Multi-parameter Inverse Problems Research Articles

DEVELOPMENT OF A METHODOLOGY FOR MONITORING THE STATE OF GAS HYDRATE DEPOSITS OF THE EAST SIBERIAN SHELF

One of the possible mechanisms causing significant emissions of methane into the atmosphere within the Arctic shelf may be the decomposition of gas hydrates. Their accumulations within the Arctic shelf formed almost simultaneously with the formation of permafrost, which contributed to the emergence of a zone of stable existence of gas hydrates. The subsequent flooding of the Arctic shelf led to the degradation of the permafrost and the violation of the conditions for the existence of hydrates. To assess the state of the stability zone, methods of mathematical numerical modeling are used. Standard seismic methods are widely used to localize gas hydrates, but monitoring their physical state requires the development of fundamentally new approaches based on solving multiparameter inverse seismic problems. In particular, the degree of attenuation of seismic energy is one of the objective parameters for assessing the consolidation of gas hydrates: the closer they are to the beginning of decomposition, the higher the attenuation, and hence the lower the quality factor. Thus, the methods of seismic monitoring of the state of gas hydrates in order to predict the possibility of developing dangerous scenarios should be based on solving a multi-parameter inverse seismic problem. This publication is devoted to the presentation of this approach.

Regularization of anisotropic full-waveform inversion with multiple parameters by adversarial neural networks

The anisotropic full-waveform inversion (FWI) is a seismic inverse problem for multiple parameters, which aims to simultaneously reconstruct the vertical velocity and the anisotropic parameters of the earth’s subsurface. This multiparameter inverse problem suffers from two issues. First, the objective function of the data fitting is less sensitive to the anisotropic parameters. Second, the crosstalk effect among the different parameters worsens the model update in the iterative inversion. We have developed a method that statistically regularizes the anisotropic FWI using Wasserstein adversarial networks, by penalizing the Wasserstein distance between the distribution of the current model parameters and that of the parameters at the borehole locations. The regularizer can mitigate the issues of anisotropic FWI with multiple parameters and therefore it also can be applied to other inverse problems with multiple parameters.

3-D complex resistivity imaging using controlled source electromagnetic data: a multistage procedure using a second order polynomial parametrization

SUMMARY In some Earth materials, significant induced polarization (IP) phenomena are occurring when an electric perturbation is applied. These mechanisms are described by a frequency-dependent complex resistivity (CR). The study of the CR spectral signature allows to access indirectly to several properties of interest of the subsurface linked to the interaction between the pore space and fluids. CR is usually studied using the electrical method with a direct current approximation, neglecting by the way electromagnetic (EM) induction that can occur in the data. However, EM induction increases with frequency and offset, resulting in limitations at high frequencies or for the investigation of deep target. We implemented a frequency-dependent CR in a 3-D finite-differences (FD) modelling and inversion code for frequency domain controlled-source electromagnetic (CSEM) data to take into consideration IP information contained in EM data or reciprocally. The CSEM methods are resistivity imaging techniques using multifrequency EM fields that fully take into account EM induction with large investigation depth. Following a preliminary sensitivity study, a multistage inversion framework was designed to constrain the multiparameter inverse problem. Furthermore, to manage the increasing number of parameters, a second-order polynomial parametrization is used to describe independently frequency variation of CR norm and phase. We demonstrate the method through 1-D and 3-D synthetic data inversions for a deep-target model. We show that we were able to recover the CR and its frequency variation from CSEM data in the IP/EM coupling domain for 1-D targets. The problem of deep polarizable 3-D targets is more challenging and the resolution of the recovered CR spectrum was impacted. Nevertheless, we retrieved from a model containing several polarizable anomalies some crucial information allowing the discrimination of the targets from the non-polarizable background and from different spectral CR signatures. Our inversion strategy allows thus accessing to IP parameters of the medium in an extended frequency domain by fully taking EM induction information into account.

