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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.  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.  1996.  Improved Fourier terrain correction, Part II. Geophysics. 61:365-372.   10.1190/1.1443965   AbstractWebsite

Fourier methods for potential fields have always been developed with the simplification that the calculation surface is a level plane. The Fourier approach can be extended to deal with an uneven observation surface. I consider the case of terrain correction for gravity surveys, in which the attraction of a variable-thickness layer is calculated at points on its upper surface. The main idea is to use a power series in topographic height that is then converted into a series of convolutions. To avoid convergence problems, a cylindrical zone around the observer must be removed from the Fourier treatment and its contribution computed directly. The resultant algorithm is very fast: in an example based on a recent survey, the new method is shown to be more than 300 times faster than a calculation based on summing contributions from a column of material under each topographic grid point.

Rygaard-Hjalsted, C, Constable CG, Parker RL.  1997.  The influence of correlated crustal signals in modelling the main geomagnetic field. Geophysical Journal International. 130:717-726.   10.1111/j.1365-246X.1997.tb01866.x   AbstractWebsite

Algorithms used in geomagnetic main-field modelling have for the most part treated the noise in the field measurements as if it were white. A major component of the noise consists of the field due to magnetization in the crust and it has been realized for some time that such signals are highly correlated at satellite altitude. Hence approximation by white noise, while of undoubted utility, is of unknown validity. Langel, Estes & Sabaka (1989) were the first to evaluate the influence of correlations in the crustal magnetic field on main-field models. In this paper we study two plausible statistical models for the crustal magnetization described by Jackson (1994), in which the magnetization is a realization of a stationary, isotropic, random process. At a typical satellite altitude the associated fields exhibit significant correlation over ranges as great as 15 degrees or more, which introduces off-diagonal elements into the covariance matrix, elements that have usually been neglected in modelling procedures. Dealing with a full covariance matrix for a large data set would present a formidable computational challenge, brit fortunately most of the entries in the covariance matrix are so small that they can be replaced by zeros. The resultant matrix comprises only about 3 per cent non-zero entries and thus we can take advantage of efficient sparse matrix techniques to solve the numerical system. We construct several main-field models based on vertical-component data from a selected 5 degrees by 5 degrees data set derived from the Magsat mission. Models with and without off-diagonal terms are compared. For one of the two Jackson crustal models, k(3), we find significant changes in the main-field coefficients, with maximum discrepancies near degree 11 of about 27 per cent. The second crustal spectrum gives rise to much smaller effects for the data set used here, because the correlation lengths are typically shorter than the data spacing. k(4) also significantly underpredicts the observed magnetic spectrum around degree 15. We conclude that there is no difficulty in computing main-field models that include off-diagonal terms in the covariance matrix when sparse matrix techniques are employed; we find that there may be important effects in the computed models, particularly if we wish to make full use of dense data sets. Until a definitive crustal field spectrum has been determined, the precise size of the effect remains uncertain. Obtaining such a statistical model should be a high priority in preparation for the analysis of future low-noise satellite data.

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, Daniell GJ.  1979.  Interpretation of borehole magnetometer data. Journal of Geophysical Research. 84:5467-5479.   10.1029/JB084iB10p05467   AbstractWebsite

Samples recovered from Deep-Sea Drilling Project (DSDP) holes into the basaltic layer strongly suggest that a good model of the magnetization would be a spatially random one. To interpret future magnetometer observations made in the holes, we develop a theory relating the observed fields to the magnetization autocorrelation tensor, a function completely describing the second-order statistics of the medium. We examine special models with long- and short-range order and conclude that certain properties of the medium are highly desirable in making interpretation possible. Among these are that the autocorrelation tensor be independent of direction in a horizontal plane and that the direction of magnetization be uniform. Furthermore, useful interpretation is feasible only if three components of the magnetic field are measured, and it is preferable (but not essential) that the magnetometer is absolutely orientated about a vertical axis. We show that only in special circumstances (e.g., horizontal layering of the medium) does the magnetic field correlate directly with the average magnetization in a region surrounding the magnetometer. We analyze the natural remanent magnetization data from leg 37 of the DSDP to obtain partial information on the spatial statistics. There is no significant order vertically on scales longer than 10 m, but in two of the five holes there is correlation in the range 0.3–2 m. In three holes there is evidence of fairly good uniformity of direction, a highly desirable property for interpretational purposes. Unfortunately, drill hole samples give us no information about the behavior of the autocorrelation tensor as a function of horizontal distance.

Parker, RL.  1971.  The inverse problem of electrical conductivity in the mantle. Geophysical Journal of the Royal Astronomical Society. 22:121-138.   10.1111/j.1365-246X.1971.tb03587.x   AbstractWebsite

The general method of Backus and Gilbert is applied to the inverse problem of electrical conductivity in the mantle. The data of Banks are reworked to obtain a new model and its uncertainties. It is concluded that the levelling-off in conductivity previously obtained is only barely resolvable with current observations. A great improvement in the accuracy of the model could be expected if more precise data were available in the same frequency range.

