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Van Beusekom, AE, Parker RL, Bank RE, Gill PE, Constable S.  2011.  The 2-D magnetotelluric inverse problem solved with optimization. Geophysical Journal International. 184:639-650.   10.1111/j.1365-246X.2010.04895.x   AbstractWebsite

P>The practical 2-D magnetotelluric inverse problem seeks to determine the shallow-Earth conductivity structure using finite and uncertain data collected on the ground surface. We present an approach based on using PLTMG (Piecewise Linear Triangular MultiGrid), a special-purpose code for optimization with second-order partial differential equation (PDE) constraints. At each frequency, the electromagnetic field and conductivity are treated as unknowns in an optimization problem in which the data misfit is minimized subject to constraints that include Maxwell's equations and the boundary conditions. Within this framework it is straightforward to accommodate upper and lower bounds or other conditions on the conductivity. In addition, as the underlying inverse problem is ill-posed, constraints may be used to apply various kinds of regularization. We discuss some of the advantages and difficulties associated with using PDE-constrained optimization as the basis for solving large-scale nonlinear geophysical inverse problems. Combined transverse electric and transverse magnetic complex admittances from the COPROD2 data are inverted. First, we invert penalizing size and roughness giving solutions that are similar to those found previously. In a second example, conventional regularization is replaced by a technique that imposes upper and lower bounds on the model. In both examples the data misfit is better than that obtained previously, without any increase in model complexity.

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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.

Constable, CG, Tauxe L, Parker RL.  1998.  Analysis of 11 Myr of geomagnetic intensity variation. Journal of Geophysical Research-Solid Earth. 103:17735-17748.   10.1029/98jb01519   AbstractWebsite

We have conducted a detailed exploratory analysis of an II million year long almost continuous record of relative geomagnetic paleointensity from a sediment core acquired on Deep Sea Drilling Project Leg 73, at Site 522 in the South Atlantic. We assess the quality of the paleointensity record using spectral methods and conclude that the relative intensity record is minimally influenced by climate variations. Isothermal remanence is shown to be the most effective normalizer for these data, although both susceptibility and anhysteretic remanence are also adequate. Statistical analysis shows that the paleointensity variations follow a gamma distribution, and are compatible with predictions from modified paleosecular variation models and global absolute paleointensity data. When subdivided by polarity interval, the variability in paleointensity is proportional to the average, and further, the average is weakly correlated with interval length. Spectral estimates for times from 28.77 until 22.74 Ma, when the reversal rate is about 4 Myr(-1), are compatible with a Poisson model in which the spectrum of intensity variations is dominated by the reversal process in the frequency range 1-50 Mgr(-1) In contrast, between 34.7 and 29.4 Ma, when the reversal rate is about 1.6 Myr(-1), the spectra indicate a different secular variation regime. The magnetic field is stronger, and more variable, and a strong peak in the spectrum occurs at about 8 Myr(-1). This peak magi be a reflection of the same signal as recorded by the small variations known as tiny wiggles seen in marine magnetic anomaly profiles.

Parker, RL, Zumberge MA.  1989.  An Analysis of geophysical experiments to test Newton's law of gravity. Nature. 342:29-32.   10.1038/342029a0   AbstractWebsite

Signals reported as evidence for a non-newtonian 'fifth' force at a North Carolina television tower and elsewhere can be explained in a conventional way by postulating small density variations underground. The assumptions employed in earlier analyses which pointed to a failure of the inverse square law are examined and found to be difficult to justify.

Klitgord†, KD, Mudie JD, Huestis SP, Parker RL.  1975.  An analysis of near-bottom magnetic anomalies: Sea-floor spreading and the magnetized layer. Geophysical Journal of the Royal Astronomical Society. 43:387-424.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1975.tb00641.x   AbstractWebsite

Near-bottom magnetic data over six oceanic ridge segments in the East Pacific are inverted, giving magnetization solutions with alternate positive and negative bands which correspond to geomagnetic field reversals. We estimate the average half-width of the crustal formation zone to be 2–3 km, based on the transition widths between these bands. The solutions show a narrow region of high magnetization centred directly over the centre of spreading, superimposed on a more gradual decrease of magnetization amplitudes with age. Both features are attributed to weathering of highly magnetized pillow lavas. We demonstrate that the short wavelength (<3km) anomalies are largely due to topography. Distances to reversal boundaries give distance vs age curves for each ridge which show that spreading changes occur as sudden accelerations typically separated by several million years of very constant motion. These rate changes are probably accompanied by shifts in the locations of poles of relative motion, which are necessary in a system of more than two interacting plates. Palaeomagnetic data and reversal boundary locations from near-bottom and surface data are combined to give spreading half-rates and a refined time scale for the past 6 My. Widespread spreading rate variations occurred at 2–3 MyBP and about 5 MyBP, possibly as a response to large scale changes in the plate pattern.

