A compressed implicit Jacobian scheme for 3D electromagnetic data inversion

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. F173-F183 ◽  
Author(s):  
Maokun Li ◽  
Aria Abubakar ◽  
Jianguo Liu ◽  
Guangdong Pan ◽  
Tarek M. Habashy
Keyword(s):  
Electric Fields ◽  
Jacobian Matrix ◽  
Iterative Solvers ◽  
Computational Time ◽  
Memory Usage ◽  
Inversion Algorithm ◽  
Linear System Of Equations ◽  
Electromagnetic Data ◽  
Data Inversion ◽  
Computational Overhead

We developed a compressed implicit Jacobian scheme for the regularized Gauss-Newton inversion algorithm for reconstructing 3D conductivity distributions from electromagnetic data. In this algorithm, the Jacobian matrix, whose storage usually requires a large amount of memory, is decomposed in terms of electric fields excited by sources located and oriented identically to the physical sources and receivers. As a result, the memory usage for the Jacobian matrix reduces from O(NFNSNRNP) to O[NF(NS + NR)NP], where NF is the number of frequencies, NS is the number of sources, NR is the number of receivers, and NP is the number of conductivity cells to be inverted. When solving the Gauss-Newton linear system of equations using iterative solvers, the multiplication of the Jacobian matrix with a vector is converted to matrix-vector operations between the matrices of the electric fields and the vector. In order to mitigate the additional computational overhead of this scheme, these fields are further compressed using the adaptive cross approximation (ACA) method. The compressed implicit Jacobian scheme provides a good balance between memory usage and computational time and renders the Gauss-Newton algorithm more efficient. We demonstrated the benefits of this scheme using numerical examples including both synthetic and field data for both crosswell and controlled-source electromagnetic (CSEM) applications.

scholarly journals 2.5D Inversion Algorithm of Frequency-Domain Airborne Electromagnetics with Topography

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Jianjun Xi ◽  
Wenben Li
Keyword(s):  
Frequency Domain ◽  
Large Scale ◽  
Jacobian Matrix ◽  
Sparse Matrix ◽  
Inversion Algorithm ◽  
Linear System Of Equations ◽  
Electromagnetic Data ◽  
Electric And Magnetic Fields ◽  
Airborne Electromagnetic ◽  
Airborne Electromagnetic Data

We presented a 2.5D inversion algorithm with topography for frequency-domain airborne electromagnetic data. The forward modeling is based on edge finite element method and uses the irregular hexahedron to adapt the topography. The electric and magnetic fields are split into primary (background) and secondary (scattered) field to eliminate the source singularity. For the multisources of frequency-domain airborne electromagnetic method, we use the large-scale sparse matrix parallel shared memory direct solver PARDISO to solve the linear system of equations efficiently. The inversion algorithm is based on Gauss-Newton method, which has the efficient convergence rate. The Jacobian matrix is calculated by “adjoint forward modelling” efficiently. The synthetic inversion examples indicated that our proposed method is correct and effective. Furthermore, ignoring the topography effect can lead to incorrect results and interpretations.

2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. F165-F177 ◽  
Author(s):  
A. Abubakar ◽  
T. M. Habashy ◽  
V. L. Druskin ◽  
L. Knizhnerman ◽  
D. Alumbaugh
Keyword(s):  
Cost Function ◽  
Regularization Parameter ◽  
A Priori ◽  
Low Frequency ◽  
Inversion Algorithm ◽  
Forward Algorithm ◽  
Linear System Of Equations ◽  
Controlled Source ◽  
Data Inversion ◽  
The Cost

We present 2.5D fast and rigorous forward and inversion algorithms for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements. The forward algorithm is based on a finite-difference approach in which a multifrontal LU decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single-source experiment for each frequency of operation. When the size of the linear system of equations is large, the use of this noniterative solver is impractical. Hence, we use the optimal grid technique to limit the number of unknowns in the forward problem. The inversion algorithm employs a regularized Gauss-Newton minimization approach with a multiplicative cost function. By using this multiplicative cost function, we do not need a priori data to determine the so-called regularization parameter in the optimization process, making the algorithm fully automated. The algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image. To increase the robustness of the algorithm, we also constrain the minimization and use a line-search approach to guarantee the reduction of the cost function after each iteration. To demonstrate the pros and cons of the algorithm, we present synthetic and field data inversion results for crosswell and controlled-source EM measurements.