Subsea Methane Hydrates: Origin and Monitoring the Impacts of Global Warming

The East Siberian Arctic shelf is the area where the largest natural gas reserves are concentrated. The formation of permafrost of the Arctic shelf during the Ice Age contributed to the emergence of a zone of stable existence of gas hydrates in the sedimentary layer, and subsequent flooding of the shelf led to its gradual degradation, the thawing of gas hydrates and the subsequent emissions of methane into the atmosphere. In the first part of the paper, we use mathematical modeling to study the processes of the formation of subsea permafrost on the Arctic shelf considering changes in the sea levels over the past 200 thousand years. Numerical simulations show the influence of climate warming up to 2200 on the degradation of subsea permafrost and the violation of the conditions for the stable existence of methane hydrates in bottom sediments using the example of the East Siberian shelf. The second part of the paper proposes a method for seismic monitoring of the state of gas hydrates based on a solution of multi-parameter inverse seismic problems. In particular, the degree of attenuation of seismic energy is one of the objective parameters for assessing the consolidation of gas hydrates: the closer they are to the beginning of decomposition, the higher the attenuation and, hence, the lower the quality factor. In this publication, we do not solve a multi-parameter inverse seismic problem for a real geological object. This would be impossible due to the lack of necessary data. Instead, we focus on substantiating the possibility of correct solutions for the problem of the reconstruction of the absorption and velocities for a viscoelastic medium in relation to the problem of monitoring the state of gas hydrate deposits. As noted in a range of publications, the thawing of gas hydrates leads to an increase in the fluid saturation of the geological medium followed by an increase in the absorption of seismic energy—that is, a decrease in the quality factor. Thus, the methods of seismic monitoring of the state of gas hydrates to predict the possibility of developing dangerous scenarios should be based on solving a multi-parameter inverse seismic problem. This publication is devoted to the presentation of this approach.

Recent advances in phase-array based wideband multimodal dispersion curves extraction and plate-waveguide parameter inversion

Multimodal ultrasonic guided waves with rich dispersion information have been widely applied for waveguide evaluation. However, there are still some challenges to extract multimode dispersion curves using the traditional pitch-catch method, including lack of spatial information and poor signal-to-noise ratio. Phase array has been recently developed for measuring ultrasonic guided waves, which brings many advantages for wideband multimodal dispersion curves extraction and parameter inversion. In the last years, basing on a multi-emitter and multi-receiver measurement, we proposed some array dispersive signal processing strategies for analyzing guided waves in long cortical bone, including the sparse singular vector decomposition (sparse-SVD) method for enhancing wavenumber estimation resolution and the low-amplitude mode extraction, Radon transform and dispersive Radon transform (DRT) with the capability to project temporal array dispersive signals on the space of parameters of interest for solving the inverse problem, and our recent work of deep neural networks for solving the intractable multiparameter inverse problem from the array signals to the waveguide elasticity. In the talk, we present (1) recent advances in array signal processing for extracting wideband multimodal dispersion curves; (2) some new perspectives to retrieve the waveguide parameters; (3) some pilot clinical results of long cortical bone evaluation using ultrasonic guided waves.

Fracture parameters estimation from azimuthal seismic data in orthorhombic medium

At present, approaches based on the AVAaz (P-wave amplitude versus the incident angle and azimuth) inversion are highly pivotal for estimating fracture parameters (namely fracture density and orientation) in vertically transverse isotropic background with parallel vertical fractures (namely VFTI medium, which is a common kind of orthorhombic rock). Nevertheless, these approaches involve solving a multi-parameter nonlinear inverse problem and the azimuthal variation of P-wave amplitude induced by fractures is easily concealed by noise and strong background information. Besides, unknown relationships of the fracture density with weakness parameters in the general VFTI medium will produce the uncertainty of predicting the fracture density by the weakness parameters or their related combinations. To solve these problems, we demonstrate the relationships of the weakness parameters of the VFTI medium with the fracture density and infill by a numerical experiment and conclude the tangential weakness parameters and their related combinations are preferable fracture density indicators. On this basis, we present a method integrating an azimuthally anisotropic elastic impedance (azimuthal AEI) equation and K-L (Karhunen-Loève) transform to robustly and reasonably estimate fracture density and orientation. The azimuthal AEI equation incorporating the prior fracture orientation and Fourier series weakens the 90° ambiguity of fracture orientation estimation and decouples weak fracture information from strong background information to enhance prediction stability. And the K-L transform improves the azimuthal variation of the P-wave amplitude induced by the fractures. Moreover, another simpler indicator of the fracture density, the standard deviation of the natural logarithm of the azimuthal AEI, is given to simplify the estimation process. The synthetic model and field data confirm the feasibility and stability of these proposed mechanisms and techniques.

Simultaneous Inversion for Reflectivity and Q Using Nonstationary Seismic Data With Deep-Learning-Based Decoupling

Building reflectivity and quality factor (Q) using nonstationary post-stack seismic data is important for vertical resolution enhancement of seismic data and reservoir identification. However, it is well-known that both reflectivity and Q affect the waveform of seismic data, leading to the fact that simultaneously estimating them is a strong ill-posed multi-parameter inverse problem which faces the crosstalk problem. In this paper, we propose a new method for simultaneous inversion of reflectivity and Q. A deep-learning-based data decoupling operator is proposed to decouple the effects of the two parameters on nonstationary seismic data. Based on the decoupled data, we transform the original multi-parameter inverse problem into two independent singe-parameter inverse problems that are immune to crosstalk and can build reasonable initial models for reflectivity and Q. Then alternative iteration is conducted to update the two built initial models to obtain the final models. A few well-logs are used to train the deep learning architecture and specific regularization terms are constructed for the inverse problem to ensure physically reasonable results. Synthetic and field data examples verify the effectiveness of the proposed method and its advantages over a conventional model-driven joint inversion method.

