Proposed Inversion Scheme Research Articles

A unified inversion scheme to process multifrequency measurements of various dispersive electromagnetic properties

Multi-frequency measurement of a dispersive electromagnetic (EM) property, such as electrical conductivity, dielectric permittivity, or magnetic permeability, is commonly analyzed for purposes of material characterization. Such an analysis requires inversion of the multi-frequency measurement based on a specific relaxation model, such as Cole-Cole model or Pelton's model. We develop a unified inversion scheme that can be coupled to various type of relaxation models to independently process multi-frequency measurement of varied EM properties for purposes of improved EM-based geomaterial characterization. The proposed inversion scheme is firstly tested in few synthetic cases in which different relaxation models are coupled into the inversion scheme and then applied to multi-frequency complex conductivity, complex resistivity, complex permittivity, and complex impedance measurements. The method estimates up to seven relaxation-model parameters exhibiting convergence and accuracy for random initializations of the relaxation-model parameters within up to 3-orders of magnitude variation around the true parameter values. The proposed inversion method implements a bounded Levenberg algorithm with tuning initial values of damping parameter and its iterative adjustment factor, which are fixed in all the cases shown in this paper and irrespective of the type of measured EM property and the type of relaxation model. Notably, jump-out step and jump-back-in step are implemented as automated methods in the inversion scheme to prevent the inversion from getting trapped around local minima and to honor physical bounds of model parameters. The proposed inversion scheme can be easily used to process various types of EM measurements without major changes to the inversion scheme.

3D gravity data-space inversion with sparseness and bound constraints

One of the most remarkable basis of the gravity data inversion is the recognition of sharp boundaries between an ore body and its host rocks during the interpretation step. Therefore, in this work, it is attempted to develop an inversion approach to determine a 3D density distribution that produces a given gravity anomaly. The subsurface model consists of a 3D rectangular prisms of known sizes and positions and unknown density contrasts that are required to be estimated. The proposed inversion scheme incorporates the Cauchy norm as a model norm that imposes sparseness and the depth weighting of the solution. A physicalbound constraint is enforced using a generic transformation of the model parameters. The inverse problem is posed in the data space, leading to a smaller dimensional linear system of equations to be solved and a reduction in the computation time. For more efficiency, the low-dimensional linear system of equations is solved using a fast iterative method such as Lanczos Bidiagonalization. The tests carried out on the synthetic data show that the sparse data-space inversion produces blocky and focused solutions. The results obtained for the 3D inversion of the field gravity data (Mobrun gravity data) indicate that the sparse data-space inversion could produce the density models consistent with the true structures.

Determination of the normalized-surface-height autocorrelation function of a two-dimensional randomly rough dielectric surface by the inversion of light-scattering data

An expression is obtained on the basis of phase perturbation theory for the contribution to the mean differential reflection coefficient from the in-plane co-polarized component of the light scattered diffusely from a two-dimensional randomly rough dielectric surface when the latter is illuminated by $s$-polarized light. This result forms the basis for an approach to inverting experimental light-scattering data to obtain the normalized-surface-height autocorrelation function of the surface. Several parametrized forms of this correlation function, and the minimization of a cost function with respect to the parameters defining these representations, are used in the inversion scheme. This approach also yields the rms height of the surface roughness, and the dielectric constant of the dielectric substrate if it is not known in advance. The input data used in validating this inversion consist of computer simulation results for surfaces defined by exponential and Gaussian surface-height correlation functions, without and with the addition of multiplicative noise, for a single or multiple angles of incidence. The reconstructions obtained by this approach are quite accurate for weakly rough surfaces, and the proposed inversion scheme is computationally efficient.

Inversion of arbitrary segmented loop source TEM data over a layered earth

The loop source TEM method has been widely used in the detection of a mined out area in China. In the cases that the laying of traditional rectangle or square transmitting loop is limited due to the presence of obstacle on the path of the loop, the changing of the shape of the transmitting loop to bypass the obstacle is a labor saving solution. A numeric integration scheme is proposed to calculate the response and Jacobian of the segmented loop source from that of an electric dipole source. The comparison of forward response between the segmented loop and square loop shows the effect of loop geometry on the decay curves. In order to interpret the data from an irregular source loop, this paper presents an inversion scheme that incorporate the effect of loop geometry. The proposed inversion scheme is validated on the synthetic data, and then applied to the field data. The result reveals that the developed inversion scheme is capable of interpreting the segmented loop source TEM field data.

