A Prescaled Multiplicative Regularized Gauss-Newton Inversion

A multiplicative regularized Gauss-Newton inversion algorithm is proposed for shape and location reconstruction of homogeneous targets with known permittivities. The data misfit cost functional is regularized with two different multiplicative regularizers. The first regularizer is the weighted -norm total variation which provides an edge-preserving regularization. The second one imposes a priori information about the permittivities of the objects being imaged. Using both synthetically and experimentally collected data sets, we show that the proposed algorithm is robust in reconstructing the shape and location of homogeneous targets.

Since the late 1990s, the continuous development and application of the Controlled-Source Electromagnetic (CSEM) technology have vastly improved its usage in exploration (e.g., Constable, 2010; Loseth et al., 2015; Zach et al., 2009). CSEM inversion results are the main input for interpretation of resistivity data, especially potential fluid anomalies. Complementary to other geophysical methods, e.g. Amplitude versus Offset (AVO), it can therefore be a valuable tool in hydrocarbon exploration. Yet, the unconstrained nature of the subsurface and non-unique¬ness of inversions is a tough predicament. For example, several features in the subsurface could cause high resistivity, such as abundant limestone, salt, basalt, mature source rocks or hydro¬carbon-filled sandstones (Baltar and Barker, 2015). As a result of enhanced data quality, data processing and inversion capabilities, together with improved workflows and accumulated experience, CSEM interpretation has become increasingly more robust (Buland et al., 2011; Karman et al., 2013; Loseth et al., 2014). The 3D Gauss-Newton (GN) inversion algorithm is a more robust method compared to other inversion types. GN thus provides a new, powerful methodology for generating marine CSEM resistivity images. Nonetheless, it has formerly been com¬putationally exhaustive to apply. With the Hoop area, Barents Sea, recently being nominated for the Barents Sea exploration licensing round, a tight deadline was given for the CSEM inversion of an expansive 11,000 km2, over 130 blocks (refer to Figure 1). Adopt¬ing the flexible and agile cloud system into the CSEM workflow, it has improved the 3D GN work efficiency and provided overall consistent results when compared with other CSEM inversion schemes. This paper will illustrate how a systemic workflow of CSEM 3D GN inversions together with efficient cloud computing, could lead to a more consistent interpretation of this data.

This paper describes the linear and nonlinear inversion algorithms for reconstructing two inhomogeneous dielectric targets in an anechoic chamber in a controlled environment. The applied algorithms are the linear multiple signal classification (MUSIC) algorithm combined with the time reversal (TR) imaging and two nonlinear iterative algorithms-two domain integral equation based methods, namely the contrast source inversion (CSI) algorithm and the multiplicative regularized contrast source inversion (MR-CSI) algorithm. The inversion results, which are presented and compared for the transverse magnetic (TM) and transverse electric (TE) polarization datasets using the multi-frequency multi-bistatic measurements, verify the capability, the accuracy and the efficiency of the MR-CSI algorithm.

The weighted L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm total variation multiplicative regularized Gauss-Newton inversion method, recently developed for inversion of low-frequency deep electromagnetic geophysical measurements, is used for microwave biomedical imaging. This inversion algorithm automatically adjusts the regularization weight and provides edge-preserving characteristics. The accuracy of this method is demonstrated by inverting experimental data of a human forearm and synthetic data taken from brain and breast models, both assuming two-dimensional (2D) transverse magnetic illumination.

Regularization inversion uses constraints and a regularization factor to solve ill-posed inversion problems in geophysics. The choice of the regularization factor and of the initial model is critical in regularization inversion. To deal with these problems, we propose a multiobjective particle swarm inversion (MOPSOI) algorithm to simultaneously minimize the data misfit and model constraints, and obtain a multiobjective inversion solution set without the gradient information of the objective function and the regularization factor. We then choose the optimum solution from the solution set based on the trade-off between data misfit and constraints that substitute for the regularization factor. The inversion of synthetic two-dimensional magnetic data suggests that the MOPSOI algorithm can obtain as many feasible solutions as possible; thus, deeper insights of the inversion process can be gained and more reasonable solutions can be obtained by balancing the data misfit and constraints. The proposed MOPSOI algorithm can deal with the problems of choosing the right regularization factor and the initial model.