3D sequential inversion of frequency-domain airborne electromagnetic data to determine conductive and magnetic heterogeneities

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. E357-E369 ◽  
Author(s):  
Kyubo Noh ◽  
Seokmin Oh ◽  
Soon Jee Seol ◽  
Joongmoo Byun
Keyword(s):  
Frequency Domain ◽  
Input Data ◽  
Computational Time ◽  
Induction Effect ◽  
Inversion Algorithm ◽  
Phase Component ◽  
Data Set ◽  
Conductivity Model ◽  
Electromagnetic Data ◽  
Phase Data

We have developed two inversion workflows that sequentially invert conductivity and susceptibility models from a frequency-domain controlled-source electromagnetic data set. Both workflows start with conductivity inversion using electromagnetic (EM) kernel and out-of-phase component data, which is mainly sensitive to conductivity, and then we adopt the susceptibility inversion using in-phase component data. The difference between these two workflows is in the susceptibility inversion algorithm: One uses an EM kernel and a conductivity model as the input model; the other uses a magnetostatic kernel and a conductivity model to generate the appropriate input data. Because the appropriate input data for magnetostatic inversion should not contain the EM induction effect, the in-phase induction effect is simulated through the conductivity model obtained by inverting out-of-phase data and subtracting them from observed in-phase data to generate an “induction-subtracted” in-phase data set that becomes input data for magnetostatic inversion. For magnetostatic inversion, we used a linear magnetostatic kernel to enable rapid computation. Then, we applied the two inversion workflows to a field data set of a DIGHEM survey, and we successfully reconstructed the conductivity and susceptibility models from each workflow using two zones within the data sets, in which conductive and susceptible anomalies were present. One important finding is that the susceptibility inversion results obtained from two different workflows are very similar to each other. However, computational time can be significantly saved with linear magnetostatic inversion. We found out how the results of the conductivity and susceptibility models could be well-imaged using a sequential inversion workflow and also how magnetostatic inversion could be used efficiently for airborne EM data inversion.

A 3D parametric inversion algorithm for triaxial induction data

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. G1-G9 ◽  
Author(s):  
Aria Abubakar ◽  
Tarek M. Habashy ◽  
Vladimir Druskin ◽  
Leonid Knizhnerman ◽  
Sofia Davydycheva
Keyword(s):  
Jacobian Matrix ◽  
Regularization Parameter ◽  
Scattering Problem ◽  
Synthetic Data ◽  
Inverse Scattering Problem ◽  
Single Frequency ◽  
Computational Time ◽  
Inversion Algorithm ◽  
Induction Logging ◽  
Parametric Inversion

We develop a parametric inversion algorithm to determine simultaneously the horizontal and vertical resistivities of both the formation and invasion zones, invasion radius, bed boundary upper location and thickness, and relative dip angle from electromagnetic triaxial induction logging data. This is a full 3D inverse scattering problem in transversally isotropic media. To acquire sufficient sensitivity to invert for all of these parameters, we collect the data using a multicomponent, multispacing induction array. For each transmitter-receiver spacing this multicomponent tool has sets of three orthogonal transmitter and receiver coils. At each logging point single-frequency data are collected at multiple spacings to obtain information at different depths of investigation. This inversion problem is solved iteratively with a constrained regularized Gauss-Newton minimization scheme. As documented in the literature, the main computational bottleneck when solving this full 3D inverse problem is the CPU time associated with constructing the Jacobian matrix. In this study, to achieve the inversion results within a reasonable computational time, we implement a dual grid approach wherein the Jacobian matrix is computed using a very coarse optimal grid. Furthermore, to regularize the inversion process we use the so-called multiplicative regularization technique. This technique automatically determines the regularization parameter. Synthetic data tests indicate the developed inversion algorithm is robust in extracting formation and invasion anisotropic resistivities, invasion radii, bed boundary locations, relative dip, and azimuth angle from multispacing, multicomponent induction logging data.