Full-waveform inversion in three-dimensional PML-truncated elastic media

We are concerned with high-fidelity subsurface imaging of the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Specifically, we attempt to image the spatial distribution of the Lamé parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous formations, using surficial measurements of the soil’s response to probing elastic waves. We use the complete waveform response of the medium to drive the inverse problem , by using a partial-differential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the misfit between the observed response of the medium at select measurement locations, and a computed response corresponding to a trial distribution of the Lamé parameters. We discuss strategies that lend algorithmic robustness to our proposed inversion scheme. To limit the computational domain to the size of interest, we employ perfectly-matched-layers (PMLs). In order to resolve the forward problem, we use a recently developed hybrid finite element approach, where a displacement–stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. Time-integration is accomplished by using an explicit Runge–Kutta scheme, which is well-suited for large-scale problems on parallel computers. We verify the accuracy of the material gradients obtained via our proposed scheme, and report numerical results demonstrating successful reconstruction of the two Lamé parameters for both smooth and sharp profiles.

Modeling and Inverse Compensation of Nonmonotonic Hysteresis in VO $_2$-Coated Microactuators

Vanadium dioxide ( <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm VO_2}$</tex></formula> ) undergoes a thermally induced solid-to-solid phase transition, which can be exploited for actuation purposes. <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm VO_2}$</tex></formula> -coated silicon cantilevers demonstrate abrupt curvature changes when their temperature is varied across the phase transition. Unlike the monotonic hysteresis phenomena observed in many other smart materials, the curvature–temperature hysteresis of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\rm VO_2}$</tex></formula> actuators is nonmonotonic due to competing mechanisms associated with the material’s phase transition and the different thermal expansion coefficients of the materials that form the bilayered cantilever. Motivated by the underlying physics, a novel model for the nonmonotonic hysteresis that combines a monotonic Preisach hysteresis operator and a quadratic operator is presented. A constrained least-squares scheme is proposed for model identification, and an effective inverse control scheme is presented for hysteresis compensation. For comparison purposes, a Preisach operator with a signed density function and a single-valued polynomial model are considered. Experimental results show that, for a 300- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu \hbox{m}$</tex></formula> -long actuator, the largest modeling errors with the proposed model, the signed Preisach operator, and the polynomial approximation are 46.8, 80.3, and 483 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm m}^{-1}$</tex></formula> , respectively, over the actuated curvature range of [ <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm -}$</tex></formula> 104, 1846] <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm m}^{-1}$</tex></formula> . In addition, both the largest tracking error and root-mean-square error under the proposed inversion scheme are only around 10% of those under the polynomial-based inversion scheme.

On the correct estimation of gap fraction: How to remove scattered radiation in gap fraction measurements?

Correct estimates of gap fraction are essential for quantifying canopy architectural variables, such as leaf area and clumping indices, which modify land–atmosphere interactions. However, gap fraction measurements from optical sensors are contaminated by radiation that is scattered by plant elements and ground surfaces. In this study, we propose a simple one-dimensional, invertible, bidirectional transmission model to remove scattering effects from gap fraction measurements. To evaluate how well the proposed model computes scattered radiance under a variety of ecosystem conditions, we compared simulated scattered radiance by the proposed model to a more sophisticated three-dimensional model in four ecosystem types (oak-grass savanna, birch, pine, and spruce stands). The simple model showed good agreement with the three-dimensional model in the scattering factor (scattered radiation from leaves normalized by sky radiation), except for highly reflective stems such as birch. The simple model showed that the scattering factor is highest when the leaf area index (LAI) is low (1–2m2m−2) in a non-clumped canopy, potential errors in estimating the LAI increase with an increase in LAI, and bright land surfaces (e.g., snow and bright soil) and bright stems (e.g., birch) can contribute significantly to scattering effects. By applying the simple model with LAI-2200 data collected in an oak-grass savanna woodland, we found that the scattering factor causes significant underestimation of the LAI (up to 26% for sunny conditions, 7.7% for diffuse sky conditions) and significant overestimation of the apparent clumping index (up to 14% for sunny conditions, 4.3% for diffuse sky conditions). The LAI is underestimated because of the effect of scattered radiation on gap fraction estimates, which cause overestimation of the clumping index. Even under highly diffuse sky conditions, errors in LAI estimates due to scattering effects are not always negligible (up to 7.7% underestimation). The proposed inversion scheme provides an opportunity to quantify gap fractions, LAI, and apparent clumping index even under sunny conditions.

