Maokun Li

Inversion algorithms for large-scale geophysical electromagnetic measurements

Low-frequency surface electromagnetic prospecting methods have been gaining a lot of interest because of their capabilities to directly detect hydrocarbon reservoirs and to compliment seismic measurements for geophysical exploration applications. There are two types of surface electromagnetic surveys. The first is an active measurement where we use an electric dipole source towed by a ship over an array of seafloor receivers. This measurement is called the controlled-source electromagnetic (CSEM) method. The second is the Magnetotelluric (MT) method driven by natural sources. This passive measurement also uses an array of seafloor receivers. Both surface electromagnetic methods measure electric and magnetic field vectors. In order to extract maximal information from these CSEM and MT data we employ a nonlinear inversion approach in their interpretation. We present two types of inversion approaches. The first approach is the so-called pixel-based inversion (PBI) algorithm. In this approach the investigation domain is subdivided into pixels, and by using an optimization process the conductivity distribution inside the domain is reconstructed. The optimization process uses the Gauss–Newton minimization scheme augmented with various forms of regularization. To automate the algorithm, the regularization term is incorporated using a multiplicative cost function. This PBI approach has demonstrated its ability to retrieve reasonably good conductivity images. However, the reconstructed boundaries and conductivity values of the imaged anomalies are usually not quantitatively resolved. Nevertheless, the PBI approach can provide useful information on the location, the shape and the conductivity of the hydrocarbon reservoir. The second method is the so-called model-based inversion (MBI) algorithm, which uses a priori information on the geometry to reduce the number of unknown parameters and to improve the quality of the reconstructed conductivity image. This MBI approach can also be used to refine the conductivity image obtained using the PBI approach. The MBI also adopts the multiplicative regularized Gauss–Newton method. The unknown parameters that govern the location and the shape of an anomaly are the locations of the user-defined nodes for the boundaries of the probed region, whereas the unknown parameter that describes the physical property is the conductivity. We will show some inversion results of synthetic and field data to demonstrate the advantages of both the PBI and MBI approaches. We further demonstrate that by combining both inversion algorithms we can arrive at a better interpretation of both CSEM and MT data.

Applying Divergence-Free Condition in Solving the Volume Integral Equation

Applying divergence-free condition to volume integral equation solver will be discussed. Three schemes are available: basis reduction scheme, minimal complete volume loop basis set and expanded volume loop basis set. All of them will generate smaller matrix equations than the SWG basis. The first two schemes generate poorly-conditioned matrices that are hard to solve by iterative solvers. The expanded loop basis set is easier to solve iteratively in spite of the existence of a null space in the matrix. Moreover, the construction of the expanded loop basis set is much easier than the other two schemes.

Application of the Multiplicative Regularized Gauss–Newton Algorithm for Three-Dimensional Microwave Imaging

We apply the so-called multiplicative regularized Gauss-Newton inversion algorithm for solving three-dimensional electromagnetic microwave inverse problems. This inversion algorithm automatically adjusts the regularization parameter and when combined with the total variation type regularization function, it can provide inversion results with excellent edge-preserving characteristics. In addition, in order to deal with an extensive memory requirement for the Gauss-Newton method, we employ an implicit Jacobian calculation scheme. By using this scheme we do not have to explicitly store the Jacobian matrix. Hence, we are able to significantly reduce the memory requirement of the Gauss-Newton method albeit at an additional computational overhead. Furthermore, in order to be able to handle a large scale computational problem, both the forward and the inversion algorithms are parallelized using the MPI library, where we obtain a nearly linear speedup factor. We demonstrate efficiency and robustness of this algorithm by inverting synthetic data, Fresnel experimental data, and biomedical experimental data.

A programmable metasurface with dynamic polarization, scattering and focusing control.

Diverse electromagnetic (EM) responses of a programmable metasurface with a relatively large scale have been investigated, where multiple functionalities are obtained on the same surface. The unit cell in the metasurface is integrated with one PIN diode, and thus a binary coded phase is realized for a single polarization. Exploiting this anisotropic characteristic, reconfigurable polarization conversion is presented first. Then the dynamic scattering performance for two kinds of sources, i.e. a plane wave and a point source, is carefully elaborated. To tailor the scattering properties, genetic algorithm, normally based on binary coding, is coupled with the scattering pattern analysis to optimize the coding matrix. Besides, inverse fast Fourier transform (IFFT) technique is also introduced to expedite the optimization process of a large metasurface. Since the coding control of each unit cell allows a local and direct modulation of EM wave, various EM phenomena including anomalous reflection, diffusion, beam steering and beam forming are successfully demonstrated by both simulations and experiments. It is worthwhile to point out that a real-time switch among these functionalities is also achieved by using a field-programmable gate array (FPGA). All the results suggest that the proposed programmable metasurface has great potentials for future applications.

A contrast source inversion method in the wavelet domain

We present a contrast source inversion (CSI) algorithm using truncated wavelet representations. Specifically, we represent the unknown contrast sources and the contrast function in terms of the wavelet basis functions. In order to reduce the number of wavelet coefficients for these unknowns, we apply a progressive multiscale truncation scheme based on the reconstructed contrast function. This approach increases the robustness of the CSI algorithm for noisy or limited data, and decreases the computation time as well as the memory usage. We tested the wavelet-domain CSI method using both synthetic and experimental data. The numerical experiments show that similar results with the regular spatial-domain CSI method are obtained when the number of (independent) measurement data points is comparable to the number of the unknowns in the contrast function. The advantages of the wavelet-domain CSI method become apparent as we deal with cases where the number of measurement data points is smaller than the number of unknowns in the contrast function.

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

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 m...

Application of the multiplicative regularized contrast source inversion method on 3D experimental Fresnel data

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.

A compressed implicit Jacobian scheme for 3D electromagnetic data inversion

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 us...

A 1-Bit

An electronically reconfigurable reflectarray antenna (RRA) with 10 × 10 elements is presented with a detailed design procedure for an improved beam-scanning performance. The element, designed at Ku band using a simple patch structure with one PIN diode and two substrate layers, can be electronically controlled to generate two states with 180° phase difference and low reflection loss. A reflectarray prototype is fabricated and experimentally studied for proof of principle. The limitations of the small aperture size are analyzed in detail, and synthetic optimizations of both feed location and aperture phase distribution are used to improve the beam-scanning performance of the prototype. Experimental results agree well with the full-wave simulations, and scan beams within ±50° range are obtained with a maximum aperture efficiency of 17.9% at 12.5 GHz. Consistent scan beams are obtained from 11.75 to 13.25 GHz. Furthermore, the versatile beam-forming capability of the RRA is also demonstrated by a wide-beam pattern synthesis. A fast beam-switching time (12 μs) is theoretically analyzed and verified by the measurement.

10 \times 10

We present a three-dimensional (3D) model-based inversion algorithm for inverting electromagnetic data. In our approach, the models are described by points in the 3D space and the so-called radial basis functions are used as the interpolation functions for connecting these points. The use of the radial basis functions renders the surface of the target intrinsically smooth. The L2-norm and weighted L2-norm regularization schemes are employed to constrain the curvature and to further smooth the surface of the target. We tested this inversion algorithm using both synthetic and experimental data sets. The inversion results demonstrate that both the shapes and material properties of the target are well reconstructed. As a complementary algorithm to the pixel-based (tomographic) inversion algorithms, the model-based inversion algorithm has fewer parameters to invert; hence, it has a higher computational efficiency. The model-based inversion algorithm also gives the users the freedom to construct an initial model based on their knowledge and other related information. It helps to reduce the non-uniqueness in the data interpretation.