Crosswell electromagnetic inversion using integral and differential equations

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 899-911 ◽  
Author(s):  
Gregory Newman
Keyword(s):  
Inverse Problem ◽  
Field Data ◽  
Green's Functions ◽  
Green’S Functions ◽  
Salt Water ◽  
Single Frequency ◽  
Model Parameters ◽  
Inversion Algorithm ◽  
Before And After ◽  
Using Data

The crosswell electromagnetic (EM) inverse problem is solved with an integral‐equation (IE) formulation using successive Born approximations in the frequency domain. Because the inverse problem is nonlinear, the predicted fields and Green’s functions are continually updated. Updating the fields and Green’s functions relates small changes in the predicted data to small changes in the model parameters through Fréchet kernels. These fields and Green functions are calculated with an efficient 3-D finite‐difference solver. Since the resistivity is invariant along strike, the 3-D fields are integrated along strike so the 2-D kernels can be assembled. At the early stages of the inversion, smoothing of the electrical conductivity stabilizes the inverse solution when it is far from convergence. As the solution converges, this smoothing is relaxed and more effort is made to reduce the data misfit. Bounds on the conductivity are included in the solution to eliminate unrealistic estimates. The robustness of the inversion scheme has been demonstrated with synthetic and field data that are underdetermined from the standpoint of the smooth models being sought. Two synthetic examples with added Gaussian noise were considered, including data arising from an IE solver. This IE solver is different from the one embedded in the inversion algorithm and has provided a stronger check on the scheme. The synthetic examples show it is more difficult to reconstruct a target’s conductivity than its geometry at a single frequency. The inversion scheme has been successfully tested using data collected at the Richmond‐field site near Berkeley, California, where it has imaged a salt water plume injected into the interwell region. The data in this experiment consisted of two sets of measurements, taken before and after the injection of 50 000 gallons of 1 Ωm salt water. Findings show that underdetermined inversion using small amounts of field data can be sufficient to produce useful, but smoothed, maps of the conductivity. The data in this instance need be only single frequency and single component.

Inversion of helicopter electromagnetic data to a magnetic conductive layered earth

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1211-1223 ◽  
Author(s):  
Haoping Huang ◽  
Douglas C. Fraser
Keyword(s):  
Magnetic Permeability ◽  
Field Data ◽  
Model Parameters ◽  
Relative Importance ◽  
Magnetic Polarization ◽  
Inversion Algorithm ◽  
Layered Earth ◽  
Electromagnetic Data ◽  
Airborne Electromagnetic ◽  
True Values

Inversion of airborne electromagnetic (EM) data for a layered earth has been commonly performed under the assumption that the magnetic permeability of the layers is the same as that of free space. The resistivity inverted from helicopter EM data in this way is not reliable in highly magnetic areas because magnetic polarization currents occur in addition to conduction currents, causing the inverted resistivity to be erroneously high. A new algorithm for inverting for the resistivity, magnetic permeability, and thickness of a layered model has been developed for a magnetic conductive layered earth. It is based on traditional inversion methodologies for solving nonlinear inverse problems and minimizes an objective function subject to fitting the data in a least‐squares sense. Studies using synthetic helicopter EM data indicate that the inversion technique is reasonably dependable and provides fast convergence. When six synthetic in‐phase and quadrature data from three frequencies are used, the model parameters for two‐ and three‐layer models are estimated to within a few percent of their true values after several iterations. The analysis of partial derivatives with respect to the model parameters contributes to a better understanding of the relative importance of the model parameters and the reliability of their determination. The inversion algorithm is tested on field data obtained with a Dighem helicopter EM system at Mt. Milligan, British Columbia, Canada. The output magnetic susceptibility‐depth section compares favorably with that of Zhang and Oldenburg who inverted for the susceptibility on the assumption that the resistivity distribution was known.

3D modeling and inversion of VLF and VLF-R electromagnetic data

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. WB219-WB231 ◽  
Author(s):  
P. Kaikkonen ◽  
S. P. Sharma ◽  
S. Mittal
Keyword(s):  
Field Data ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Low Frequency ◽  
Reliability And Validity ◽  
Physical Parameters ◽  
Model Parameters ◽  
Data Inversion ◽  
Inversion Procedure ◽  
Using Data

Three-dimensional linearized nonlinear electromagnetic inversion is developed for revealing the subsurface conductivity structure using isolated very low frequency (VLF) and VLF-resistivity anomalies due to conductors that may be arbitrarily directed towards the measuring profiles and the VLF transmitter. We described the 3D model using a set of variables in terms of geometric and physical parameters. These model parameters were then optimized (parametric inversion) to obtain their best estimates to fit the observations. Two VLF transmitters, i.e., the [Formula: see text], [Formula: see text] (“E”) and the [Formula: see text], [Formula: see text] (“H”) polarizations, respectively, can be considered jointly in inversion. After inverting several noise-free and noisy synthetic data, the results revealed that the estimated model parameters and the functionality of the approach were very good and reliable. The inversion procedure also worked well for the field data. The reliability and validity of the results after the field data inversion have been checked using data from a shear zone associated with uranium mineralization.

