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Shure, L, Parker RL.  1981.  An alternative explanation for intermediate-wavelength magnetic anomalies. Journal of Geophysical Research. 86:1600-1608.   10.1029/JB086iB12p11600   AbstractWebsite

Harrison and Carle [this issue] and others have examined very long profiles of the magnetic field and have calculated one-dimensional power spectra. In these they expect to see, but do not find, a minimum in power at intermediate wavelengths, between 65 and 150 km. Conventional one-dimensional models of the field predict very little power in this band, which lies between the spectral peaks arising from sources in the crust and the core. Mantle sources or high-intensity, long-wavelength magnetizations have been proposed to account for the observations. An alternative, more plausible explanation is that one-dimensional spectra of two-dimensional fields contain contributions from wavenumbers in the perpendicular (i.e., nonsampled) direction. Unless the seafloor spreading anomalies are perfectly lineated at right angles to the profile, some low-wavenumber energy must be attributed to this effect; we propose that such directional aliasing is a major factor in the power spectra. To support this idea, we discuss theoretical models and analyze a large-scale marine survey.

Shure, L, Parker RL, Backus GE.  1982.  Harmonic splines for geomagnetic modelling. Physics of the Earth and Planetary Interiors. 28:215-229.   10.1016/0031-9201(82)90003-6   AbstractWebsite

Which features of a geomagnetic field model on the surface of the core are really necessary in order to fit, within observational error, the field observations at and above the Earth's surface? To approach this question, we define ‘roughness’ in various ways as a norm on an appropriate Hilbert space of field models which is small when the field is smooth on the core surface. Then, we calculate the model with least norm (the smoothest model) which fits the data, sources outside the core being treated as noise. Sample calculations illustrate the effects of noise, of the choice of norm and of an uneven distribution of observing stations.

Shure, L, Parker RL, Langel RA.  1985.  A preliminary harmonic spline model from Magsat data. Journal of Geophysical Research-Solid Earth and Planets. 90:1505-1512.   10.1029/JB090iB13p11505   AbstractWebsite

We present a preliminary main field model for 1980 derived from a carefully selected subset of Magsat vector measurements using the method of harmonic splines. This model (PHS (80) for preliminary harmonic splines) is the smoothest model (in the sense that the rms radial field at the core surface is minimum) consistent with the measurements (with an rms misfit of 10 nT to account for crustal and external fields as well as noise in the measurement procedure). Therefore PHS (80) is more suitable for studies of the core than models derived with the traditional least squares approach (e.g., GSFC (9/80)). We compare characteristics of the harmonic spline spectrum, topology of the core field and especially the null-flux curves (loci where the radial field is zero) and the flux through patches bounded by such curves. PHS (80) is less complex than GSFC (9/80) and is therefore more representative of that part of the core field that the data constrain.

Stark, PB, Parker RL.  1987.  Velocity bounds from statistical estimates of τ(p) and X (p). Journal of Geophysical Research-Solid Earth and Planets. 92:2713-2719.   10.1029/JB092iB03p02713   AbstractWebsite

We present a new technique for constructing the narrowest corridor containing all velocity profiles consistent with a finite collection of τ(p) data and their statistical uncertainties. Earlier methods for constructing such bounds treat the confidence interval for each τ datum as a strict interval within which the true value might lie with equal probability, but this interpretation is incompatible with the estimation procedure used on the original travel time observations. The new approach, based upon quadratic programming (QP), shares the advantages of the linear programming (LP) solution: it can invert τ(p) and X(p) data concurrently; it permits the incorporation of constraints on the radial derivative of velocity for spherical earth models; and theoretical results about convergence and optimality can be obtained for the method. We compare P velocity bounds for the core obtained by QP and LP. The models produced by LP predict data values at the ends of the confidence intervals; these values are unlikely according to the proper statistical dstribution of errors. For this reason the LP velocity bounds can be wider than those given by QP, which takes better account of the statistics. Sometimes, however, the LP bounds are more restrictive because LP never permits the predictions of the models to lie outside the confidence intervals even though occasional excursions are expected. The QP bounds grow narrower at lower levels of confidence, but the corridors at 95% and 99.9% are virtually indistinguishable: The data must be improved substantially to make a significant change in the velocity bounds.

