Export 51 results:
Sort by: [ Author  (Asc)] Title Type Year
A B C D E F G H I J K L M N O [P] Q R S T U V W X Y Z   [Show ALL]
Parker, RL, Denham CR.  1979.  Interpolation of unit vectors. Geophysical Journal of the Royal Astronomical Society. 58:685-687.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1979.tb04802.x   AbstractWebsite

Summary. Time series of unit vectors occur in geophysics as palaeomagnetic poles or poles of relative motion in plate tectonics, and it is often required to trace a smooth curve through the individual points. A simple method is given for interpolating such time series based on cubic splines. The curve obtained is smooth (e.g. possesses continuous curvature) and does not depend on the choice of coordinate axes. An extension with the same desirable properties is given for the case where the given data are inexact.

Parker, RL, Oldenburg DW.  1973.  Thermal model of ocean ridges. Nature Physical Science. 242:137-139.   10.1038/physci242137a0   AbstractWebsite

McKenzie's model of crustal creation at the ocean ridges1,2 and its derivatives3,4 predicts such features as the topography and high heat flow of the ridges. In spite of this success there are some unsatisfactory aspects of the model; for example, the arbitrary temperature distribution in the intrusive zone gives rise to infinite heat generation and the lithospheric thickness is a free parameter not determined by the physics. We offer here a simple refinement of McKenzie's model that overcomes these difficulties. The essential difference stems from the inclusion of terms in the boundary conditions to account for the evolution of latent heat in places where the plate is growing. We first describe the physical basis of the model.

Parker, RL, Booker JR.  1996.  Optimal one-dimensional inversion and bounding of magnetotelluric apparent resistivity and phase measurements. Physics of the Earth and Planetary Interiors. 98:269-282.   10.1016/s0031-9201(96)03191-3   AbstractWebsite

The properties of the log of the admittance in the complex frequency plane lead to an integral representation for one-dimensional magnetotelluric (MT) apparent resistivity and impedance phase similar to that found previously for complex admittance. The inverse problem of finding a one-dimensional model for MT data can then be solved using the same techniques as for complex admittance, with similar results. For instance, the one-dimensional conductivity model that minimizes the chi(2) misfit statistic for noisy apparent resistivity and phase is a series of delta functions. One of the most important applications of the delta function solution to the inverse problem for complex admittance has been answering the question of whether or not a given set of measurements is consistent with the modeling assumption of one-dimensionality. The new solution allows this test to be performed directly on standard MT data, Recently, it has been shown that induction data must pass the same one-dimensional consistency test if they correspond to the polarization in which the electric field is perpendicular to the strike of two-dimensional structure, This greatly magnifies the utility of the consistency test. The new solution also allows one to compute the upper and lower bounds permitted on phase or apparent resistivity at any frequency given a collection of MT data, Applications include testing the mutual consistency of apparent resistivity and phase data and placing bounds on missing phase or resistivity data, Examples presented demonstrate detection and correction of equipment and processing problems and verification of compatibility with two-dimensional B-polarization for MT data after impedance tensor decomposition and for continuous electromagnetic profiling data.

Parker, RL.  1984.  The inverse problem of resistivity sounding. Geophysics. 49:2143-2158.   10.1190/1.1441630   AbstractWebsite

The electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two‐layer systems. This new representation is called the bilayer expansion for the Green’s function. It enables the forward problem of resistivity sounding to be solved very efficiently, even for complicated profiles. Also, the bilayer expansion facilitates the solution of the resistivity inverse problem: the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model. Specifically, I consider models comprising uniformly conducting layers of equal thickness; for a given finite data set a quadratic program can be used to find the best‐fitting model in this class for any specified thickness. As the thickness is reduced, models of this kind can approximate arbitrary profiles with unlimited accuracy. If there is a model that satisfies the data well, there are other models equally good or better whose variation takes place in an infinitesimally thin zone near the surface, below which there is a perfectly conducting region. This extraordinary class of solutions underscores the serious ambiguity in the interpretation of apparent resistivity data. It is evident that strong constraints from outside the electrical data set must be applied if reliable solutions are to be discovered. Previous work seems to have given a somewhat overly optimistic impression of the resolving abilities of this kind of data. I consider briefly a regularization technique designed to maximize the smoothness of models found with the bilayer inversion.

Parker, RL.  1977.  Understanding inverse theory. Annual Review of Earth and Planetary Sciences. 5:35-64.   10.1146/annurev.ea.05.050177.000343   AbstractWebsite
Parker, RL.  2003.  Ideal bodies for Mars magnetics. Journal of Geophysical Research-Planets. 108   10.1029/2001je001760   AbstractWebsite

[1] The high-amplitude magnetic anomalies observed by the Mars Global Surveyor imply the presence of a large intensity of magnetization in the Martian crust. We investigate the mathematical question of determining the distribution of magnetization that has the smallest possible intensity, without any assumptions about the direction of magnetization. The greatest lower bound on intensity found in this way depends on an assumed layer thickness. An analytical expression is discovered for the optimal magnetization, and numerical methods are described for solving the equations that determine the distribution. Some relatively small scale numerical calculations illustrate the theory. These calculations enable us to conclude, for example, that if the magnetization of Mars is confined to a 50-km thick layer, it must be magnetized with an intensity of at least 4.76 A/m.