VENDOR-NEUTRAL STOCHASTIC INVERSION OF LWD DEEP AZIMUTHAL RESISTIVITY DATA AS A STEP TOWARD EFFICIENCY STANDARDIZATION OF GEOSTEERING SERVICES

2021 ◽  
Author(s):  
Mikhail Sviridov ◽  
◽  
Anton Mosin ◽  
Sergey Lebedev ◽  
Ron Thompson ◽  
...  
Keyword(s):  
Markov Chains ◽  
Quality Indicators ◽  
Parallel Execution ◽  
Oil Field ◽  
Model Complexity ◽  
Computational Time ◽  
Model Parameters ◽  
Formation Model ◽  
Inversion Algorithm ◽  
Azimuthal Resistivity

While proactive geosteering, special inversion algorithms are used to process the readings of logging-while-drilling resistivity tools in real-time and provide oil field operators with formation models to make informed steering decisions. Currently, there is no industry standard for inversion deliverables and corresponding quality indicators because major tool vendors develop their own device-specific algorithms and use them internally. This paper presents the first implementation of vendor-neutral inversion approach applicable for any induction resistivity tool and enabling operators to standardize the efficiency of various geosteering services. The necessity of such universal inversion approach was inspired by the activity of LWD Deep Azimuthal Resistivity Services Standardization Workgroup initiated by SPWLA Resistivity Special Interest Group in 2016. Proposed inversion algorithm utilizes a 1D layer-cake formation model and is performed interval-by-interval. The following model parameters can be determined: horizontal and vertical resistivities of each layer, positions of layer boundaries, and formation dip. The inversion can support arbitrary deep azimuthal induction resistivity tool with coaxial, tilted, or orthogonal transmitting and receiving antennas. The inversion is purely data-driven; it works in automatic mode and provides fully unbiased results obtained from tool readings only. The algorithm is based on statistical reversible-jump Markov chain Monte Carlo method that does not require any predefined assumptions about the formation structure and enables searching of models explaining the data even if the number of layers in the model is unknown. To globalize search, the algorithm runs several Markov chains capable of exchanging their states between one another to move from the vicinity of local minimum to more perspective domain of model parameter space. While execution, the inversion keeps all models it is dealing with to estimate the resolution accuracy of formation parameters and generate several quality indicators. Eventually, these indicators are delivered together with recovered resistivity models to help operators with the evaluation of inversion results reliability. To ensure high performance of the inversion, a fast and accurate semi-analytical forward solver is employed to compute required responses of a tool with specific geometry and their derivatives with respect to any parameter of multi-layered model. Moreover, the reliance on the simultaneous evolution of multiple Markov chains makes the algorithm suitable for parallel execution that significantly decreases the computational time. Application of the proposed inversion is shown on a series of synthetic examples and field case studies such as navigating the well along the reservoir roof or near the oil-water-contact in oil sands. Inversion results for all scenarios confirm that the proposed algorithm can successfully evaluate formation model complexity, recover model parameters, and quantify their uncertainty within a reasonable computational time. Presented vendor-neutral stochastic approach to data processing leads to the standardization of the inversion output including the resistivity model and its quality indicators that helps operators to better understand capabilities of tools from different vendors and eventually make more confident geosteering decisions.

Calibration and data reduction for X-hotwires using cross validation

2021 ◽  
pp. 095441002110403
Author(s):  
Mohan Vijaya Anoop ◽  
Budda Thiagarajan Kannan
Keyword(s):  
Cross Validation ◽  
Low Flow ◽  
Calibration Data ◽  
Inversion Algorithm ◽  
Present Scheme ◽  
Wire Probe ◽  
Data Inversion ◽  
Turbulent Jet Flow ◽  
High Reynolds ◽  
Yaw Angle

A strategy for calibration of X-wire probes and data inversion is described in this article. The approach used has elements of full velocity vs yaw-angle calibration with robust curve fitting. The responses of an X-wire probe placed in a calibration jet are recorded for a set of velocity and yaw inputs followed by fitting cross-validated splines. These spline functions trained from calibration data are evaluated for the probe responses during measurement. X-wire probes are calibrated for low to moderate velocities (0.65 m/s to 32 m/s) and yaw angles in the range −40° to 40° and comparisons with conventional interpolation schemes are made. The proposed algorithm can be extended to calibration of other multiple wire probes and for higher velocities. Some measurements in a single round turbulent jet flow at high Reynolds number using the proposed inversion algorithm are also presented. The present scheme is found to perform better particularly at low flow magnitudes and/or extreme flow angles than the schemes used previously.

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.

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