Parker, RL.  2014.  Anomalous phases in TE-mode magnetotellurics. Geophysics. 79:E75-E79.   10.1190/geo2013-0325.1   AbstractWebsite

We examined the transverse electric mode of 2D magnetotelluric sounding for a simple system comprising a laterally variable thin sheet over an insulator terminated by a perfectly conducting base. We found, by asymptotic analysis and a numerical example, that the phase of the c response, or that of the corresponding entry in the impedance tensor, is completely unrestricted. This behavior is unlike that of 1D systems or transverse magnetic mode induction, where the phase is confined to a single quadrant.

Parker, RL, Shure L, Hildebrand JA.  1987.  The application of inverse theory to seamount magnetism. Reviews of Geophysics. 25:17-40.   10.1029/RG025i001p00017   AbstractWebsite

The traditional least squares method for modeling seamount magnetism is often unsatisfactory because the models fail to reproduce the observations accurately. We describe an alternative approach permitting a more complex internal structure, guaranteed to generate an external field in close agreement with the observed anomaly. Potential field inverse problems like this one are fundamentally incapable of a unique solution, and some criterion is mandatory for picking a plausible representative from the infinite-dimensional space of models all satisfying the data. Most of the candidates are unacceptable geologically because they contain huge magnetic intensities or rapid variations of magnetization on fine scales. To avoid such undesirable attributes, we construct the simplest type of model: the one closest to a uniform solution as measured by the norm in a specially chosen Hilbert space of magnetization functions found by a procedure called seminorm minimization. Because our solution is the most nearly uniform one we can say with certainty that any other magnetization satisfying the data must be at least as complex as ours. The theory accounts for the complicated shape of seamounts, representing the body by a covering of triangular facets. We show that the special choice of Hilbert space allows the necessary volume integrals to be reduced to surface integrals over the seamount surface, and we present numerical techniques for their evaluation. Exact agreement with the magnetic data cannot be expected because of the error of approximating the shape and because the measured fields contain noise of crustal, ionospheric, and magnetospheric origin. We examine the potential size of the various error terms and find that those caused by approximation of the shape are generally much smaller than the rest. The mean magnetization is a vector that can in principle be discovered from exact knowledge of the external field of the seamount; this vector is of primary importance for paleomagnetic work. We study the question of how large the uncertainty in the mean vector may be, based on actual noise, as opposed to exact, data; the uncertainty can be limited only by further assumptions about the internal magnetization. We choose to bound the rms intensity. In an application to a young seamount in the Louisville Ridge chain we find that remarkably little nonuniformity is required to obtain excellent agreement with the observed anomaly while the uniform magnetization gives a poor fit. The paleopole position of ordinary least squares solution lies over 30° away from the geographic north, but the pole derived from our seminorm minimizing model is very near the north pole as it should be. A calculation of the sensitivity of the mean magnetization vector to the location of the magnetic observations shows that the data on the perimeter of the survey were given the greatest weight and suggests that enlargement of the survey area might further improve the reliability of the results.

McMillan, DG, Constable CG, Parker RL.  2004.  Assessing the dipolar signal in stacked paleointensity records using a statistical error model and geodynamo simulations. Physics of the Earth and Planetary Interiors. 145:37-54.   10.1016/j.pepi.2004.02.011   AbstractWebsite