Parker, RL.  1972.  Inverse theory with grossly inadequate data. Geophysical Journal of the Royal Astronomical Society. 29:123-138.   10.1111/j.1365-246X.1972.tb02203.x   AbstractWebsite

When only a few observations are available as data for an inverse problem, it is proposed that the best way to use them is to obtain bounds on various functionals of the structure. To do this, the model is found that has the smallest (or largest) value of the functional. In this way, for example, equations are derived for finding the model value that is exceeded somewhere by all structures satisfying the data, and thus this value must be exceeded in the Earth itself. The same techniques can be used to derive conditions for the existence of a solution, when a certain data set is given; this is an important problem in non-linear inverse theory.Three examples are given, including the non-linear problem of electrical conductivity in the mantle. There, one- and two-data problems are solved and, by means of the existence theory, self-consistency criteria are defined for amplitude and phase measurements and for amplitude measurements at two different frequencies.

Parker, PR, Zumberge MA, Parker RL.  1995.  A new method for fringe-signal processing in absolute gravity meters. Manuscripta Geodaetica. 20:173-181. AbstractWebsite

In all modern absolute gravity meters, an interferometer illuminated with a stabilized laser tracks the motion of a freely falling retroreflector. The value of gravity is measured by timing the passage of interference fringes. Typically, the sinusoidal fringe signal is converted to a series of pulses, a subset of which are input to a time digitizer. In our new system, the fringe signal is digitized with a fast analog-to-digital converter and fit to an increasing-frequency sine wave. In addition to being smaller and less expensive, the system should eliminate some potential systematic errors that may result from imperfect zero-crossing discrimination and pulse pre-scaling.

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.

Parker, RL.  2011.  New analytic solutions for the 2-D TE mode MT problem. Geophysical Journal International. 186:980-986.   10.1111/j.1365-246X.2011.05091.x   AbstractWebsite

A closed-form solution is given for a 2-D, transverse electric mode, magnetotelluric (MT) problem. The model system consists of a finite vertical thin conductor with variable integrated conductivity over a perfectly conducting base. A notable property of the solution is that the frequency response possesses a single pole in the complex plane. Systems with finitely many resonances play a central role in the 1-D MT inverse problem based on finite data sets, but until now, no 2-D system of this kind was known. The particular model is shown to be just one of a large class of thin conductors with same the property, and further examples are given. The solutions of the induction problem for members of this family can often be written in compact closed form, making them the simplest known solutions to the 2-D MT problem.

Parker, RL, Huestis SP.  1974.  The inversion of magnetic anomalies in the presence of topography. Journal of Geophysical Research. 79:1587-1593.   10.1029/JB079i011p01587   AbstractWebsite

The inversion of magnetic anomalies in terms of an irregular layer of magnetized material is studied, and an efficient procedure for constructing solutions is developed. Even when magnetic orientation and layer thickness are known, the solution is not unique because of the existence of a magnetization (called the magnetic annihilator) that produces no observable magnetic field. We consider an example of near-bottom marine data and discuss methods for overcoming the problem of nonuniqueness.

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.  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.

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.

Parker, RL.  1980.  The inverse problem of electromagnetic induction: Existence and construction of solutions based on incomplete data. Journal of Geophysical Research. 85:4421-4428.   10.1029/JB085iB08p04421   AbstractWebsite

A theory is described for the inversion of electromagnetic response data associated with one-dimensional electrically conducting media. The data are assumed to be in the form of a collection of (possibly imprecise) complex admittances determined at a finite number of frequencies. We first solve the forward problem for conductivity models in a space of functions large enough to include delta functions. Necessary and sufficient conditions are derived for the existence of solutions to the inverse problem in this space. The approach relies on a representation of real-part positive functions due to Cauer and an application of Sabatier's theory of constrained linear inversion. We find that delta-function models are fundamental to the problem. When existence of a solution has been established for a given set of data, actual conductivities fitting the measurements may be explicitly constructed for various special classes of functions. For a solution in delta functions or homogeneous layers a development as a continued fraction is the essential idea; smoothly varying models are found with an adaption of Weidelt's analytic solution.

Parker, RL.  1974.  A new method for modeling marine gravity and magnetic anomalies. Journal of Geophysical Research. 79:2014-2016.   10.1029/JB079i014p02014   AbstractWebsite

A very fast technique involving Fourier transformation can find the gravity or magnetic anomaly of an irregular crustal model as observed on a plane above the material. It is shown how the method can be used to invert the magnetic field data to obtain a magnetization model, but the model so obtained is not unique. The normal restrictions placed on the magnetization models lead to a family of solutions with one degree of freedom.

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.

