Continuation of Three‐dimensional Potential Fields Measured on an Uneven Surface†
01 Aug 1974-Geophysical Journal International (Blackwell Publishing Ltd)-Vol. 38, Iss: 2, pp 299-314
TL;DR: In this paper, the authors describe a method for the continuation of 3D potential fields measured on an uneven surface, which is based on a technique proposed earlier by the authors: first they represent potential functions as a sum of elementary interpolating functions; then an inversion technique gives the continued field as a very simple linear combination of the observed values.
Abstract: Summary This paper describes a method for the continuation of three-dimensional potential fields measured on an uneven surface; this method is based on a technique proposed earlier by the authors : first we represent potential functions as a sum of elementary interpolating functions; then an inversion technique gives the continued field as a very simple linear combination of the observed values. The method is of interest in several geophysical problems: for example, aeromagnetic surveys made at different altitudes can be joined with no edge effect. This case and others, both theoretical and real, are presented in the paper.
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TL;DR: In this article, a new approach to the topographic correction for terrestrial heat flow measurements is presented, which features calculation of a Fourier series fit to the surface temperature-surface elevation data where the surface temperatures are based on a model that includes surface temperature variations due to microclimate variations.
Abstract: A new approach to the topographic correction for terrestrial heat flow measurements is presented. The approach features calculation of a Fourier series fit to the surface temperature-surface elevation data where the surface temperatures are based on a model that includes surface temperature variations due to microclimate variations. The mathematics of the terrain correction problem are similar to the upward (away from source) continuation problem in gravity and magnetics so several solutions, in addition to the Fourier series approach, are available in the literature that allow an accurate calculation of the correction provided the surface boundary condition is properly specified. However, the usual boundary condition applied, a linear relation between ground surface temperature and elevation, is shown to be inadequate for drill holes in the depth range 30–200 m no matter how low the topographic relief. Thus a model of ground surface temperature is developed that includes the effects of elevation, slope orientation, and slope angle. Because of the effects of microclimate, the classical models that have isothermal surfaces that generally parallel the topographic surface are significantly in error in many cases, and the patterns of isotherms near the topographic surface are more complicated than was previously recognized. This complexity causes gradient variations with depth in 30- to 200-m holes that have not been previously recognized as being related to topographic effects. Because the temperature effects of slope orientation and inclination do not scale with respect to the magnitude of the relief, significant terrain corrections may be required even in areas of relatively low relief. The application of the technique is illustrated by application to a line dipole hill and a group of drill holes near Wilbur, Washington. In addition, several examples of two-dimensional terrain effects and one example of three-dimensional terrain effects are illustrated for topographic sections in the northwestern United States. In the United States, most ‘anomalous’ gradients in the upper 100–200 m of drill holes in impermeable rocks can be explained by a combination of topographic and microclimatic effects, without resorting to temporal climatic changes or unknown types of water effects. The depth of the holes necessary for reliable heat flow measurements in such settings is a signal to noise problem where the noise is the effect at depth of the microclimatically related surface temperature variations, coupled with the topographic effect, and the signal is a temperature increase at any depth due to the background geothermal gradient. Typically, the noise has decreased to a few degrees centigrade per kilometer within the depth range 100–200 m. Thus the general conclusion has been that these depths of holes are required for reliable heat flow values. In fact, when linear temperature-depth data are observed in shallower holes or when appropriate corrections are made, reliable measure ments in impermeable rocks may be consistently made in holes no deeper than 100 m.
141 citations
01 Jan 1988
TL;DR: In this paper, the authors consider the thermal regime of the Earth to be quasi-steady state over times of the order of the thermal time constant for the crust, that is a few hundred thousand to a few million years.
Abstract: Although on the geological time and space scales the geothermal regime of the Earth is, strictly speaking, both a transient and three dimensional phenomenon, on the global scale by far the most important component of heat transfer is radial. Furthermore, over times of the order of the thermal time constant for the crust, that is a few hundred thousand to a few million years, we consider the thermal regime to be quasi-steady state.
135 citations
TL;DR: In this article, an equivalent source algorithm is described for continuing either one-dimensional or two-dimensional potential fields between arbitrary surfaces, where the dipole surface is approximated as a set of plane faces with constant moments over each face.
Abstract: An equivalent source algorithm is described for continuing either one‐ or two‐dimensional potential fields between arbitrary surfaces. In the two‐dimensional case, the dipole surface is approximated as a set of plane faces with constant moments over each face. In the one‐dimensional case, the plane faces of the dipole surface reduce to straight line segments. Application of the algorithm to model and field examples of aeromagnetic data shows the method to be effective and accurate even when the terrain has strong topographic relief and is composed of highly magnetic volcanic rocks.
83 citations
28 Jan 2014
TL;DR: In this paper, a Gaussian linear inversion approach is proposed to extract exhumation rates from spatially distributed low temperature thermochronometric data, which is based on a linear forward model.