Stacks of globally distributed relative paleointensity records from sediment cores are used to study temporal variations in the strength of the geomagnetic dipole. We assess the intrinsic accuracy and resolution of such stacks, which may be limited by errors in paleointensity, non-dipole field contributions, and the age scales assigned to each sediment core. Our approach employs two types of simulations. Numerical geodynamo models generate accurate predictions of time series of magnetic variations anywhere in the world. The predicted variations are then degraded using an appropriate statistical model to simulate expected age and paleointensity errors. A series of experiments identify the major contributors to error and loss of resolution in the resulting stacks. The statistical model simulates rock magnetic and measurement errors in paleointensity, and age errors due to finite sampling and approximations inherent in interpolation, incomplete or inaccurate tie point information, and sedimentation rate variations. Data sampling and interpolation to a designated age scale cause substantial decorrelation, and control the maximum level of agreement attainable between completely accurate records. The particular method of interpolation appears to have little effect on the coherence between accurate records, but denser tie point data improve the agreement. Age errors decorrelate geomagnetic signals, usually at shorter periods, although they can destroy coherence over a broad range of periods. The poor correlation between neighboring paleomagnetic records often observed in real data can be accounted for by age errors of moderate magnitude. In a global dataset of 20 records, modeled after the SINT800 compilation and spanning 300 kyr, our results show that dipole variations with periods longer than about 20 kyr can be recovered by the stacking process. Reasonable contributions to error in the paleointensity itself have a modest influence on the result, as do non-dipole field contributions whose effect is minor at periods longer than 10 kyr. Modest errors in the ages of tie points probably account for most of the degradation in geomagnetic signal. Stacked sedimentary paleomagnetic records can be improved by denser temporal sampling and careful selection of independent high-quality tie points. (C) 2004 Elsevier B.V. All rights reserved.

Parker, RL, Song YQ.  2005.  Assigning uncertainties in the inversion of NMR relaxation data. Journal of Magnetic Resonance. 174:314-324.   10.1016/j.jmr.2005.03.002   AbstractWebsite

Recovering the relaxation-time density function (or distribution) from NMR decay records requires inverting a Laplace transform based on noisy data, an ill-posed inverse problem. An important objective in the face of the consequent ambiguity in the solutions is to establish what reliable information is contained in the measurements. To this end we describe how upper and lower bounds on linear functionals of the density function, and ratios of linear functionals, can be calculated using optimization theory. Those bounded quantities cover most of those commonly used in the geophysical NMR, such as porosity, T-2 log-mean, and bound fluid volume fraction, and include averages over any finite interval of the density function itself. In the theory presented statistical considerations enter to account for the presence of significant noise in the signal, but not in a prior characterization of density models. Our characterization of the uncertainties is conservative and informative; it will have wide application in geophysical NMR and elsewhere. © 2005 Elsevier Inc. All rights reserved.

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Parker, RL.  1974.  Best bounds on density and depth from gravity data. Geophysics. 39:644-649.   10.1190/1.1440454   AbstractWebsite

Gravity data cannot usually be inverted to yield unique structures from incomplete data; however, there is a smallest density compatible with the data or, if the density is known, a deepest depth of burial. A general theory is derived which gives the greatest lower bound on density or the least upper bound on depth. These bounds are discovered by consideration of a class of “ideal” bodies which achieve the extreme values of depth or density. The theory is illustrated with several examples which are solved by analytic methods. New maximum depth rules derived by this theory are, unlike some earlier rules of this type, optimal for the data they treat.

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.

Huestis, SP, Parker RL.  1977.  Bounding the thickness of the oceanic magnetized layer. Journal of Geophysical Research. 82:5293-5303.   10.1029/JB082i033p05293   AbstractWebsite

We present a theory for placing a lower bound on the thickness of the oceanic magnetized layer using magnetic anomaly observations and estimates of the intensity of magnetization; the theory makes only a minimum number of assumptions regarding the spatial distribution of the magnetization. The principle of the method is based upon the fact that thin layers imply high magnetizations. We show how to calculate the source distribution that has minimum intensity yet fits the data and is confined to a given thickness layer; because the minimum intensity must be a monotonically decreasing function of layer thickness, it follows that an upper bound on the intensity allows us to obtain a lower limit on the thickness. The practical calculations are performed by using linear programing. The method is applied to two sets of near-bottom magnetic profiles, one on the Galápagos Spreading Center at 86°W and the other set on the Pacific-Antarctic Ridge at 51°S. In the first area we conclude that the magnetic layer must be at least 450 m thick, and in the other a crossing of the Jaramillo event indicates that the magnetic layer is probably more than 1000 m in thickness.