Stark, PB, Parker RL.  1987.  Smooth profiles from τ(p) and X(p) data. Geophysical Journal of the Royal Astronomical Society. 89:997-1010.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1987.tb05205.x   AbstractWebsite

Summary. We reduce the problem of constructing a smooth, 1-D, monotoni-cally increasing velocity profile consistent with discrete, inexact τ (p) and X(p) data to a quadratic programming problem with linear inequality constraints. For a finite-dimensional realization of the problem it is possible to find a smooth velocity profile consistent with the data whenever such a profile exists. We introduce an unusual functional measure of roughness equivalent to the second central moment or ‘Variance’ of the derivative of depth with respect to velocity for smooth profiles, and we prove that its minimal value is unique. In our experience, solutions minimizing this functional are very smooth in the sense of the two-norm of the second derivative and can be constructed inexpensively by solving one quadratic programming problem. Still smoother models (in more traditional measures) may be generated iteratively with additional quadratic programs. All the resulting models satisfy the τ (p) and X(p) data and reproduce travel-time data remarkably well, although sometimes τ (p) data alone are insufficient to ensure arrivals at large X; then an X(p) datum must be included.

Stark, PB, Parker RL.  1995.  Bounded-variable least-squares: an algorithm and applications. Computational Statistics. 10:129-141. AbstractWebsite

The Fortran subroutine BVLS (bounded variable least-squares) solves linear least-squares problems with upper and lower bounds on the variables, using an active set strategy. The unconstrained least-squares problems for each candidate set of free variables are solved using the QR decomposition. BVLS has a ''warm-start'' feature permitting some of the variables to be initialized at their upper or lower bounds, which speeds the solution of a sequence of related problems. Such sequences of problems arise, for example, when BVLS is used to find bounds on linear functionals of a model constrained to satisfy, in an approximate l(p)-norm sense, a set of linear equality constraints in addition to upper and lower bounds. We show how to use BVLS to solve that problem when p = 1, 2, or infinity, and to solve minimum l(1) and l(infinity) fitting problems. FORTRAN 77 code implementing BVLS is available from the statlib gopher at Carnegie Mellon University.

Stark, PB, Parker RL, Masters G, Orcutt JA.  1986.  Strict bounds on seismic velocity in the spherical earth. Journal of Geophysical Research-Solid Earth and Planets. 91:13892-13902.   10.1029/JB091iB14p13892   AbstractWebsite

We address the inverse problem of finding the smallest envelope containing all velocity profiles consistent with a finite set of imprecise τ(p) data from a spherical earth. Traditionally, the problem has been attacked after mapping the data relations into relations on an equivalent flat earth. Of the two contemporary direct methods for finding bounds on velocities in the flat earth consistent with uncertain τ(p) data, a nonlinear (NL) approach descended from the Herglotz-Wiechert inversion and a linear programming (LP) approach, only NL has been used to solve the spherical earth problem. On the basis of the finite collection of τ(p) measurements alone, NL produces an envelope that is too narrow: there are numerous physically acceptable models that satisfy the data and violate the NL bounds, primarily because the NL method requires continuous functions as bounds on τ(p) and thus data must be fabricated between measured values by some sort of interpolation. We use the alternative LP approach, which does not require interpolation, to place optimal bounds on the velocity in the core. The resulting velocity corridor is disappointingly wide, and we therefore seek reasonable physical assumptions about the earth to reduce the range of permissible models. We argue from thermodynamic relations that P wave velocity decreases with distance from the earth's center within the outer core and quite probably within the inner core and lower mantle. We also show that the second derivative of velocity with respect to radius is probably not positive in the core. The first radial derivative constraint is readily incorporated into LP. The second derivative constraint is nonlinear and cannot be implemented exactly with LP; however, geometrical arguments enable us to apply a weak form of the constraint without any additional computation. LP inversions of core τ(p) data using the first radial derivative constraint give new, extremely tight bounds on the P wave velocity in the core. The weak second derivative constraint improves them slightly.