Abstract: . We present a formal inverse procedure to extract exhumation rates from spatially distributed low temperature thermochronometric data. Our method is based on a Gaussian linear inversion approach in which we define a linear problem relating exhumation rate to thermochronometric age with rates being parameterized as variable in both space and time. The basis of our linear forward model is the fact that the depth to the "closure isotherm" can be described as the integral of exhumation rate, ..., from the cooling age to the present day. For each age, a one-dimensional thermal model is used to calculate a characteristic closure temperature, and is combined with a spectral method to estimate the conductive effects of topography on the underlying isotherms. This approximation to the four-dimensional thermal problem allows us to calculate closure depths for data sets that span large spatial regions. By discretizing the integral expressions into time intervals we express the problem as a single linear system of equations. In addition, we assume that exhumation rates vary smoothly in space, and so can be described through a spatial correlation function. Therefore, exhumation rate history is discretized over a set of time intervals, but is spatially correlated over each time interval. We use an a priori estimate of the model parameters in order to invert this linear system and obtain the maximum likelihood solution for the exhumation rate history. An estimate of the resolving power of the data is also obtained by computing the a posteriori variance of the parameters and by analyzing the resolution matrix. The method is applicable when data from multiple thermochronometers and elevations/depths are available. However, it is not applicable when there has been burial and reheating. We illustrate our inversion procedure using examples from the literature.
52 citations
TL;DR: In this article, the problem of upward and downward continuation of a two-dimensional field to a level surface everywhere below the observation locations has been studied, and the solution estimates must be weighted averages of the field not only on this level, but also on a line passing between the observations and sources.
Abstract: Summary. The formalism of Backus & Gilbert is applied to the problems of upward and downward continuation of harmonic functions. We first treat downward continuation of a two-dimensional field to a level surface everywhere below the observation locations; the calculation of resolving widths and solution estimates is a straightforward application of Backus-Gilbert theory. The extension to the downward continuation of a three-dimensional field uses a delta criterion giving resolving areas rather than widths. A feature not encountered in conventional Backus-Gilbert problems is the requirement of an additional constraint to guarantee the existence of the resolution integrals. Finally, we consider upward continuation of a two-dimensional field to a level above all observations. We find that solution estimates must be weighted averages of the field not only on this level, but also on a line passing between the observations and sources. Weighting on the lower line may be traded off against resolution on the upper level.
43 citations
References
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TL;DR: In this paper, the authors provide a broad overview of Fourier Transform and its relation with the FFT and the Hartley Transform, as well as the Laplace Transform and the Laplacian Transform.
Abstract: 1 Introduction 2 Groundwork 3 Convolution 4 Notation for Some Useful Functions 5 The Impulse Symbol 6 The Basic Theorems 7 Obtaining Transforms 8 The Two Domains 9 Waveforms, Spectra, Filters and Linearity 10 Sampling and Series 11 The Discrete Fourier Transform and the FFT 12 The Discrete Hartley Transform 13 Relatives of the Fourier Transform 14 The Laplace Transform 15 Antennas and Optics 16 Applications in Statistics 17 Random Waveforms and Noise 18 Heat Conduction and Diffusion 19 Dynamic Power Spectra 20 Tables of sinc x, sinc2x, and exp(-71x2) 21 Solutions to Selected Problems 22 Pictorial Dictionary of Fourier Transforms 23 The Life of Joseph Fourier
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3,156 citations
TL;DR: In this article, the authors show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail, and the shortest length scale which the given data can resolve at any particular depth.
Abstract: A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of interia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail; and how to determine the shortest length scale which the given data can resolve at any particular depth. We apply the general theory to the linear problem of finding the depth-variation of a frequency-independent local Q from the observed quality factors Q of a finite number of normal modes. We also apply the theory to the non-linear problem of finding density vs depth from the total mass, moment, and normal-mode frequencies, in case the compressional and shear velocities are known.
1,371 citations
TL;DR: In this article, it was shown that a given set G of measured gross Earth data permits such a construction of localized averages, and if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount.
Abstract: A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We suppose that a finite set G of gross Earth data has been measured, that the measurements are inaccurate, and that the variance matrix of the errors of measurement can be estimated. We show that some such sets G of measurements determine the structure of the Earth within certain limits of error except for fine-scale detail. That is, from some setsG it is possible to compute localized averages of the Earth structure at various depths. These localized averages will be slightly in error, and their errors will be larger as their resolving lengths are shortened. We show how to determine whether a given set G of measured gross Earth data permits such a construction of localized averages, and, if so, how to find the shortest length scale over which G gives a local average structure at a particular depth if the variance of the error in computing that local average from G is to be less than a specified amount. We apply the general theory to the linear problem of finding the depth variation of a frequency-independent local elastic dissipation ( Q ) from the observed damping rates of a finite number of normal modes. We also apply the theory to the nonlinear problem of finding density against depth from the total mass, moment and normal-mode frequencies, in case the compressional and shear velocities are known.
1,291 citations
TL;DR: In this paper, the authors prove that the collection of Earth models which yield the physically observed values of any independent set of gross Earth data is either empty or infinite dimensional, and exploit this very high degree of non-uniqueness in real geophysical inverse problems to generate computer programs which iteratively produce Earth models to fit given gross earth data and satisfy other criteria.
Abstract: Summary
A gross datum of the Earth is a single measurable number describing some property of the whole Earth, such as mass, moment of inertia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We prove that the collection of Earth models which yield the physically observed values of any independent set of gross Earth data is either empty or infinite dimensional. We exploit this very high degree of non-uniqueness in real geophysical inverse problems to generate computer programs which iteratively produce Earth models to fit given gross Earth data and satisfy other criteria. We describe techniques for exploring the collection of all Earth models which fit given gross Earth data. Finally, we apply the theory to the normal modes of elastic-gravitational oscillation of the Earth.
867 citations