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Parker, RL, Gee JS.  2002.  Calibration of the pass-through magnetometer - II. Application. Geophysical Journal International. 150:140-152.   10.1046/j.1365-246X.2002.01692.x   AbstractWebsite

We describe the experimental procedure we use to calibrate a cryogenic pass-through magnetometer. The procedure is designed to characterize the magnetometer sensitivity as a function of position within the sensing region. Then we extend a theory developed in an earlier paper to cover inexact observations and apply it to the data set. The theory allows the calculation of a smooth, harmonic, internally consistent interpolating function for each of the nine components of the response tensor of the magnetometer. With these functions we can calculate the response to a dipole source in any orientation and position, and predict the magnetometer signal from any kind of specimen. The magnetometer in the paleomagnetic laboratory onboard the research vessel Joides Resolution is the subject of one such experiment and we present the results. The variation with position of sensitivity is displayed in a series of plane slices through the magnetometer. We discover from the calibration model that the X and Z coils are misaligned so that the magnetic centre of the coils is displaced from the geometric centre by approximately 0.7 cm. We synthesize the signal expected from the magnetometer when a variety of simple cores are measured. We find that, unless appropriate corrections are made, changes in magnetization direction can appear as variations in magnetic intensity, and conversely, fluctuations in the magnetization strength can produce apparent swings in declination and inclination. The magnitude of these effects is not small and is certainly worth taking into account in the interpretation of records from this kind of instrument. In a pilot study on data from a core measured with the shipboard magnetometer, we observe some large distortions, particularly in declination, that are attributable to uncorrected instrumental effects.

Parker, RL.  2000.  Calibration of the pass-through magnetometer—I. Theory. Geophysical Journal International. 142:371-383.   10.1046/j.1365-246x.2000.00171.x   AbstractWebsite

By studying a simple model of a pass-through magnetometer we show that there are circumstances in which misleading results might arise if the spatial sensitivity of the instrument is not properly corrected. For example, if the core sample is not correctly centred, or the magnetometer itself is misaligned, serious distortion can appear in the inferred inclination distribution. The possibility of such errors warrants a thorough study of laboratory instruments and, as a first step, we require a spatial calibration, that is, an estimate of the sensitivity of the various coils to samples placed anywhere in the sensing region. Only when this information is available for laboratory magnetometers will it be possible to calculate suitable corrections. The fact that laboratory magnetometers employ superconducting material makes inferring the response from the geometry of the coils impractical because the field from a specimen is modified inside the instrument by image currents flowing in the superconducting elements. To overcome this obstacle we treat a very general calibration problem. We show that the sensitivity of a particular coil as a function of position obeys Laplace's equation, and therefore the description in space of the sensitivity is mathematically exactly the same as modelling the geomagnetic field. A calibration experiment consists of several hundred measurements performed on a tiny dipole sample, systematically positioned throughout the sensing volume of the instrument. From such observations we aim to construct a harmonic interpolating function that represents the response in the measurement region. The natural geometry for the problem is that of a cylinder, so we work from the cylindrical harmonic expansion of an equivalent magnetic field. Cylindrical harmonic expansions take the form of an infinite set of unknown functions, not just a collection of coefficients as with spherical harmonics. To build a suitable interpolating function from them we appeal to the principles of spline interpolation by constructing a model that minimizes some measure of response complexity. We examine in detail two such measures. The first corresponds to magnetic field energy; the second is a more abstract norm that smoothes more heavily than the energy norm, and whose Gram matrix elements can be found without recourse to lengthy numerical procedures. The second norm promises to form the basis of a practical programme of calibration.

Parker, RL.  2010.  Can a 2-D MT frequency response always be interpreted as a 1-D response? Geophysical Journal International. 181:269-274.   10.1111/j.1365-246X.2010.04512.x   AbstractWebsite

Weidelt and Kaikkonen showed that in the transverse magnetic (TM) mode of magnetotellurics it is not always possible to match exactly the 2-D response at a single site with a 1-D model, although a good approximation usually seems possible. We give a new elementary example of this failure. We show for the first time that the transverse electric (TE) mode responses can also be impossible to match with a 1-D response, and that the deviations can be very large.

