Inhomogeneous Earth Models Research Articles

Coseismic gravitational curvatures changes in a spherical symmetric Earth model

The occurrence of subduction earthquakes usually leads to considerable localized mass migration changes. To improve the detection of earthquake subductions as well as the constraint of fault parameters, we derive expressions that describe changes of the gravitational curvatures (GC), i.e., the third-order derivatives of the Earth's gravitational potential, caused by a point dislocation while adopting a spherical symmetric Earth model. As a 3-D tensor matrix, the GC have twenty-seven components of those seven are independent. First, we investigate the dislocation Love numbers of the Earth's gravitational potential and derive the Green's functions of GC caused by four independent point sources in a spherical inhomogeneous Earth model. We then present the GC changes in a half-space Earth model. Furthermore, we conduct a sensitivity study by using three physical quantities that involve gravitation, gravitational gradients, and GC to compare their abilities in a seismic source depth detection. Our numerical results reveal that changes in the GC are more sensitive to a medium information about the field source compared to gravitation and gravitational gradients. This finding indicates that GC measurements could provide a more detailed information about slip fault parameters when considering a heterogeneous slip. Despite a widespread application of gravity gradients in Earth science, especially after launching the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission, measurements of the third-order derivatives of the gravitational potential have an enormous potential in the study of the solid Earth, although further work is needed in terms of instrument design and development.

An algorithm for fast 3D inversion of surface electrical resistivity tomography data: application on imaging buried antiquities

In this work a new algorithm for the fast and efficient 3D inversion of conventional 2D surface electrical resistivity tomography lines is presented. The proposed approach lies on the assumption that for every surface measurement there is a large number of 3D parameters with very small absolute Jacobian matrix values, which can be excluded in advance from the Jacobian matrix calculation, as they do not contribute significant information in the inversion procedure. A sensitivity analysis for both homogeneous and inhomogeneous earth models showed that each measurement has a specific region of influence, which can be limited to parameters in a critical rectangular prism volume. Application of the proposed algorithm accelerated almost three times the Jacobian (sensitivity) matrix calculation for the data sets tested in this work. Moreover, application of the least squares regression iterative inversion technique, resulted in a new 3D resistivity inversion algorithm more than 2.7 times faster and with computer memory requirements less than half compared to the original algorithm. The efficiency and accuracy of the algorithm was verified using synthetic models representing typical archaeological structures, as well as field data collected from two archaeological sites in Greece, employing different electrode configurations. The applicability of the presented approach is demonstrated for archaeological investigations and the basic idea of the proposed algorithm can be easily extended for the inversion of other geophysical data.

Effects of lateral inhomogeneity in a spherical Earth on gravity Earth tides

Molodenskiy (1977, 1980) presented the disturbance method to deal with the effect of a lateral inhomogeneous structure on tidal gravimetric factors. Those studies introduced expressions to calculate the effects caused by seismic waves, but ignored the contribution of lateral density changes. The present study develops the theory by reformulating the expressions with new notation and calculation. We present novel expressions for computing the effects of lateral density inhomogeneity and introduce a new three‐dimensional lateral inhomogeneous Earth model (Zhao, 2001) to compute these effects. Our numerical results show that the effects of density are of the same level as those of seismic waves: they are not negligible. The effects of the lateral inhomogeneous structure calculated for the real three‐dimensional inhomogeneous model are much less, by a factor of about 0.2, than those of the simple Ocean‐Land model presented in Molodenskiy and Kramer (1980). Collecting contributions from the seismic wave and density models, we obtain the completed total effect of the real three‐dimensional inhomogeneous Earth structure on semidiurnal gravimetric factors, with a magnitude of about −0.16 to 0.1%. This result is less than, but almost of the same order as that of Earth’s elliptical effect (ca. 0.7%; Dehant, 1995). Finally, we calculate the corresponding effects on tidal gravity for all three kinds of Earth tide: semidiurnal, diurnal, and long period ones. Compared to the tidal gravity changes, the gravity variations caused by the increments are about 0.15% for the semidiurnal tide and 0.1% for the diurnal and long period tides.