Inversion of borehole weak motion records observed in Istanbul (Turkey)

SUMMARY The estimation of shear wave velocity and attenuation in near-surface geology is of primary importance in engineering seismology. In fact, such knowledge is essential for site response studies when preparing improved seismic hazard scenarios. In this study, we propose a linear inversion of the spectra of a deconvolved wavefield collected by a borehole array in Istanbul, Turkey. The spectra are calculated using as a reference the recordings collected by a sensor at the surface. This allows us to minimize the effect of the deconvolution filter on the peaks’ amplitudes. The feasibility of the proposed inversion scheme and the parametrization of the velocity and attenuation models were tested and assessed using synthetic data. The real data inversion is carried out using observation from three weak motion events. Several starting models are used for each event, changing the values of the quality factor Qs, whereas the starting Vs profiles are fixed to what was obtained by the analysis of the deconvolved wavefield in the time domain. Our results showed that fairly well-constrained Vs models can be obtained while the variability of the Qs profile, both between different inversions of the same data set, and between the inversions of different data sets, might be large. This indicates that Qs variations observed in models derived by strong motion data inversions should be considered with care. However, the high agreement of the estimated soil profiles in the uppermost layer, both when using different events and different starting models, suggests that any variation in the S-wave velocity that might be estimated when using strong motion data should be reliable.

Inversion of potential field data using the structural index as weighting function rate decay

Nonparametric inverse methods provide a general framework for solving potential-field problems. The use of weighted norms leads to a general regularization problem of Tikhonov form. We present an alternative procedure to estimate the source susceptibility distribution from potential field measurements exploiting inversion methods by means of a flexible depth-weighting function in the Tikhonov formulation. Our approach improves the formulation proposed by Li and Oldenburg (1996, 1998), differing significantly in the definition of the depth-weighting function. In our formalism the depth weighting function is associated not to the field decay of a single block (which can be representative of just a part of the source) but to the field decay of the whole source, thus implying that the data inversion is independent on the cell shape. So, in our procedure, the depth-weighting function is not given with a fixed exponent but with the structural index N of the source as the exponent. Differently than previous methods, our choice gives a substantial objectivity to the form of the depth-weighting function and to the consequent solutions. The allowed values for the exponent of the depth-weighting function depend on the range of N for sources: 0 ≤N≤ 3 (magnetic case). The analysis regarding the cases of simple sources such as dipoles, dipole lines, dykes or contacts, validate our hypothesis. The study of a complex synthetic case also proves that the depth-weighting decay cannot be necessarily assumed as equal to 3. Moreover it should not be kept constant for multi-source models but should instead depend on the structural indices of the different sources. In this way we are able to successfully invert the magnetic data of the Vulture area, Southern Italy. An original aspect of the proposed inversion scheme is that it brings an explicit link between two widely used types of interpretation methods, namely those assuming homogeneous fields, such as Euler deconvolution or depth from extreme points transformation and the inversion under the Tikhonov-form including a depth-weighting function. The availability of further constraints, from drillings or known geology, will definitely improve the quality of the solution.

Robust Full Waveform Inversion Using Normalized Residual in the Frequency Domain

ABSTRACT A robust objective function for the full waveform inversion has been suggested in the frequency domain. The proposed objective function is defined as sum of complex absolute values of residual wavefields in the frequency domain. Generally, the full waveform inversion is extremely sensitive to the parameterization, frequency bandwidth, attenuation, noise and so on. Especially, noise is the most important factor in the full waveform inversion. In the frequency-domain waveform inversion, the attenuation of wavefields is one of the critical parameters for successful inversion. To verify the robustness of our algorithm, the proposed inversion scheme was tested in terms of the sensitivity to attenuation and noise. The comparison examples with both the conventional l 2-norm and the logarithmic objective function demonstrated that the proposed inversion algorithm is robust to noise and less sensitive to attenuation.

A linear sampling method for the inverse transmission problem in near-field elastodynamics

Elastic-wave shape reconstruction of buried penetrable scatterers from near-field surface measurements is examined within the framework of the linear sampling method. The proposed inversion scheme is based on a linear integral equation of the first kind whose solution becomes unbounded as the (trial) source point of the reference Green's function approaches the boundary of an elastic scatterer from its interior. We provide a comprehensive theoretical setting to establish (i) the necessary transmission problems for near-field elastodynamics and (ii) solvability properties of the postulated linear equation in the context of penetrable obstacles. A set of numerical results with simply and multiply connected elastic scatterers is included to illustrate the performance of the reconstruction technique.