Parker, RL.  1997.  Coherence of signals from magnetometers on parallel paths. Journal of Geophysical Research-Solid Earth. 102:5111-5117.   10.1029/96jb03803   AbstractWebsite

During a recent marine magnetic survey of the Juan de Fuca Rise, two magnetometers were towed near the seafloor, one about 300 m above the other. To understand how to interpret the records, we investigate a simple statistical model: two magnetometers moving on parallel paths above a statistically stationary source, with known spectrum. Magnetometers on paths normal to perfectly lineated magnetic anomalies will measure signals that have unit coherence at all wavelengths. Departure of the system from this ideal state can be diagnosed by a; lower coherence, and something about the across-track structure can be learned from the shape of the coherence spectrum. We calculate the power and cross spectra of the profile signals in terms of the two-dimensional power spectrum of the field just above the source region; hence we obtain the coherence and phase spectra. For the special case of a white source spectrum we find surprisingly high coherences. A set of inequalities between the spectral estimates is derived and can be used to check the consistency of the measured signals with the model assumptions. The theory is applied to a magnetic traverse of the Juan de Fuca Rise when two near-bottom magnetometers were deployed. The key results are these: in the wavelength range above about 1 km the observed coherency is substantially higher than that from the disordered field model, consistent with the highly lineated structures observed at the surface over all ocean ridge systems. On scales between 500 m and 1 km the coherence falls to levels indistinguishable from those given by an isotropic flat spectrum, implying that on these scales there is little or no across-track lineation. This finding means that the resolution of paleomagnetic field behavior based on seafloor data in this area is no better than 36,000 years.

Prieto, GA, Thomson DJ, Vernon FL, Shearer PM, Parker RL.  2007.  Confidence intervals for earthquake source parameters. Geophysical Journal International. 168:1227-1234.   10.1111/j.1365-246X.2006.03257.x   AbstractWebsite

We develop a method to obtain confidence intervals of earthquake source parameters, such as stress drop, seismic moment and corner frequency, from single station measurements. We use the idea of jackknife variance combined with a multitaper spectrum estimation to obtain the confidence regions. The approximately independent spectral estimates provide an ideal case to perform jackknife analysis. Given the particular properties of the problem to solve for source parameters, including high dynamic range, non-negativity, non-linearity, etc., a log transformation is necessary before performing the jackknife analysis. We use a Student's t distribution after transformation to obtain accurate confidence intervals. Even without the distribution assumption, we can generate typical standard deviation confidence regions. We apply this approach to four earthquakes recorded at 1.5 and 2.9 km depth at Cajon Pass, California. It is necessary to propagate the errors from all unknowns to obtain reliable confidence regions. From the example, it is shown that a 50 per cent error in stress drop is not unrealistic, and even higher errors are expected if velocity structure and location errors are present. An extension to multiple station measurement is discussed.

Gill, AE, Parker RL.  1970.  Contours of “h cosec θ” for the world's oceans. Deep-Sea Research. 17:823-&.   10.1016/0011-7471(70)90044-6   AbstractWebsite

Contours of d = h cosec θ are presented for the worlds oceans, where h is the depth of the ocean and θ the latitude. This quantity is the distance between the ocean surface and the ocean floor in the direction of the axis of rotation of the earth. The inverse is proportional to 2Ω/d = f/h where Ω is the rate of rotation of the earth and f = 2Ω sinθ is the Coriolis parameter. The quantity f/h may be interpreted as the potential vorticity of the ocean in the absence of motion relative to the rotating earth.

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Constable, C, Parker R.  1991.  Deconvolution of long-core paleomagnetic measurements: Spline therapy for the linear problem. Geophysical Journal International. 104:453-468.   10.1111/j.1365-246X.1991.tb05693.x   AbstractWebsite

The magnetization of long cores of sedimentary material is often measured in a pass-through magnetometer, whose output is the convolution of the desired function with the broad impulse response of the system. Because of inevitable measurement noise and the inherent poor conditioning of the inverse problem, any attempt to estimate the true magnetization function from the observations must avoid unnecessary amplification of small-scale features which would otherwise dominate the model with deceptively large undulations. We propose the construction of the smoothest possible magnetization model satisfying the measured data to within the observational error. By means of a cubic spline basis in the representations of both the unknown magnetization and the empirically measured response, we facilitate the imposition of maximum smoothness on the unknown magnetization. For our purposes, the smoothest model is the one with the smallest 2-norm of the second derivative, the same criterion used in the construction of cubic spline interpolators. The approach is tested on a marine core that was subsequently sectioned and measured in centimetre-sized individual specimens, with highly satisfactory results. An empirical estimate of the resolution of the method indicates a three-fold improvement in the processed record over the original signal. We illuminate the behaviour of the numerical scheme by showing the relation between our smoothness-maximizing procedure and a more conventional filtering approach. Our solution can indeed be approximated by convolution with a special set of weights, although the approximation may be poor near the ends of the core. In an idealized system we study the question of convergence of the deconvolution process, by whether the model magnetization approaches the true one when the experimental error and other system parameters are held constant, while the spacing between observations is allowed to become arbitrarily small. We find our procedure does in fact converge (under certain conditions) but only at a logarithmic rate. This suggests that further significant improvement in resolution cannot be achieved by increased measurement density or enhanced observational accuracy.

