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.

Joint MT and CSEM data inversion using a multiplicative cost function approach

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. F203-F214 ◽  
Author(s):  
A. Abubakar ◽  
M. Li ◽  
G. Pan ◽  
J. Liu ◽  
T. M. Habashy
Keyword(s):  
Cost Function ◽  
Large Scale ◽  
Joint Inversion ◽  
A Priori ◽  
Upper And Lower Bounds ◽  
Model Parameters ◽  
Data Sets ◽  
Inversion Algorithm ◽  
Earth Resistivity ◽  
The Cost

We have developed an inversion algorithm for jointly inverting controlled-source electromagnetic (CSEM) data and magnetotelluric (MT) data. It is well known that CSEM and MT data provide complementary information about the subsurface resistivity distribution; hence, it is useful to derive earth resistivity models that simultaneously and consistently fit both data sets. Because we are dealing with a large-scale computational problem, one usually uses an iterative technique in which a predefined cost function is optimized. One of the issues of this simultaneous joint inversion approach is how to assign the relative weights on the CSEM and MT data in constructing the cost function. We propose a multiplicative cost function instead of the traditional additive one. This function does not require an a priori choice of the relative weights between these two data sets. It will adaptively put CSEM and MT data on equal footing in the inversion process. The inversion is accomplished with a regularized Gauss-Newton minimization scheme where the model parameters are forced to lie within their upper and lower bounds by a nonlinear transformation procedure. We use a line search scheme to enforce a reduction of the cost function at each iteration. We tested our joint inversion approach on synthetic and field data.

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.

Imaging quasi‐layered conductive structures by simple processing of transient electromagnetic data

Geophysics ◽  
1987 ◽  
Vol 52 (4) ◽  
pp. 545-554 ◽  
Author(s):  
James Macnae ◽  
Yves Lamontagne
Keyword(s):  
Delay Time ◽  
A Priori ◽  
Step Response ◽  
Layered Earth ◽  
Electromagnetic Data ◽  
Apparent Conductivity ◽  
Data Inversion ◽  
Transient Electromagnetic ◽  
Static Zone ◽  
The Cost

An “imaged” conductivity section of a layered earth can be obtained by simple transformation of step‐response electromagnetic data measured in the quasi‐static zone. This method of data transformation is presented as an alternative to conventional apparent conductivity transformations. At each delay time, the variation of the step response as a function of geometry (transmitter and receiver location) is transformed to an equivalent reference depth h, which can be related to the depth of electromagnetic field diffusion. The behavior of h as a function of delay time is nearly independent of the source‐receiver geometry. The slowness dt/dh divided by the magnetic permeability is almost exactly proportional to the cumulative conductance measured from the surface down to a depth h. Thus we can estimate an apparent conductivity, which we call the “imaged conductivity,” at depth to be [Formula: see text]. The cost of this transformation is a fraction of the cost of conventional data inversion, and it does not require an a priori constraint on the number of parameters used in the inversion. The empirically developed technique was used successfully to process UTEM field data measured over a quasi‐layered earth.

Simultaneous reconstruction of 1-D susceptibility and conductivity from electromagnetic data

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 33-47 ◽  
Author(s):  
Zhiyi Zhang ◽  
Douglas W. Oldenburg
Keyword(s):  
Cost Function ◽  
Objective Functions ◽  
Inversion Algorithm ◽  
Data Set ◽  
Circular Loop ◽  
Simultaneous Inversion ◽  
The Earth ◽  
Simultaneous Reconstruction ◽  
Single Data ◽  
The Cost

In this paper, we develop an inversion algorithm to simultaneously recover 1-D distributions of electric conductivity and magnetic susceptibility from a single data set. The earth is modeled as a series of homogeneous layers of known thickness with constant but unknown conductivities and susceptibilities. The medium of interest is illuminated by a horizontal circular loop source located above the surface of the earth. The secondary signals from the earth are received by a circular loop receiver located some distance from the source. The model objective function in the inversion, which we refer to as the cost function, is a weighted sum of model objective functions of conductivity and susceptibility. We minimize this cost function subject to the data constraints and show how the choice of weights for the model objective functions of conductivity and susceptibility affects the results of the inversion through 1-D synthetic examples. We also invert 3-D synthetic and field data. From these examples we conclude that simultaneous inversion of electromagnetic (EM) data can provide useful information about the conductivity and susceptibility distributions.