Tight-Binding Model Study of Tunneling Spectra of Monolayer Graphene-on-Substrate

2018 ◽  
Vol 17 (04) ◽  
pp. 1760027 ◽  
Author(s):  
Himanshu Sekhar Gouda ◽  
Sivabrata Sahu ◽  
G. C. Rout
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Tight Binding ◽  
Mean Field Approximation ◽  
Coulomb Energy ◽  
Electron Interaction ◽  
Model Parameters ◽  
Tight Binding Model ◽  
Binding Model ◽  
Model Study

We report here the theoretical model study of antiferromagnetic ordering in graphene. We propose a tight-binding model Hamiltonian describing electron hopping up to third-nearest neighbors in graphene. The Hamiltonian describing inequivalence of [Formula: see text] and [Formula: see text] sublattices in graphene-on-substrate is incorporated. The Hubbard-type repulsive Coulomb interaction is considered for both the sublattices with same Coulomb energy. The electron–electron interaction is considered within mean-field approximation with mean electron occupancies [Formula: see text] at [Formula: see text] sublattice and [Formula: see text] at [Formula: see text] site with [Formula: see text] and [Formula: see text] being the antiferromagnetic magnetizations at [Formula: see text] and [Formula: see text] sublattices, respectively. The total Hamiltonian is solved by Zubarev’s techniques of double time single particle Green’s functions. The magnetizations are calculated from the correlation functions corresponding to the respective Green’s functions. The temperature-dependent magnetizations are solved self-consistently taking suitable grid points for the electron momentum. Finally, the electron density of states (DOS) which is proportional to imaginary part of the electron Green’s functions is calculated and computed numerically at a given temperature varying different model parameters for the system. The conductance spectra show a gap near the Dirac point due to substrate-induced gap and magnetic gap, while the van Hove singularities split into eight peaks due to two different sublattice magnetizations and two different spin orientations of the electron in graphene-on-substrate.

Uncertainty Space Expansion: A Consistent Integration of Measurement Errors in Linear Inversion

SPE Journal ◽  
2020 ◽  
Vol 25 (06) ◽  
pp. 3317-3331
Author(s):  
Pipat Likanapaisal ◽  
Hamdi A. Tchelepi
Keyword(s):  
Inverse Problem ◽  
Measurement Errors ◽  
History Matching ◽  
Model Parameters ◽  
State Variables ◽  
Inversion Algorithm ◽  
Additional Information ◽  
Uncertainty Space ◽  
Reservoir Simulator ◽  
New Formula

Summary In general, a probabilistic framework for a modeling process involves two uncertainty spaces: model parameters and state variables (or predictions). The two uncertainty spaces in reservoir simulation are connected by the governing equations of flow and transport in porous media in the form of a reservoir simulator. In a forward problem (or a predictive run), the reservoir simulator directly maps the uncertainty space of the model parameters to the uncertainty space of the state variables. Conversely, an inverse problem (or history matching) aims to improve the descriptions of the model parameters by using the measurements of state variables. However, we cannot solve the inverse problem directly in practice. Numerous algorithms, including Kriging-based inversion and the ensemble Kalman filter (EnKF) and its many variants, simplify the system by using a linear assumption. The purpose of this paper is to improve the integration of measurement errors in the history-matching algorithms that rely on the linear assumption. The statistical moment equation (SME) approach with the Kriging-based inversion algorithm is used to illustrate several practical examples. In the Motivation section, an example of pressure conditioning has a measurement that contains no additional information because of its significant measurement error. This example highlights the inadequacy of the current method that underestimates the conditional uncertainty for both model parameters and predictions. Accordingly, we derive a new formula that recognizes the absence of additional information and preserves the unconditional uncertainty. We believe this to be the consistent behavior to integrate measurement errors. Other examples are used to validate the new formula with both linear and nonlinear (i.e., the saturation equation) problems, with single and multiple measurements, and with different configurations of measurement errors. For broader applications, we also develop an equivalent formula for algorithms in the Monte Carlo simulation (MCS) approach, such as EnKF and ensemble smoother (ES).