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.  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.  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, 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.

Oldenburg, DW, Whittall KP, Parker RL.  1984.  Inversion of ocean bottom magnetotelluric data revisited. Journal of Geophysical Research. 89:1829-1833.   10.1029/JB089iB03p01829   AbstractWebsite

Three ocean bottom magnetotelluric data sets from sites on the Pacific plate are reinterpreted. The initial analysis found a correlation between the lithospheric age and the depth to a conductive zone beneath each site. That work also suggested that the resistivity increased below the conductor. This analysis, which includes new methods for constructing one-dimensional conductivity models, shows that the postulated increase in resistivity is not demanded by the data. It also reveals an unexpectedly large nonuniqueness inherent in the interpretation of these data. The previously reported trends with lithospheric age still exist, but they are not as strong as initially believed. Finally, it is shown rigorously that the different age sites are distinct in that no one-dimensional model can account for all three data sets.

Parker, RL.  1998.  Inversion of on-axis magnetic anomalies. Geophysical Journal International. 134:617-624.   10.1111/j.1365-246X.1998.tb07143.x   AbstractWebsite

The theory for recovering crustal magnetization from along-strike and, especially, axial magnetic profiles is examined. We develop a conventional Fourier technique that takes into account the special magnetic cross-section at a ridge axis including the thinning of layer 2A. Such an approach might be completely inappropriate because it is assumed that the observation path is perpendicular to all the magnetic variability, whereas in fact the path lies in the direction of least magnetic variation. To study this question and to overcome possible deficiencies, we consider a statistical modification of the theory in which the magnetization is treated as a planar stationary process in a thin layer with known power spectrum. The relationship between two signals is studied: the magnetic anomaly on a straight path at the sea surface, and the magnetization in the crust immediately under the observation track. The coherence between the two signals can be calculated, as well as the transfer function between them. We test the ideas with data from a long axial magnetic profile on the southern East Pacific Rise compiled by Gee & Kent. A model power spectrum is estimated from these data: the spectrum is red and, as expected, highly elongated perpendicular to the strike of the ridge. We find strong coherence (gamma(2) > 0.8) between the magnetic anomaly and the subtrack magnetization for wavelengths longer than 50 km, but coherence falls sharply for smaller scales. The naive, 1-D filter theory incorrectly predicts a close relationship clown to much finer scales (3 km). Calculations for hypothetical surveys off-axis predict that there is always a band of high coherence, but only for an on-axis survey does the good correlation extend to infinite wavelength. We conclude that, in a wide variety of circumstances, the magnetic anomaly and the subtrack magnetization are highly correlated in a particular wavelength interval that depends on the shape of the power spectrum.

McNutt, MK, Parker RL.  1978.  Isostasy in Australia and evolution of the compensation mechanism. Science. 199:773-775.   10.1126/science.199.4330.773   AbstractWebsite

A linear transfer function analysis has been applied to gravity and topographic data from Australia to calculate the isostatic response function of Dorman and Lewis. The Australian response function is considerably different from that calculated for the United States. The differences can be explained on the basis of an apparent evolution of the isostatic compensation mechanism in which viscoelastic creep occurs in the lithosphere and relaxes the initial long-wavelength elastic stresses.

Banks, RJ, Parker RL, Huestis SP.  1977.  Isostatic compensation on a continental scale: local versus regional mechanisms. Geophysical Journal of the Royal Astronomical Society. 51:431-452.: Blackwell Publishing Ltd   10.1111/j.1365-246X.1977.tb06927.x   AbstractWebsite

Summary. Using the techniques of linear and quadratic programming, it can be shown that the isostatic response function for the continental United States, computed by Lewis & Dorman (1970), is incompatible with any local compensation model that involves only negative density contrasts beneath topographic loads. We interpret the need for positive densities as indicating that compensation is regional rather than local. The regional compensation model that we investigate treats the outer shell of the Earth as a thin elastic plate, floating on the surface of a liquid. The response of such a model can be inverted to yield the absolute density gradient in the plate, provided the flexural rigidity of the plate and the density contrast between mantle and topography are specified. If only positive density gradients are allowed, such a regional model fits the United States response data provided the flexural rigidity of the plate lies between 1021 and 1022 N m. The fit of the model is insensitive to the mantle/ load density contrast, but certain bounds on the density structure can be established if the model is assumed correct. In particular, the maximum density increase within the plate at depths greater than 34 kin must not exceed 470 kg m−3; this can be regarded as an upper bound on the density contrast at the Mohorovicic discontinuity. The permitted values of the flexural rigidity correspond to plate thicknesses in the range 5–10 km, yet deformations at depths greater than 20 km are indicated by other geophysical data. We conclude that the plate cannot be perfectly elastic; its effective elastic moduli must be much smaller than the seismically determined values. Estimates of the stress-differences produced in the earth by topographic loads, that use the elastic plate model, together with seismically determined elastic parameters, will be too large by a factor of four or more.