Solving Multiphysics, Multiparameter, Multimodal Inverse Problems: An Application to NMR Relaxation in Porous Media

A general and robust Bayesian optimization framework for the extraction of intrinsic physical properties from an integration of pore-scale forward modeling and experimental measurements of macroscopic system responses is developed. The efficiency of the scheme, which utilizes Gaussian process regression, enables the simultaneous extraction of multiple intrinsic physical properties with a minimal number of function evaluations. Here it is applied to nuclear magnetic resonance (NMR) relaxation responses, paving the way for inverse problem approaches to digital rock physics given its general nature. NMR relaxation responses of fluids in porous media may be described by sums of multiexponential decays resulting in a relaxation time distribution. The shape of this distribution is dependent on intrinsic physical system properties, but also effects like diffusion coupling between different relaxation regimes in heterogeneous porous materials. Forward models based on high-resolution images are employed to naturally incorporate structural heterogeneity and diffusive motion without limiting assumptions. Extracting the required multiple intrinsic parameters of the system poses an ill-conditioned multiphysics multiparameter inverse problem where multiple scales are covered by the underlying microstructure. Exploration of the multidimensional search space given an expensive cost function makes an efficient solution strategy mandatory. We propose a workflow to match experimental measurements with simulations via Bayesian optimization, with special attention paid to the multimodal nature of the topography of the objective function using solution space partitioning. A multimodal search strategy using state-of-the-art evolutionary algorithms and gradient-based optimization algorithms guarantees that the multimodal nature is captured. The workflow is demonstrated on ${T}_{2}$ relaxation responses of Bentheimer sandstone, extracting three physical parameters simultaneously: the surface relaxivity of quartz grains, the effective transverse relaxation time, and the effective diffusion coefficient in clay regions. Multiple mathematically sound and physically plausible solutions corresponding to global minimum and multiple local minima of the objective function are identified within a limited number of function evaluations. Importantly, the shape of the experimental ${T}_{2}$ distribution is recovered almost perfectly, enabling the use of classical interpretation techniques and local analysis of responses based on numerical simulation.

Deep Learning Analysis of Ultrasonic Guided Waves for Cortical Bone Characterization.

Ultrasonic guided waves (UGWs) propagating in the long cortical bone can be measured via the axial transmission method. The characterization of long cortical bone using UGW is a multiparameter inverse problem. The optimal solution of the inverse problem often includes a complex solving process. Deep neural networks (DNNs) are essentially powerful multiparameter predictors based on universal approximation theorem, which are suitable for solving parameter predictions in the inverse problem by constructing the mapping relationship between UGW and cortical bone material parameters. In this study, we investigate the feasibility of applying the multichannel crossed convolutional neural network (MCC-CNN) to simultaneously estimate cortical thickness and bulk velocities (longitudinal and transverse). Unlike the multiparameter estimation in most previous studies, the technique mentioned in this work avoids solving a multiparameter optimization problem directly. The finite-difference time-domain (FDTD) method is performed to obtain the simulated UGW array signals for training the MCC-CNN. The network that is exclusively trained on simulated data sets can predict cortical parameters from the experimental UGW data. The proposed method is confirmed by using FDTD simulation signals and experimental data obtained from four bone-mimicking plates and from ten ex vivo bovine cortical bones. The estimated root-mean-squared error (RMSE) in the simulated test data for the longitudinal bulk velocity ( VL ), transverse bulk velocity ( VT ), and cortical thickness (Th) is 97 m/s, 53 m/s, and 0.089 mm, respectively. The predicted RMSE in the bone-mimicking phantom experiments for VL|| , VT|| , and Th is 120 m/s, 80 m/s, and 0.14 mm, respectively. The experimental dispersion trajectories are matched with the theoretical dispersion curves calculated by the predicted parameters in ex vivo bovine cortical bone experiments. Our proposed method demonstrates a feasible approach for the accurate evaluation of long cortical bones based on UGW.