The linear sampling method for the transmission problem in 2D anisotropic elasticity

In the present work, the problem of reconstructing the shape of two-dimensional elastic anisotropic inclusions embedded in isotropic media is investigated within the framework of the linear sampling method. It is well known that the latter approach has been extensively used as an inverse solver in acoustic, electromagnetic and elastic scattering problems dealing with isotropic media and only recently in anisotropic acoustics and electromagnetics. The work at hand aims at contributing to the extension of the linear sampling method to anisotropic elastic inverse scattering. As in the previous works referring to the aforementioned reconstruction method, the proposed inversion scheme is based on the unboundedness of the solution of a linear integral equation of the first kind. Numerical results are also presented for several inclusion geometries and a system thereof exhibiting the applicability of the method.

Retrieval of biophysical parameters of agricultural crops using polarimetric SAR interferometry

An existing two-layer model for forest height estimation is adapted for agricultural crops in order to develop a retrieval algorithm based on polarimetric synthetic aperture radar interferometry. This new inversion scheme is specifically tailored for vertically oriented agricultural crops, with extinction coefficients dependent on the wave polarization. Physical parameters of the vegetation scene are estimated from the location of the measured coherences in the complex plane. The proposed inversion scheme is validated experimentally with indoor wide-band polarimetric measurements on samples of corn and rice fields. Results show that the estimates of the thickness of the vegetation layer and the ground topography are reasonably accurate for a wide range of frequencies and baselines. Moreover, some interesting results are also obtained when using only dual-polarized data, which brings up new applications for present and future spaceborne missions.

Adaptive multiscale reconstruction of buried objects

In this contribution, an adaptive multiscale approach for the localization and characterization of buried objects in a half-space is proposed. The main goal of the approach is to reduce the number of elements to be estimated and so the degrees of freedom in the unknown profile. This leads to improvement of the robustness of the inversion and to an increase in the quality of reconstruction. The proposed inversion scheme is based on an adaptive, coarse-to-fine iterative strategy using spline pyramids. The global procedure consists of sequences of non-linear inversions separated by refinement steps, which overall produces an accurate, low-order representation of the sought object.

Multifrequency reconstruction of moderately rough interfaces via quasi-ray Gaussian beams

In this paper, we present a new technique for determining the surface profile of a moderately rough interface between air and a homogeneous dielectric half-space. Based on sparsely sampled step-frequency ground penetrating radar measurements, the proposed inversion scheme uses a quasi-ray Gaussian beam fast forward model, coupled with a low-order parameterization of the surface profile in terms of B-splines. The profile estimation problem is posed as a parameter optimization problem, which is solved using a multiresolution continuation method via frequency hopping. Numerical experiments establish that the algorithm is efficient and yields accurate reconstructions throughout most of the illuminated region even in noisy environments, losing accuracy only in regions with very weak illumination.

INVER2DBASE—A program to compute basement depths of density interfaces above which the density contrast varies with depth

A computer program to invert the gravity anomalies of density interfaces above which the density contrast varies with depth, is presented. The sediment–basement interface is approximated by an N-sided polygon. A function subprogram GR2DPOL and two subroutine subprograms ZOR2DPOL and SIMEQ support the main program. Subroutine ZOR2DPOL calculates the initial depth estimates of a density interface at all anomaly points on the principal profile using the infinite slab approximation. Function subprogram GR2DPOL computes the theoretical gravity anomaly of a density interface at each anomaly point on the profile and returns to the main program. Subroutine SIMEQ is used to solve n incremental depth parts of the vertices of a density interface. Partial derivatives are calculated by a simple finite-difference method. Normal equations are constructed and solved by Marquardt's algorithm. The proposed inversion scheme is independent on the equal station spacing criteria. The validity of the algorithm is demonstrated by calculating the basement depths of the Tucson basin, southern Arizona.

Rough surface inverse scattering within the Rytov approximation

A method is presented for determining the surface profile of a deterministic, perfectly conducting rough surface boundary from fixed-frequency scattering data. The proposed inversion scheme is based on the use of a perturbation expansion of the phase of the wave scattered from the surface and, as such, is analogous to the use of the Rytov approximation in inverse scattering from penetrable scatterers. A simple algorithmic procedure is derived for determining the surface profile from the scattering amplitude of the surface, and a number of examples illustrating the procedure are presented.

Minimum cross-entropy inversion of satellite photometer data

The inversion of satellite photometry data to produce a 2-D distribution of volume emission rate of optical emissions in the upper atmosphere does not provide a unique solution, but this problem may be alleviated by the application of the minimum cross-entropy (MCE) principle. This principle provides a rational criterion of choice for the selection of that distribution of volume emission rate which is maximally noninformative a subject to the constraints imposed by the data (namely, the integrated column brightness measurements). A practical and effective iterative algorithm is developed to compute the MCE reconstruction of volume emission rates. This algorithm is then applied to both synthetic and real satellite photometer data to demonstrate the effectiveness of the proposed inversion scheme.