3D controlled-source electromagnetic imaging of gas hydrates: Insights from the Pelotas Basin offshore Brazil

2019 ◽  
Vol 7 (4) ◽  
pp. SH111-SH131 ◽  
Author(s):  
Raghava Tharimela ◽  
Adolpho Augustin ◽  
Marcelo Ketzer ◽  
Jose Cupertino ◽  
Dennis Miller ◽  
...  
Keyword(s):  
Gas Hydrates ◽  
Gas Hydrate ◽  
A Priori ◽  
Inversion Algorithm ◽  
Near Surface ◽  
Bottom Simulating Reflector ◽  
Controlled Source ◽  
Lateral Extent ◽  
Priori Information ◽  
Pelotas Basin

Mapping of natural gas hydrate systems has been performed successfully in the past using the controlled-source electromagnetic (CSEM) method. This method relies on differentiating resistive highly saturated free gas or hydrate-bearing host sediment from a less resistive low-saturated gas or brine-bearing host sediments. Knowledge of the lateral extent and resistivity variations (and hence the saturation variations) within sediments that host hydrates is crucial to be able to accurately quantify the presence of saturated gas hydrates. A 3D CSEM survey (PUCRS14) was acquired in 2014 in the Pelotas Basin offshore Brazil, with hydrate resistivity mapping as the main objective. The survey was acquired within the context of the CONEGAS research project, which investigated the origin and distribution of gas hydrate deposits in the Pelotas Basin. We have inverted the acquired data using a proprietary 3D CSEM anisotropic inversion algorithm. Inversion was purely CSEM data driven, and we did not include any a priori information in the process. Prior to CSEM, interpretation of near-surface geophysical data including 2D seismic, sub-bottom profiler, and multibeam bathymetry data indicated possible presence of gas hydrates within features identified such as faults, chimneys, and seeps leading to pockmarks, along the bottom simulating reflector and within the gas hydrate stability zone. Upon integration of the same with CSEM-derived resistivity volume, the interpretation revealed excellent spatial correlation with many of these features. The interpretation further revealed new features with possible hydrate presence, which were previously overlooked due to a lack of a clear seismic and/or multibeam backscatter signature. In addition, features that were previously mapped as gas hydrate bearing had to be reinterpreted as residual or low-saturated gas/hydrate features, due to the lack of significant resistivity response associated with them. Furthermore, we used the inverted resistivity volume to derive the saturation volume of the subsurface using Archie’s equation.

Inversion for Fire Heat Release Rate Using Transient Heat Flux Data

2017 ◽  
Author(s):  
Andrew Kurzawski ◽  
Ofodike A. Ezekoye
Keyword(s):  
Cost Function ◽  
Heat Release ◽  
Heat Release Rate ◽  
Release Rate ◽  
Important Variable ◽  
Heat Fluxes ◽  
Data Sets ◽  
Inversion Algorithm ◽  
Gas Measurement ◽  
The Cost

The heat-release rate (HRR) of a burning item is key to understanding the thermal effects of a fire on its surroundings. It is, perhaps, the most important variable used to characterize a burning fuel packet and is defined as the rate of energy released by the fire. HRR is typically determined using a gas measurement calorimetry method. In this study, an inversion algorithm is presented for conducting calorimeter on fires with unknown HRRs located in a compartment. The algorithm compares predictions of a forward model with observed heat fluxes from synthetically generated data sets to determine the HRR that minimizes a cost function. The effects of tuning a weighting parameter in the cost function and the issues associated with two different forward models of a compartment fire are examined.

scholarly journals Statistically optimized inversion algorithm for enhanced retrieval of aerosol properties from spectral multi-angle polarimetric satellite observations

2011 ◽  
Vol 4 (5) ◽  
pp. 975-1018 ◽  
Author(s):  
O. Dubovik ◽  
M. Herman ◽  
A. Holdak ◽  
T. Lapyonok ◽  
D. Tanré ◽  
...  
Keyword(s):  
A Priori ◽  
Algorithm Design ◽  
Statistical Optimization ◽  
Satellite Observations ◽  
Spatial And Temporal Variability ◽  
Inversion Algorithm ◽  
Simultaneous Inversion ◽  
Data Inversion ◽  
Retrieval Scheme ◽  
Aerosol Properties