The creation of a surrogate model for restoring surface profiles of the electrophysical characteristics of cylindrical objects

One of the stages of solving the multi-parameter inverse problem of eddy current measurements for identification of the radial profiles of the electrophysical characteristics of the cylindrical control objects by the approximation method is shown. The method involves the use of a priori information on testing cylindrical control objects obtained by mathematical modelling using the “accurate” Uzal-Dodd-Deeds electrodynamic model. It is proposed to use a high-performance, in a computational sense, surrogate model of the control process as a carrier of a priori information. The analysis of methods of constructing surrogate models is carried out, and, taking into account the identified advantages in the framework of the research, the neural network method of creating metamodels is selected. The characteristic features for the construction of metamodels, consisting in the need to use complex-valued neural networks, are noted, as well as ways to solve this problem are indicated. Metamodels are based on split complex-valued neural networks. The computational technology for constructing surrogate models in the Python 3 environment using the Keras open-source library has been debugged. The objective numerical statistical indicators to verify the adequacy of metamodels are presented. Based on specific data, the created surrogate models are verified and the accuracy of the approximation of the response surfaces is estimated using the previously presented indicators. Ways of generalizing the method to more complicated cases of creating metamodels for measuring problems characterized by the presence of response hypersurfaces with a greater dimensionality of space are indicated. Attention is focused on the universality of the approach acceptable for almost any complex hypersurface topology.

Reconstruction of surface radial profiles of the electrophysical characteristics of cylindrical objects during eddy current measurements with a priori data. The selection formation for the surrogate model construction

The prospects of solving the multi-parameter inverse problem of eddy current measurements are shown. This problem is the identification of the radial profiles of the electrophysical characteristics of cylindrical control objects by the approximation method. The method involves the use of a priori information about the testing objects obtained as a result of mathematical modelling. The problem statement is completed, the main stages of its effective solution are determined. A universal mathematical model of the measurement process is presented, a set of programs for its implementation in the Python 3 is created. The adequacy of the created software is proved by calculations on the test models for simple cases when analytical solutions are known. A series of numerical experiments using a set of programs are carried out. The experiments have shown the possibility of distinction of distribution profile types of material characteristics of modelling objects. The preparatory stage of constructing an approximation surrogate model for the case of the dependence of the output signal of the eddy current probe on the electrical conductivity and magnetic permeability at a fixed excitation frequency has been completed. The preparatory stage includes the Design of Computer Experiment creation based on Sobol’s LPt-sequences and the formation of training sample based on an “accurate” electrodynamic model. The ways of generalization to more complex cases of solving measurement problems are indicated.

The Resolution Matrix in Tomographic Multiplexing: Optimization of Inter-Parameter Cross-Talk, Relative Quantitation, and Localization.

We use a resolution matrix-based Bayesian framework to compare inversion methods for tomographic fluorescence lifetime multiplexing in a diffuse medium, such as biological tissue. We consider three inversion methods; an asymptotic time domain (ATD) approach, based on a multiexponential analysis of time domain data, a direct time domain (DTD) approach, which is a minimum error solution, and a cross-talk constrained time domain (CCTD) inversion, which is a solution to an optimization problem that minimizes both error and cross-talk. We compare these methods using Monte Carlo simulations and time domain fluorescence measurements with tissue-mimicking phantoms. The ATD approach provides high accuracy of relative quantitation and spatial localization of two fluorophores embedded in a 18-mm thick turbid medium, with concentration ratios of up to 1:4.25. DTD leads to significant errors in relative quantitation and localization. CCTD provides improved quantitation accuracy over DTD, and better spatial resolution compared to ATD. We present a rigorous theoretical basis for these results and provide a complete derivation of the CCTD estimator. The Bayesian analysis also leads to a formula for rapid computation of the DTD inverse operator for large-scale tomography measurements. The ATD and CCTD inversion methods provide significant advantages over DTD for accurately estimating multiple overlapping fluorophores. Time domain fluorescence tomography, using zero cross-talk estimators, can serve as a powerful tool for quantifying multiple fluorescently labeled biological processes. The Bayesian framework presented here can be applied to general multiparameter inverse problems for the quantitative estimation of multiple overlapping parameters.

Study of Efficiency of Dividing the Problem Space as a Means to Improve Solution of Multi-parameter Inverse Problem by Adaptive Methods

The considered multi-parameter inverse problem is determination of concentrations of salts or ions in multi-component water solutions of inorganic salts by Raman spectroscopy with subsequent spectra analysis by a non-linear adaptive method (multilayer perceptron type artificial neural networks (ANN) and by a linear adaptive method (partial least squares (PLS) method based on principal component analysis). Dividing the problem space into parts by data clustering simplifies the problem within each cluster but reduces the number of samples. This study compares efficiency of application of this approach for problems with different complexity (determination of concentrations of five salts, or ten salts, or ten ions) and with various distributions of samples over concentration range of the components. Based on experimental results, limitations and areas of application of the approach are discussed.