Horizontal magnetic dipole over an inhomogeneous earth model with exponential variation of conductivity

The paper outlines the formulation of the problem of calculating the electromagnetic field components due to a horizontal magnetic dipole placed over a multilayered earth model with one of the layers having exponential variation of conductivity with depth. Analytical solutions and numerical computations are performed for three-layered earth models possessing the described conductivity variation in the transition layer. It is assumed that the conduction currents dominate the displacement currents. Results presented here show the influence of the transition layer thickness and the conductivity contrast between the top and the bottom layers on electric and magnetic field components. The results show a characteristic dependence on the conductivity inhomogeneity.

Modes to replace transitional asymptotic ray fields in a vertically inhomogeneous earth model

Summary. Asymptotic ray theory (ART) fails in transition regions near critically reflected, bottom glancing or caustic-forming rays in a vertically inhomogeneous layered earth. These deficiencies are repaired here by replacing the transitional ray fields with guided modes plus truncation remainders. Exact ray-mode equivalences and their high-frequency asymptotic approximations are formulated, and their validity and efficiency are verified by numerical comparisons for SH motion in a two-layer earth model comprised of an inhomogeneous sediment above an homogeneous semi-infinite bedrock.

Theoretical body wave interactions with upper mantle structure

Long‐period seismograms are generated by a spectral method for body waves interacting with the upper mantle discontinuities that define the low‐velocity and transition zones of the spherically symmetric but radially inhomogeneous earth models, PEM‐C and T7. A complex velocity profile incorporates the SL8 Q model. Models PEM‐C and T7 predict distinct differences in displacement waveforms for P waves that interact with the 400‐ and 670‐km discontinuities. The waveforms demonstrate that later arrivals such as interference head waves can be important constraints upon earth models. Theoretical seismograms were computed by a spectral method that modeled the velocity profile with arbitrarily thick inhomogeneous layers (full wave theory). The seismograms agreed well with those generated by methods that approximate the velocity profile with thin homogeneous layers (the Cagniard and reflectivity methods) provided the displacements were convolved with source‐time functions and a long‐period instrument response. Minor differences among the results of the methods do not affect the interpretation of observed waveform data but do help define the limitations of the methods. For example, the thin‐layer methods suffer from numerical limitations on layer thickness and the number of layers needed to approximate the earth model. The full wave theory may occasionally also require the use of thin layers when the zero‐order asymptotic theory it uses fails to describe wave propagation properly in layers having rapid velocity variation with depth.

Eigenvibrations of laterally inhomogeneous Earth models

Summary. A realistic three-dimensional Earth having both radial and lateral variations in seismic properties has been modelled using finite elements. Numerical algorithms are produced that yield efficient estimations of the non-degenerate eigenfunctions and eigenfrequencies of this laterally in-homogeneous spheroid. These serve as quite independent alternatives to previous mathematical methods used to study this problem. The algorithms rely upon Rayleigh's Variational Principle to solve the finite element eigen-problem. The Earth model used is perfectly elastic and non-rotating. The finite element formulation is demonstrated numerically in the case of eigenvibrations of low order torsional oscillations, 0T2 to 0T10, but also applies to the spheroidal oscillations. Theoretical splitting of the eigenfrequencies due to the Earth's ellipticity and to the structural contrasts between continents and oceans is obtained. Eigenfunctions of a realistic laterally inhomogeneous Earth model are computed for this suite of torsional oscillations. It is seen that the relative positions of the continents establish patterns of displacement which may provide explanations for the variations in eigen-spectra observed from one seismographic station to another. The deviations in eigenfrequencies due to the structural contrasts between the continents and the oceans are found to increase from 0.01 per cent of the unperturbed frequency for 0T2 to 0.1 per cent for 0T10. Over the same range of eigenvibrations, the effects on the eigenfrequencies of the Earth's ellipticity are about 0.2 per cent. The amplitudes of the eigenfunctions of these oscillations are seen to have significant global variation. The lower order modes show a latitudinal dependence of the amplitudes that would be expected were the Earth a simple ellipsoid. The higher order modes have more longitudinal amplitude dependence as a result of the increased perturbation effects of the continents. This work demonstrates that the finite element method can be used as a practical scheme for calculating theoretical eigenspectra for global seismographic stations.