Abstract. The proposed development is an attempt to enhance aerosol retrieval by emphasizing statistical optimization in inversion of advanced satellite observations. This optimization concept improves retrieval accuracy relying on the knowledge of measurement error distribution. Efficient application of such optimization requires pronounced data redundancy (excess of the measurements number over number of unknowns) that is not common in satellite observations. The POLDER imager on board the PARASOL micro-satellite registers spectral polarimetric characteristics of the reflected atmospheric radiation at up to 16 viewing directions over each observed pixel. The completeness of such observations is notably higher than for most currently operating passive satellite aerosol sensors. This provides an opportunity for profound utilization of statistical optimization principles in satellite data inversion. The proposed retrieval scheme is designed as statistically optimized multi-variable fitting of all available angular observations obtained by the POLDER sensor in the window spectral channels where absorption by gas is minimal. The total number of such observations by PARASOL always exceeds a hundred over each pixel and the statistical optimization concept promises to be efficient even if the algorithm retrieves several tens of aerosol parameters. Based on this idea, the proposed algorithm uses a large number of unknowns and is aimed at retrieval of extended set of parameters affecting measured radiation. The algorithm is designed to retrieve complete aerosol properties globally. Over land, the algorithm retrieves the parameters of underlying surface simultaneously with aerosol. In all situations, the approach is anticipated to achieve a robust retrieval of complete aerosol properties including information about aerosol particle sizes, shape, absorption and composition (refractive index). In order to achieve reliable retrieval from PARASOL observations even over very reflective desert surfaces, the algorithm was designed as simultaneous inversion of a large group of pixels within one or several images. Such multi-pixel retrieval regime takes advantage of known limitations on spatial and temporal variability in both aerosol and surface properties. Specifically the variations of the retrieved parameters horizontally from pixel-to-pixel and/or temporary from day-to-day are enforced to be smooth by additional a priori constraints. This concept is expected to provide satellite retrieval of higher consistency, because the retrieval over each single pixel will be benefiting from coincident aerosol information from neighboring pixels, as well, from the information about surface reflectance (over land) obtained in preceding and consequent observations over the same pixel. The paper provides in depth description of the proposed inversion concept, illustrates the algorithm performance by a series of numerical tests and presents the examples of preliminary retrieval results obtained from actual PARASOL observations. It should be noted that many aspects of the described algorithm design considerably benefited from experience accumulated in the preceding effort on developments of currently operating AERONET and PARASOL retrievals, as well as several core software components were inherited from those earlier algorithms.

Catalog-Based Customization

1999 ◽  
Author(s):  
Bala Chidambaram ◽  
Alice M. Agogino
Keyword(s):  
Cost Function ◽  
Cost Estimation ◽  
A Priori ◽  
Production Costs ◽  
Estimation Methods ◽  
Optimization Formulation ◽  
Catalog Design ◽  
Standard Designs ◽  
Approximate Cost ◽  
The Cost

Abstract This paper develops a new method for implementing mass-customization, namely, the customization around standard products, or catalog-based customization. The method addresses the customization requirements of a class of products that are complex in configuration, multi-functional and structurally similar. We formulate catalog-based customization as an optimization problem consistent with the manufacturer’s goal of incurring minimal costs in the redesign of existing standard components, while meeting customer specifications and satisfying design constraints. The ‘catalog-based’ nature of the formulation raises concomitant issues of cost function development and problem simplification/solution. We identify the generational structure as best suited to exploit the cost data in existing catalogs and construct a product cost function. The cost-estimation methods used by the generational structure in the construction are identified as weight-based — for modeling the material costs, and methods based on similarity principles and regression analyses — for the production costs. The optimization formulation of catalog-based customization may be simplified by an a priori identification of a standard catalog design as the customization basis. This is accomplished with function costing — a cost-estimation hypothesis that uses product functionality to develop an approximate cost-estimate. The function-costing estimate is also used to abstract features from the standard base design into the optimization formulation. The preferred solution strategy for the optimization formulation is identified as genetic algorithms. We apply the customization method developed to Brushless D.C. Permanent Magnet (BDCPM) motors and obtain optimal minimal cost custom designs (from the standard designs of a BDCPM motor family) for different sets of customer requirements.

