Confidence Set Inference with a Prior Quadratic Bound

Abstract

Summary 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)= (y  1  (0),…, y   D  (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.