Alexei Kuvshinov

Three-dimensional induction logging problems, Part I: An integral equation solution and model comparisons

A 3‐D frequency‐domain solution based on a volume integral equation approach has been implemented to simulate induction log responses. In our treatment of the problem, we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy. The solution process uses a Krylov subspace iteration to solve the scattering equation, which is based on the modified iterative dissipative method. Internal consistency checks and comparisons with mode matching and finite‐difference solutions for vertical borehole models demonstrate the accuracy of the solution.There are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries. To simulate such responses requires the numerical solution of Maxwells equations in three dimensions along with independent tests to validate the solution approach and its accuracy. In this paper, we compare two independent 3‐D frequency‐domain solutions for the ...

A global model of mantle conductivity derived from 5 years of CHAMP, Ørsted, and SAC-C magnetic data

[1]xa0We present a global 1-D conductivity model which is obtained by analysis of five years (2001–2005) of simultaneous magnetic data from the three satellites Orsted, CHAMP and SAC-C. After removal of core and crustal fields as predicted by a recent field model we used non-polar scalar and vector observations from the night-side sector, and interpret the field residuals in terms of a large-scale contribution from the magnetospheric ring current and its induced counterpart. We then derive transfer functions between internal (induced) and external expansion coefficients of the magnetic potential and provide globally-averaged C-responses in the period range between 14 hours and 4 months. Since the satellite responses are probably influenced by induction in the oceans for periods shorter than a few days, we correct the data for this effect. Interpreting the corrected responses yields a 1-D conductivity model which is rather similar to models derived from ground-based data.

The Swarm End-to-End mission simulator study: A demonstration of separating the various contributions to Earth’s magnetic field using synthetic data

Swarm, a satellite constellation to measure Earth’s magnetic field with unpreceded accuracy, has been selected by ESA for launch in 2009. The mission will provide the best ever survey of the geomagnetic field and its temporal evolution, in order to gain new insights into the Earth system by improving our understanding of the Earth’s interior and climate. An End-to-End mission performance simulation was carried out during Phase A of the mission, with the aim of analyzing the key system requirements, particularly with respect to the number of Swarm satellites and their orbits related to the science objectives of Swarm. In order to be able to use realistic parameters of the Earth’s environment, the mission simulation starts at January 1, 1997 and lasts until re-entry of the lower satellites five years later. Synthetic magnetic field values were generated for all relevant contributions to Earth’s magnetic field: core and lithospheric fields, fields due to currents in the ionosphere and magnetosphere, due to their secondary, induced, currents in the oceans, lithosphere and mantle, and fields due to currents coupling the ionosphere and magnetosphere. Several independent methods were applied to the synthetic data to analyze various aspects of field recovery in relation to different number of satellites, different constellations and realistic noise sources. This paper gives an overview of the study activities, describes the generation of the synthetic data, and assesses the obtained results.

3-D electromagnetic induction studies using the Swarm constellation: Mapping conductivity anomalies in the Earth's mantle

An approach is presented to detect deep-seated regional conductivity anomalies by analysis of magnetic observations taken by low-Earth-orbiting satellites. The approach deals with recovery of C-responses on a regular grid and starts with a determination of time series of external and internal coefficients of the magnetic potential. From the coefficients, time series of the magnetic vertical component and of the horizontal divergence of the horizontal components are synthesized on the grid and the C-responses are determined by means of signal processing of the corresponding time series. For validation of the approach, 3 years of realistic synthetic data at simulated orbits of the forthcoming Swarm constellation of 3 satellites have been used. To obtain the synthetic data for a given 3-D conductivity Earth’s model a time-domain scheme has been applied which relies on a Fourier transformation of the inducing field, and on a frequency domain forward modelling. The conductivity model consists of a thin surface layer of realistic conductance and a 3-D mantle that incorporates a hypothetic deep regional anomaly beneath the Pacific Ocean plate. To establish the ability of the approach to capture the geometry of the mantle heterogeneities used in the forward approach, numerical experiments have been undertaken using various satellite combinations, sampling periods of the resulting time series, and numbers of internal coefficients. The possibility of the approach to map anomalies in the mantle using satellite data that contain contributions from the core and lithosphere, from the magnetosphere and ionosphere (and their Earth-induced counterparts), as well as payload noise has been investigated. The model studies have shown that C-responses obtained on a regular grid might be used to map regional deep-seated conductivity anomalies. Moreover, it has been demonstrated that these C-responses are successfully recovered from magnetic data collected by the proposed Swarm constellation of 3 satellites.

Electromagnetic induction in the oceans and the anomalous behaviour of coastal C‐responses for periods up to 20 days

[1]xa0Electromagnetic transfer functions at coastal sites are known to be strongly distorted by the conductivity of the seawater. This ocean effect is generally considered to be small for periods greater than a few days. We revise this statement by detailed and systematic model studies in the period range from 1 to 64 days, with subsequent comparison of the modelled and observed electromagnetic responses. The conductivity model consists of a radial symmetric (1-D) section that is overlaid by a thin spherical surface shell, the conductance of which is compiled using the NOAA ETOPO topography/bathymetry and map of sediment thicknesses. The simulations were performed for spatial resolutions of the surface shell of 5° × 5°, 2° × 2° and 1° × 1°, respectively, and for two, continental and oceanic, underlying 1-D conductivity models. The inducing source is described by the spherical harmonic P10 in dipole coordinates. Comparisons are made for the coastal geomagnetic observatories Apia, Hermanus, Kakioka, Kanoya, and Simosato where an anomalous behaviour of the local response has been previously detected. From the comparison of observed and modelled responses we conclude that peculiarities in the observed coastal responses in the period range from 1 to 20 days can be explained to a large amount by induction in the oceans. We show that correction for the ocean effect results in responses that are much better interpretable by 1-D conductivity models compared to the uncorrected responses.