PLAUSIBLE IMAGE MATCHING: DETERMINING DENSE AND SMOOTH MAPPING BETWEEN IMAGES WITHOUT A PRIORI KNOWLEDGE

2005 ◽  
Vol 19 (04) ◽  
pp. 565-583 ◽  
Author(s):  
SHUNTARO YAMAZAKI ◽  
KATSUSHI IKEUCHI ◽  
YOSHIHISA SHINAGAWA
Keyword(s):  
Cost Function ◽  
Vector Fields ◽  
A Priori ◽  
Image Features ◽  
A Priori Knowledge ◽  
Linear Filters ◽  
Smooth Mapping ◽  
Highly Nonlinear ◽  
The Cost ◽  
Priori Knowledge

This paper presents a method for automatic determination of dense and smooth mapping between two images without a priori knowledge of either the camera pose or the objects in the images. We designed an algorithm to find the mapping between a pair of arbitrary images, and accomplish automatic image morphing. In order to extract image features which look natural to human, we use a set of linear filters similar to those that are used in early vision. Then the derived vector fields consisting of filter responses are matched with each other through a minimization of the cost function which expresses the similarity of transformed images and mapping smoothness, in a multiresolutional hierarchy. Since the cost function in general is highly nonlinear, we avoid excessive distortion in the estimated mapping by providing a local convexity of mapping in nonlinear optimization. In this paper, a variety of experimental results are discussed for various data sets, including images of rotating objects, static objects, human faces and texture patterns, to demonstrate the performance of the proposed method.

scholarly journals Statistically optimized inversion algorithm for enhanced retrieval of aerosol properties from spectral multi-angle polarimetric satellite observations

2010 ◽  
Vol 3 (6) ◽  
pp. 4967-5077 ◽  
Author(s):  
O. Dubovik ◽  
M. Herman ◽  
A. Holdak ◽  
T. Lapyonok ◽  
D. Tanré ◽  
...  
Keyword(s):  
A Priori ◽  
Algorithm Design ◽  
Statistical Optimization ◽  
Satellite Observations ◽  
Spatial And Temporal Variability ◽  
Inversion Algorithm ◽  
Simultaneous Inversion ◽  
Data Inversion ◽  
Retrieval Scheme ◽  
Aerosol Properties

Abstract. The proposed development is an attempt to enhance aerosol retrieval by emphasizing statistical optimization in inversion of advanced satellite observations. This optimization concept improves retrieval accuracy relying on the knowledge of measurement error distribution. Efficient application of such optimization requires pronounced data redundancy (excess of the measurements number over number of unknowns) that is not common in satellite observations. The POLDER imager on board of the PARASOL micro-satellite registers spectral polarimetric characteristics of the reflected atmospheric radiation at up to 16 viewing directions over each observed pixel. The completeness of such observations is notably higher than for most currently operating passive satellite aerosol sensors. This provides an opportunity for profound utilization of statistical optimization principles in satellite data inversion. The proposed retrieval scheme is designed as statistically optimized multi-variable fitting of the all available angular observations of total and polarized radiances obtained by POLDER sensor in the window spectral channels where absorption by gaseous is minimal. The total number of such observations by PARASOL always exceeds a hundred over each pixel and the statistical optimization concept promises to be efficient even if the algorithm retrieves several tens of aerosol parameters. Based on this idea, the proposed algorithm uses a large number of unknowns and is aimed on retrieval of extended set of parameters affecting measured radiation. The algorithm is designed to retrieve complete aerosol properties globally. Over land, the algorithm retrieves the parameters of underlying surface simultaneously with aerosol. In all situations, the approach is anticipated to achieve a robust retrieval of complete aerosol properties including information about aerosol particle sizes, shape, absorption and composition (refractive index). In order to achieve reliable retrieval from PARASOL observations even over very reflective desert surfaces, the algorithm was designed as simultaneous inversion of a large group of pixels within one or several images. Such, multi-pixel retrieval regime takes advantage from known limitations on spatial and temporal variability in both aerosol and surface properties. Specifically the variations of the retrieved parameters horizontally from pixel-to-pixel and/or temporary from day-to-day are enforced to be smooth by additional appropriately set a priori constraints. This concept is expected to provide satellite retrieval of higher consistency, because the retrieval over each single pixel will be benefiting from co-incident aerosol information from neighboring pixels, as well, from the information about surface reflectance (over land) obtained in preceding and consequent observations over the same pixel. The paper provides in depth description of the proposed inversion concept, illustrates the algorithm performance by a series of numerical tests and presents the examples of preliminary retrieval results obtained from actual PARASOL observations. It is should be noted that many aspects of the described algorithm design considerably benefited from experience accumulated in the preceding effort on developments of currently operating AERONET and PARASOL retrievals, as well as, several core software components were inherited from those earlier algorithms.

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