Parker, RL.  1971.  The determination of seamount magnetism. Geophysical Journal of the Royal Astronomical Society. 24:321-324.   10.1111/j.1365-246X.1971.tb02181.x   AbstractWebsite

Using a technique due to Backus the average direction of magnetization of a seamount can be found from sea-surface magnetic observations without assuming anything about the internal structure of the body. Other advantages of this approach are that the required integrals can be made very simple without approximation and that uncertainties can be estimated.

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Parker, RL, Shure L.  1982.  Efficient modeling of the Earth's magnetic field with harmonic splines. Geophysical Research Letters. 9:812-815.   10.1029/GL009i008p00812   AbstractWebsite

The construction of smooth potential field models has many geophysical applications. The recently-developed method of harmonic splines produces magnetic field models at the core surface which are maximally smooth in the sense of minimization of certain special norms. They do not exhibit the highly oscillatory fields produced by models derived from a least-squares analysis with a truncated spherical harmonic series. Modeling the data by harmonic splines requires solving a square system of equations with dimension equal to the number of data. Too many data have been collected since the 1960s for this method to be practical. We produce almost optimally smooth models by the following method. Since each spline function for the optimal model corresponds to an observation location (called a knot), we select a subset of these splines with knots well-distributed around the Earth’s surface. In this depleted basis we then find the smoothest model subject to an appropriate fit to all of the data. This reduces the computational problem to one comparable to least-squares analysis while nearly preserving the optimality inherent in the original harmonic spline models.

Parker, RL.  1967.  Electromagnetic induction in a thin strip. Geophysical Journal of the Royal Astronomical Society. 14:487-495.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1967.tb06268.x   AbstractWebsite

Electromagnetic induction in a thin strip is investigated to provide further understanding of the geomagnetic effects at an ocean edge. A solution to the equations is found by an analytic technique. It is shown that even in the case where the integrated conductivity is finite, infinite field strengths occur at the edges accompanied by rapid changes in phase.

Bullard, EC, Parker RL.  1970.  Electromagnetic induction in a thin strip. The sea Pt. 1, Vol. 4, New concepts of sea floor evolution. Regional observations concepts. :695-730., New York: Interscience Publ. Abstract
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Lowe, DAJ, Parker RL, Purucker ME, Constable CG.  2001.  Estimating the crustal power spectrum from vector Magsat data. Journal of Geophysical Research-Solid Earth. 106:8589-8598.   10.1029/2000jb900437   AbstractWebsite

The Earth's magnetic field can be subdivided into core and crustal components and we seek to characterize the crustal part through its spatial power spectrum, R-1. We process vector Magsat data to isolate the crustal field and then invert power spectral densities of flight-local components along-track for R-1 following O'Brien et al. [1999]. Our model, designated LPPC, is accurate up to approximately spherical harmonic degree 45 (lambda = 900 km): this is the resolution limit of our data and suggests that global crustal anomaly maps constructed from vector Magsat data should not contain features with wavelengths less than 900 km. We find continental power spectra to be greater than oceanic ones and attribute this to the relative thicknesses of continental and oceanic crust.

Parker, RL.  1982.  The existence of a region inaccessible to magnetotelluric sounding. Geophysical Journal of the Royal Astronomical Society. 68:165-170.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1982.tb06967.x   AbstractWebsite

Summary. The exponential attenuation of fluctuating electromagnetic fields suggests that practical magneto telluric measurements may be uninformative about the electrical conductivity at sufficiently great depths. This notion can be made precise for one-dimensional systems. Below a critical depth the conductivity function may be chosen freely without affecting the consistency of the model with the data. This depth is readily computable with quadratic or linear programming techniques and does not rely upon linearization of the equations.