Resistivity/depth mapping with airborne electromagnetic survey data

Geophysics ◽  
1983 ◽  
Vol 48 (2) ◽  
pp. 181-196 ◽  
Author(s):  
Klaus‐Peter Sengpiel
Keyword(s):  
Salt Water ◽  
Mapping Method ◽  
Shielding Effect ◽  
Single Frequency ◽  
Salt Water Intrusion ◽  
Horizontal Gradient ◽  
Model Parameters ◽  
Apparent Depth ◽  
Contour Maps ◽  
Airborne Electromagnetic

Using the homogeneous half‐space as a universal interpretation model, all of the secondary field data obtained with a single‐frequency airborne electromagnetic (EM) system that satisfies the superposed dipole condition can be converted to the model parameters [Formula: see text] (apparent resistivity) and [Formula: see text] (apparent depth). These parameters have been investigated for their behavior above various conductivity models and at various flight altitudes, first for theoretical examples and then for several applications in the field. The values of [Formula: see text] and [Formula: see text] are good approximations of the true resistivity and true depth of an extended, buried conductor only where the shielding effect of the cover is small. Moreover, a depth value has a meaning only within the lateral limits of a target conductor. A method is described to locate these lateral limits and to select acceptable depth and resistivity values by means of the “area of [Formula: see text],” which is derived from the horizontal gradient of log [Formula: see text] and the maxima of [Formula: see text]. The results of the resistivity/depth mapping method are presented in the form of two contour maps. Examples of the practical application of the method, over known sulfide ore bodies and over a salt water intrusion, show that reliable data can be obtained on the depth, dip, and extent of these kinds of conductors, as well as on the approximate resistivity of the conductors and the host rock.

Model parameters of the nonlinear stiffness of the vibrator-ground contact determined by inversion of vibrator accelerometer data

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. H25-H32 ◽  
Author(s):  
Andrey V. Lebedev ◽  
Igor A. Beresnev ◽  
Pieter L. Vermeer
Keyword(s):  
Inverse Problem ◽  
Harmonic Distortion ◽  
Source Model ◽  
Coupled Systems ◽  
Equivalent Model ◽  
Model Parameters ◽  
Accelerometer Data ◽  
Ground Contact ◽  
Force Signal ◽  
Using Data

Analysis of vibroseis data shows that harmonic distortion in the ground-force signal may exceed the primary distortions in the hydraulic system. This can be explained by the baseplate-ground contact nonlinearity created by the deformations of the contact roughness. We separated the nonlinear distortion generated in the hydraulics from that generated at the contact. We then formulated an inverse problem for resolving the parameters of the nonlinear contact rigidity, based on the equivalent model of the nonlinear source and the comparison of predicted and observed harmonic levels. The inverse problem was solved for models of bilinear contact and the contact with the rigidity smoothly varying between two asymptotic values, using data obtained on sandy soil. Rigidities changing between approximately [Formula: see text] in compression and [Formula: see text] in tension were resolved from the inversion for both models, although the smooth nonlinear-rigidity model is a better approximation. The analysis shows the adequacy of the equivalent mechanical source model used for the description of nolinear distortions in real soil-baseplate coupled systems.

scholarly journals Discrete representations for Marchenko imaging of imperfectly sampled data

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. A1-A5
Author(s):  
Kees Wapenaar ◽  
Johno van IJsseldijk
Keyword(s):  
Field Data ◽  
Green's Functions ◽  
Green’S Functions ◽  
Integral Representations ◽  
Sampled Data ◽  
Point Spread Functions ◽  
Point Spread ◽  
Discrete Representations

Marchenko imaging is based on integral representations for focusing functions and Green’s functions. In practice, the integrals are replaced by finite summations. This works well for regularly sampled data, but the quality of the results degrades in a case of imperfect sampling. We have developed discrete representations that account for imperfect sampling of the sources (or, via reciprocity, of the receivers). These representations contain point-spread functions that explain the blurring of the focusing functions and Green’s functions due to imperfect sampling. Deblurring the focusing functions and Green’s functions involves multidimensional deconvolution for the point-spread functions. The discrete representations form the basis for modifying Marchenko imaging to account for imperfectly sampled data, which is important for field data applications.

scholarly journals Combining the All-Source Green's Functions and the GPS-Derived Source Functions for Fast Tsunami Predictions—Illustrated by the March 2011 Japan Tsunami

2013 ◽  
Vol 30 (7) ◽  
pp. 1542-1554 ◽  
Author(s):  
Zhigang Xu ◽  
Y. Tony Song
Keyword(s):  
Green's Functions ◽  
Source Function ◽  
Green’S Functions ◽  
Geographic Origin ◽  
Deep Ocean ◽  
Propagation Model ◽  
Tsunami Source ◽  
Tsunami Propagation ◽  
2011 Tohoku Tsunami ◽  
Using Data

Abstract This paper proposes an effective approach on how to predict tsunamis rapidly following a submarine earthquake by combining a real-time GPS-derived tsunami source function with a set of precalculated all-source Green's functions (ASGFs). The approach uses the data from both teleseismic and coastal GPS networks to constrain a tsunami source function consisting of both sea surface elevation and horizontal velocity field, and uses the ASGFs to instantaneously transfer the source function to the arrival time series at the destination points. The ASGF can take a tsunami source of arbitrary geographic origin and resolve it as fine as the native resolution of a tsunami propagation model from which the ASGF is derived. This new approach is verified by the 2011 Tohoku tsunami using data measured by the Deep-Ocean Assessment and Reporting of Tsunamis (DART) buoys.

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