Parker, RL, Shure L.  1985.  Gravitational and magnetic fields of some simple solids of revolution. Geophysical Journal of the Royal Astronomical Society. 80:631-647.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1985.tb05115.x   AbstractWebsite

Summary. Exact spherical harmonic expansions are given for calculating the gravitational and magnetic fields associated with certain uniform solids of revolution. The figures are those made by rotating a conic section about one of its principal axes. The coefficients in the expansions can be computed accurately and efficiently and this approach leads to a very satisfactory method for calculating the fields of geological bodies with approximate circular symmetry about a vertical axis. A complete theory of convergence is given for the expansions. Somewhat unexpectedly, the sphere of convergence is determined by the location of a number of equivalent point or line sources that lie within the body or on its edges.

Parker, RL.  1977.  The Fréchet derivative for the one-dimensional electromagnetic induction problem. Geophysical Journal of the Royal Astronomical Society. 49:543-547.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1977.tb03723.x   AbstractWebsite

Summary. From consideration of the higher order terms, it is shown that the magneto-telluric response is Fréchet differentiable with respect to conductivity; this result remains valid for discontinuous profiles, which is not so in the case of the corresponding free-oscdlation problem for the elastic earth. The remainder term in the Fréchet formula is shown to be O|δσ|2 and a numerical estimate is made of the bounding constant for a restricted class of conductivity models.

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.

Parker, RL, Klitgord KD.  1972.  Magnetic upward continuation from an uneven track. Geophysics. 37:662-668.   10.1190/1.1440289   AbstractWebsite

A new method for continuing two‐dimensional potential data upward from an uneven track is developed with special emphasis on solving a particular practical problem, that of magnetic data taken near the bottom of the ocean. The method is based on the use of the Schwarz‐Christoffel transformation, which maps the original, irregular track into a horizontal straight line. It has been found to be very fast computationally and to suffer none of the restrictions found in some earlier two‐dimensional algorithms.

Parker, RL.  1995.  Improved Fourier terrain correction, Part I. Geophysics. 60:1007-1017.   10.1190/1.1443829   AbstractWebsite

A description of a new Fourier technique is given for calculating the gravitational attraction of a layer with an irregular top surface for application in the terrain correction of marine gravity surveys in shallow water. An earlier Fourier-based algorithm fails or becomes inaccurate when the peaks of the topography approach the sea surface too closely. The new approach divides the attraction into two parts: a local contribution from the material within a cylinder around each observation point and the attraction from the matter outside the cylinder. A special quadrature rule, optimized for the actual data distribution, evaluates the local contribution. The calculation of the exterior component represents the bulk of the numerical effort. Fortunately, the exterior integral possesses an expansion as a series of convolutions, and by evaluating these in the Fourier domain, the procedure can take advantage of the efficiency of the fast Fourier transform. Chebychev economization of the convolution series provides further significant improvements in computational speed. Two examples, one artificial and the other based on a survey around Guadalupe Island, illustrate the application of the new technique. Estimates of the errors from computation sources and from the inadequacies of the topographic model confirm the general accuracy of the approach, except in regions of very steep terrain.

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, Wheelock B.  2012.  Fourier domain calculation of terrain effects in marine MT. Geophysical Journal International. 189:240-250.   10.1111/j.1365-246X.2011.05350.x   AbstractWebsite

Magnetotelluric surveys on the seafloor have become an important part of marine geophysics in recent years. The distorting effects of topographic relief on the electromagnetic fields can be far-reaching, but local terrain is also important. Thus, computational techniques that can treat a large area containing fine-scale topography could find widespread application. We describe a new solution to the problem based on a well-established theory of electromagnetic induction in thin sheets. The procedure requires taking the Fourier transform of the integral equations derived by Dawson and Weaver in 1979, and by McKirdy, Weaver and Dawson in 1985. The equations in the transformed electric field are solved iteratively by a new technique. We prove the new iterative procedure is always convergent, whereas the original scheme diverges when the grid spacing of the discretization is small. We also give a means of correcting for distant features that need not be specified in as great detail. Preliminary tests confirm the new process is very efficient and that topographic data sets of several million points will be handled with ease.

Parker, RL.  1975.  The theory of ideal bodies for gravity interpretation. Geophysical Journal of the Royal Astronomical Society. 42:315-334..: Blackwell Publishing Ltd   10.1111/j.1365-246X.1975.tb05864.x   AbstractWebsite

Ambiguity in gravity interpretation is inevitable because of the fundamental incompleteness of real observations; it is, however, possible to provide rigorous limits on possible solutions even with incomplete data. In this paper a systematic theory is developed for finding such bounds, including an upper bound on depth of burial; the bounds are discovered by constructing the unique body achieving the extreme parameter, e.g. depth; such a body is called the 'ideal' one associated with the given data. Ideal bodies can also be constructed for bounding density, thickness of layer and lateral extent. General properties of ideal bodies are derived and numerical methods for modest numbers of observations are discussed. Some artificial examples, where the buried system is exactly known, are given and it is shown how relatively good bounds can be reached with only a few measurements. A Bouguer anomaly from the Swiss Alps is then considered and it is concluded that the mountain roots are unusually shallow there.