The Calculation of the Electromagnetic Fields of a Sheet Current Source with Arbitrary Spatial Intensity Distribution over a Layered Half Space--I. The General Method and Results

Summary Electromagnetic induction by a non-uniform source over a layered halfspace is studied. A method is described for solving such problems when the current intensity distribution is such that no analytic integration to obtain the field quantities is possible. A number of models are considered using a general computer program. Apparent resistivity curves are calculated and compared with previous results. Phase curves and field component profiles are calculated. Increasing interest has been shown in modelling and interpretation of electromagnetic induction anomalies during the past 15 years. Several approaches to the problem have been employed, including analytical, analogue and numerical techniques. Analytical techniques are limited in that only particular models of simple geometry may be considered. Models of more complex geometry may be studied by analogue techniques but such experimental measurements are often limited by edge effects and the range of conductivities that may be employed. The numerical techniques provide a versatile method which may be used for wide conductivity ranges and to approximate complex geometries. Studies to date have included calculations for either uniform or specific simple source geometries (e.g. a line current source or a horizontal or vertical dipole). Hermance & Peltier (1970) solved the induction problem for a line current source in the manner suggested by Price (1950). This work was then extended by Peltier & Hermance (1971) to a sheet current source with an intensity of Gaussian shape. Hibbs & Jones (1973a, b) generalized the method used by Peltier & Hermance to include non-symmetric as well as symmetric source configurations. Laterally inhomogeneous earth models were also studied by combining the approach of Peltier & Hermance with the numerical modelling method of Jones & Price (1970) as programmed by Jones & Pascoe (1971) and Pascoe & Jones (1972). The above work was limited in that only source configurations for which the source coefficient C(s) as described by Peltier & Hermance (1971) are obtained

Recent analytical investigations of electromagnetic ground wave propagation over inhomogeneous earth models

A consolidated review is presented of recent analytical studies of electromagnetic waves propagating over inhomogeneous surfaces. Emphasis is on smooth boundaries that can be characterized by a local surface impedance. A general integral equation formulation is developed for this situation. A number of special cases are then considered and various methods of solution are described. Various concrete examples are presented, particular with regard to effects that occur at coastlines. Extensions to certain types of terrain features are also treated using the closely related mode-matching method. Some controversial aspects of very recent work on the subject are descried briefly.

Propagation of Electromagnetic Waves over a Smooth Multisection Curved Earth—An Exact Theory

In this paper, we present a self-contained exposition of ground-wave propagation over a smooth inhomogeneous earth model. A factorization procedure is used to obtain an exact solution for a 2-dimensional cylindrical model which may have any number of homogeneous sections. It is demonstrated that, under most conditions in practice, the solution for the mode conversion coefficients can be simplified to a form which was derived earlier by an application of the compensation theorem. Formulas are given which permit the refinement of earlier approximate calculations if such accuracy is required.

A time integration method for computation of the intensities of seismic rays

abstract A finite-difference time integration method for the calculation of seismic ray intensity is developed. Discontinuities in the depth derivatives of the velocity-depth function at layer boundaries cause anomalies in the intensity distance curves calculated using the standard integral formulation. The time integration method overcomes these difficulties. Calculations for a simple analytic case and a Gutenberg earth model demonstrate the difficulties with the standard integral method and the superior performance of the time integration scheme. The method may also be applied to laterally inhomogeneous earth models.

Near-surface dispersion studies at Tonto Forest Seismological Observatory

abstract Four local seismic events digitally recorded on a crossed-linear array have been analyzed to evaluate the representation of wave propagation in inhomogeneous earth models in terms of “local” dispersion. The analysis techniques studied are based primarily on the correlation properties of the signals at various receivers. Frequency-wave-number spectra are computed for the events by various methods, and a comparison of the results indicates that the best method is a combination using averaged correlations computed from prewhitened data. The utility of computing dispersion curves from an estimation of the “transfer function” has been evaluated. The studies show the necessity of recording data directly “inline” in areas of rough topography between source and array if “local” dispersion is to be measured. Multichannel prediction studies indicate that the signals are highly nonstationary in space.