The inversion results of complex resistivity method are four Cole-Cole model parameters. Among the four parameters, the frequency dependence and the time constant are more difficult to invert. It is necessary to study an algorithm that can invert the four Cole-Cole model parameters at the same time. In this paper, the least squares and OCCAM inversion algorithms are used to invert four Cole-Cole model parameters. In other words, two model constraints are added to the objective function. When inverting different Cole-Cole model parameters, the real and imaginary parts of the data are weighted to adjust the proportion of real part and imaginary part of data in inversion. Firstly, the formula of the algorithm is deduced. Then the theoretical models are designed for inversion trial calculation. In the inversion process, the inversion converges steadily by adjusting the Lagrange factor, and the inversion effect is improved by adjusting the weighting coefficients of real part and imaginary part. This method can get better inversion results by adjusting the proportion of the real and imaginary parts of the data in the inversion. The model trial results show that the weighting inversion algorithm significantly improves the results of the inversion of the four Cole-Cole parameters.

Reconstructing EEG sources involves a complex pipeline, with the inverse problem being the most challenging. Multiple inversion algorithms are being continuously developed, aiming to tackle the non-uniqueness of this problem, which has been shown to be partially circumvented by including prior information in the inverse models. Despite a few efforts, there are still current and persistent controversies regarding the inversion algorithm of choice and the optimal set of spatial priors to be included in the inversion models. The use of simultaneous EEG-fMRI data is one approach to tackle this problem. The spatial resolution of fMRI makes fMRI derived spatial priors very convenient for EEG reconstruction, however, only task activation maps and resting-state networks (RSNs) have been explored so far, overlooking the recent, but already accepted, notion that brain networks exhibit dynamic functional connectivity fluctuations. The lack of a systematic comparison between different source reconstruction algorithms, considering potentially more brain-informative priors such as fMRI, motivates the search for better reconstruction models. Using simultaneous EEG-fMRI data, here we compared four different inversion algorithms (minimum norm, MN; low resolution electromagnetic tomography, LORETA; empirical Bayes beamformer, EBB; and multiple sparse priors, MSP) under a Bayesian framework (as implemented in SPM), each with three different sets of priors consisting of: (1) those specific to the algorithm; (2) those specific to the algorithm plus fMRI task activation maps and RSNs; and (3) those specific to the algorithm plus fMRI task activation maps and RSNs and network modules of task-related dFC states estimated from the dFC fluctuations. The quality of the reconstructed EEG sources was quantified in terms of model-based metrics, namely the expectation of the posterior probability P(model|data) and variance explained of the inversion models, and the overlap/proportion of brain regions known to be involved in the visual perception tasks that the participants were submitted to, and RSN templates, with/within EEG source components. Model-based metrics suggested that model parsimony is preferred, with the combination MSP and priors specific to this algorithm exhibiting the best performance. However, optimal overlap/proportion values were found using EBB and priors specific to this algorithm and fMRI task activation maps and RSNs or MSP and considering all the priors (algorithm priors, fMRI task activation maps and RSNs and dFC state modules), respectively, indicating that fMRI spatial priors, including dFC state modules, might contain useful information to recover EEG source components reflecting neuronal activity of interest. Our main results show that providing fMRI spatial derived priors that reflect the dynamics of the brain might be useful to map neuronal activity more accurately from EEG-fMRI. Furthermore, this work paves the way towards a more informative selection of the optimal EEG source reconstruction approach, which may be critical in future studies.

This paper presents the results of inversion of multi-frequency electromagnetic scattered field data, measured by the Institute of Fresnel, Marseille, France, from three-dimensional (3D) homogeneous objects, both for co-polarization and cross-polarization antenna orientations. The reconstructions were obtained by using the multiplicative regularized contrast source inversion (MR-CSI) method. The inversion results indicated that the MR-CSI is able to robustly and efficiently invert these 3D data sets using minimal a priori information.