On induction effects of geomagnetic daily variations from equatorial electrojet and solar quiet sources at low and middle latitudes

[1]xa0We investigate the spatiotemporal behavior of the magnetic vertical component, Z, of the daily ionospheric current systems: the equatorial electrojet (EEJ) and solar quiet (Sq) variations, considering induction in the mantle and oceans. The inducing EEJ and Sq current systems are provided by the comprehensive model of Sabaka et al. (2004). The three-dimensional (3-D) conductivity model of the Earth includes oceans of laterally variable conductance and a spherical conductor (1-D) underneath. Our model studies demonstrate that induction effects in Z due to the EEJ are negligible everywhere inland for all local times. At CHAMP altitude (400 km) the magnetic signal induced by EEJ above the oceans does not exceed 2–5% of the external field during local noon. This, in particular, means that considering the induction effects is not necessary when modeling the EEJ current strength from inland surface magnetic measurements and/or satellite data. As expected, induction in the oceans strongly affects the Sq field. The model studies show that the anomalous induction effect (defined as the difference between results obtained with 1-D and 3-D conductivity models) of Sq is substantial at CHAMP altitude, comprising 50% of the total field. It is therefore necessary to consider induction in the oceans when modeling Sq variations for both ground-based and satellite data. Finally, we demonstrate that the anomalous behavior of the daily variations in Z at south Indian sites, namely, a large positive prenoon peak, can be explained by 3-D induction of the Sq variations, with no contribution from the EEJ.

3-D modelling the electric field due to ocean tidal flow and comparison with observations

[1]xa0The tidal motion of the ocean water through the ambient magnetic field, generates secondary electric field. This motionally induced electric field can be detected in the sea or inland and has a potential for electrical soundings of the Earth. A first goal of the paper is to gain an understanding of the global distribution of the electric signal due to tidal ocean flow. We simulate the electric signals for two tidal constituents - lunar semidiurnal (M2) and diurnal (O1) tides. We assume a realistic Earths conductivity model with a surface thin shell and 1-D mantle underneath. Simulations demonstrate that in some coastal regions the amplitudes of the electric field can reach 100 mV/km and 10 mV/km for M2 and O1 tides respectively. The changes of lithosphere resistance produce detectable changes in the tidal electric signals. We show that our predictions are in a good agreement with observations.

Chapter 3 Modelling electromagnetic fields in a 3D spherical earth using a fast integral equation approach

Abstract We present a numerical solution for the global induction problem. The solution calculates the electromagnetic fields in spherical three-dimensional (3D) earth models that are excited by external or internal currents. The models include a number of 3D isotropic (or anisotropic) inhomogeneities that reside in a radially symmetric section. The solution exploits the modified iterative-dissipative method. This fast integral equation approach allows the recovery of an accurate solution, even for large conductivity contrasts. In order to verify our solution we compare it with a staggered-grid finite-difference solution for a model with two (surface and mantle) laterally inhomogeneous conducting thin layers, and also with a Cartesian integral equation solution for a model with deep-seated local 3D anomaly. Both comparisons demonstrate very good agreement.

Chapter 4 Modelling induction log responses in 3D geometries using a fast integral equation approach

Abstract We present modelling results for the electromagnetic logging problem. Our numerical solution calculates the log responses for three-dimensional earth models that include a deviated/tilted borehole, including invaded beds. The solution accounts for both displacement currents and electrical anisotropy of the formation. Our solution exploits the modified iterative dissipative method that is based on a convergent Neumann series expansion. To achieve solution convergence, the conventional scattering integral equation is modified using a spectral shift and a change of variable, which are based on an inequality for the Greens dyadic function. For deviated boreholes in layered formations, there are no known analytical solutions. Therefore, our solution was first validated against the mode-matching solution for a vertical borehole model in a layered formation. In this model we varied the eccentricity of the 50 MHz two-coil sonde. Comparison to the 10 kHz, 160 kHz and 5 MHz responses from a staggered-grid finite difference solution was then conducted for a 45-degree deviated borehole that crosses a formation boundary. All comparisons showed excellent agreement, and demonstrate verifiable induction log responses in the presence of displacement currents and deviated boreholes. Results for a sonde with a resistive case are also reported.

An imaging of buried anomalies, using multi-sheet inversion

The objective of this paper is to construct a stable inversion scheme to determine the conductance of a buried inhomogeneous layer masked by sedimentary cover. We employ a model which incorporates a surface sheet of conductance S1 (x, y) and a buried sheet of conductance S2 (x, y). The buried sheet is embedded at a depth d in a layered earth of conductivity σ(z). The σ(z), d and S1 (x, y) are assumed to be known. We determine S2(x, y) by inverting the monochromatic horizontal electromagnetic (EM) field measured at the earth’s surface. To do this, we use Price’s matching conditions at the sheets and a stable continuation of EM field from the earth’s surface onto the buried sheet. Employing synthetic data we demonstrate that the designed inversion scheme generates a reasonable image of the buried structure. The scheme is fast, since it doesn’t require any forward modelling.