modes in an incompressible nonviscous fluid corotating with a rigid ellipsoidal boundary, (2) partial completion of a manuscript describing a definition of helicity that resolved questions in the literature about calculating the helicities of vector fields with complicated topologies, and (3) the beginning of a reexamination of the inverse problem of inferring properties of the geomagnetic field B just outside the core-mantle boundary (CMB) from measurements of elements of B at and above the earth's surface. This last work has led to a simple general formalism for linear and nonlinear inverse problems that appears to include all the inversion schemes so far considered for the uniqueness problem in geomagnetic inversion. The technique suggests some new methods for error estimation that form part of this report

Confidence set inference with a prior quadratic bound

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

Is the Non-Dipole Magnetic Field Random?

Statistical modelling of the Earth's magnetic field B has a long history. In particular, the spherical harmonic coefficients of scalar fields derived from B can be treated as Gaussian random variables. 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(sub r) produced by the core, but the results apply to any scalar field on the core-mantle boundary (CMB) which determines B outside the CMB

The scientific case for magnetic field satellites

To make full use of modern magnetic data and the paleomagnetic record, we must greatly improve our understanding of how the geodynamo system works. It is clearly nonlinear, probably chaotic, and its dimensionless parameters cannot yet be reproduced on a laboratory scale. It is accessible only to theory and to measurements made at and above the earth's surface. These measurements include essentially all geophysical types. Gravity and seismology give evidence for undulations in the core-mantle boundary (CMB) and for temperature variations in the lower mantle which can affect core convection and hence the dynamo. VLBI measurements of the variations in the Chandler wobble and length of day are affected by, among other things, the electromagnetic and mechanical transfer of angular momentum across the CMB. Finally, measurements of the vector magnetic field, its intensity, or its direction, give the most direct access to the core dynamo and the electrical conductivity of the lower mantle. The 120 gauss coefficients of degrees up to 10 probably come from the core, with only modest interference by mantle conductivity and crustal magnetization. By contrast, only three angular accelerations enter the problem of angular momentum transfer across the CMB. Satellite measurements of the vector magnetic field are uniquely able to provide the spatial coverage required for extrapolation to the CMB, and to isolate and measure certain magnetic signals which to the student of the geodynamo represent noise, but which are of great interest elsewhere in geophysics. Here, these claims are justified and the mission parameters likely to be scientifically most useful for observing the geodynamo system are described

Explorations in combining cognitive models of individuals and system dynamics models of groups.

This report documents a demonstration model of interacting insurgent leadership, military leadership, government leadership, and societal dynamics under a variety of interventions. The primary focus of the work is the portrayal of a token societal model that responds to leadership activities. The model also includes a linkage between leadership and society that implicitly represents the leadership subordinates as they directly interact with the population. The societal model is meant to demonstrate the efficacy and viability of using System Dynamics (SD) methods to simulate populations and that these can then connect to cognitive models depicting individuals. SD models typically focus on average behavior and thus have limited applicability to describe small groups or individuals. On the other hand, cognitive models readily describe individual behavior but can become cumbersome when used to describe populations. Realistic security situations are invariably a mix of individual and population dynamics. Therefore, the ability to tie SD models to cognitive models provides a critical capability that would be otherwise be unavailable