Multi-trace acoustic impedance inversion with multiplicative regularization

Recently the drilling industry has seen many advances in the application of deep directional electromagnetic (EM) measurements for mapping deeper into the reservoir, with the latest one capable of seeing over 250 ft above and below the wellbore providing unprecedented understanding of the reservoir. This measurement technology is now being used to look ahead of bit while drilling, for exploration wells to reduce drilling risks associated with unexpectedly penetrating certain formation. With the increasing complexity of the reservoirs that the industry is targeting, there is more and more quest for expanding the reservoir mapping capability, not just a 1D approach that can only map resistive boundaries on the vertical axis or near vertical axis and assume infinite extend in all other directions, but to enable geoscientists to better steer the well and better understand the reservoir structure and fluid contact in a full three-dimensional context around the wellbore. In this communication, the authors introduce a new solution to this quest for full three-dimensional real-time reservoir mapping. The solution is composed of three parts: a set of new measurements acquired downhole and transmitted to surface in real-time, a new inversion algorithm that is model independent and therefore fit for any reservoir complexity, and a new computing paradigm that make it possible to provide answers in real-time while drilling. The new set of measurements almost doubles the number of well logs that were acquired before and greatly enriches formations evaluation around the wellbore. The new algorithm, different from all previous algorithms, is not confined to any specific forms of models, making it suitable for exploring and finding solutions in complex reservoir settings. Finally, taking advantage of the latest advances in the Cloud computing, turnaround time of the new inversion is improved by over hundred times, thanks to the scalability of the algorithm design and Cloud computing infrastructure. Combining all these together allows to achieve three-dimensional reservoir map, without having to tradeoff between high resolution and depth of investigation. The 3D reservoir map that is generated from multiple transverse 2D inversion slices in real-time, enables timely update of reservoir model as drilling progress for the operator to make informed decisions. This new technology is currently deployed in several locations around the world and in different environments. In this paper, the authors review deployment results, to illustrate the technology, from preparation to real-time execution, and finally to post-job model update. With the ability of mapping in all directions while drilling, this technology opens the door to many applications and will enable the operators to target more complex reservoirs and achieving better geosteering results where 3D mapping and steering are required. In addition to its benefits for real-time operations, the technology also enables the geoscientists to update and calibrate their reservoir models with fine and accurate details, which can further benefit multiple disciplines including drilling, completion, production and reservoir management.

An accelerated inversion algorithm using the resolvent matrix method

Classification of populations of focal mechanisms are commonly used as a step in retrieving states of stress in the lithosphere. Although such classifications are usually made according to subjective criteria, an objective one based on a classification algorithm is presented. The procedure is based on two steps. First, each mechanism is characterized by the components of its principal tensional and compressional axes and then objectively classified accordingly. Second, each mechanism is characterized by its geographic coordinates (latitude and longitude) after which each group obtained in the first step is divided into new groups according to geographic coordinates. The classification algorithm, together with a standard inversion algorithm, is applied to data from aftershocks belonging to the Irpinia (southern Italy) Earthquake of November 23, 1980. A few incompatible groups of focal mechanisms obtained by the classification algorithm show a spatial overlap. Nevertheless, the new methodology allows us to present a more accurate description of a local state of stress for an aftershock area than the stress pattern obtained from subjective classifications. The new stress pattern is compared with principal stress axes deduced from the main event and two local states of stress deduced previously from the aftershock mechanisms.

Using standard inversion algorithm, velocity and attenuation tomograms can show artifacts which compromise interpretation. These artifacts can be due to errors in borehole trajectory measurements, medium anisotropy, T<SUB>0</SUB> (initial time) or A<SUB>0</SUB> (initial amplitude) drifts. In order to cancel these artifacts, the error sources can be introduced as unknown parameters in inversion algorithms (Hollender, 1999). In this paper, we present results obtained with crosshole radar data, recorded in a limestone quarry. Using the appropriate algorithms, all the artifacts have been cancelled and tomograms show clearly subhorizontal structures in agreement with the quarry stratification. In our data set, results do not reveal significant trajectory error, and T<SUB>0</SUB> and A<SUB>0</SUB> drifts are low. However, the presence of a velocity and attenuation anisotropy appears clearly on the tomograms. In the case of attenuation tomograms, the high anisotropy rates could be explained by the cumulative effect of the partitioning of energy due to reflection and transmission mechanisms at interfaces, and medium anisotropy.

Unified systolic array architecture for finite field multiplication and inversion

Finite field arithmetic logic is central in the implementation of Reed-Solomon coders and in some cryptographic algorithms. There is a need for good multiplication and inversion algorithms that can be easily realized on VLSI chips. Massey and Omura recently developed a new multiplication algorithm for Galois fields based on a normal basis representation. In this paper, a pipeline structure is developed to realize the Massey-Omura multiplier in the finite field GF(2m). With the simple squaring property of the normal basis representation used together with this multiplier, a pipeline architecture is developed for computing inverse elements in GF(2m). The designs developed for the Massey-Omura multiplier and the computation of inverse elements are regular, simple, expandable, and therefore, naturally suitable for VLSI implementation.