Optimal estimator for tomographic fluorescence lifetime multiplexing.

We use the model resolution matrix to analytically derive an optimal Bayesian estimator for multiparameter inverse problems that simultaneously minimizes inter-parameter cross talk and the total reconstruction error. Application of this estimator to time-domain diffuse fluorescence imaging shows that the optimal estimator for lifetime multiplexing is identical to a previously developed asymptotic time-domain (ATD) approach, except for the inclusion of a diagonal regularization term containing decay amplitude uncertainties. We show that, while the optimal estimator and ATD provide zero cross talk, the optimal estimator provides lower reconstruction error, while ATD results in superior relative quantitation. The framework presented here is generally applicable to other multiplexing problems where the simultaneous and accurate relative quantitation of multiple parameters is of interest.

General approach to the formulation and solution of the multi-parameter inverse problems of atmospheric remote sensing with measurements and constraints of different types

A general approach to the formulation and solution of the multi-parameter inverse problems of atmospheric remote sensing with measurements and constraints of different types is considered, which is based on the concept of the presentation of constraints as virtual measurements of finite accuracy. The advantages of the approach are (1) the possibility to account for all kinds of available information (remote and in situ measurements, physical constraints, model predictions); (2) the unified description of different measurements, a priori information and constraints in the retrieval algorithm; (3) the possibility to use measurements and a priori information of different types in any combination and to assess individual contributions to information content. The approach can be considered as a convenient tool for implementation of different synergistic remote-sensing schemes and for utilization of maximum available information on sought parameters for the purpose of increasing the accuracy of atmospheric remote sensing and providing self-consistency of sets of retrieved parameters. The known formulae for degrees of freedom for signal, averaging kernels, error components, and parameters describing spatial resolution are generalized to the case of multiple sought variables, measurements, and constraints, and the peculiar features of such generalization are discussed. The example is given of the application of the approach to the interpretation of the middle atmosphere limb infrared radiance data obtained in CRISTA (Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere) experiments with special emphasis on the constraints describing coupling of the parameters due to physical processes.

Multiparameter identification of a lossy fluid-like object from its transient acoustic response

A transient acoustic wave multiparameter inverse problem is studied. The aim is to simultaneously retrieve three constitutive parameters and one geometrical parameter of a lossy fluid-like cylinder by nonlinear full-wave inversion of synthetic data. Contrary to most publications treating this subject, it is assumed herein that the retrieval model is not identical to the data model, this being so because some of the parameters (priors) in the retrieval model are different (by the simple fact of being more or less unknown) from their true counterparts in the data model. This so-called model discordance, which is the usual situation in real-world inverse problems, is a source of errors for the retrieval of the other parameters. Moreover, it is a source of non-uniqueness and instability, a fact revealed by the employment of a properly-tuned Nelder-Mead downhill Simplex optimization algorithm, and which requires a second-order regularization for its resolution. Retrieval errors as a function of prior uncertainty are computed, and found to be large, for various model discordance scenarios involving one or two uncertain priors, for data with and without noise.

Hybrid fitting of a hydrosystem model: Long‐term insight into the Beauce aquifer functioning (France)

This study aims at analyzing the water budget of the unconfined Beauce aquifer (8000 km2) over a 35 year period, by modeling the hydrological functioning and quantifying exchanged water fluxes inside the system. A distributed process‐based model (DPBM) is implemented to model the surface, the unsaturated zone and the aquifer subsystems. Based on an extensive literature review on multiparameter optimization and inverse problem, a pragmatic hybrid fitting method that couples manual and automatic calibration is developed. Three data subsets are used for calibration (10 year), validation (10 year) and test (35 year). The global piezometric head root‐mean‐square error is around 2.5 m for the three subsets and is rather uniformly spatially distributed over 78 piezometers. The sensitivity of the simulation to the different steps of the calibration process is investigated. The transmissivity field permits the fitting of the low‐frequency signal for long‐term filtering of the recharge signal, whereas the storage coefficient filters the signal with a higher frequency. For long‐term insight into aquifer system functioning, the priority is thus to first fit the transmissivity field and to assess the distributed aquifer recharge accurately. The fitted DPBM, coupled with a linear model of coregionalization, is then used to quantify the hydrosystem water mass balance between 1974 and 2009, indicating that there is yet no trend of water resources decrease neither due to climate nor to human activities.

Determination of photophysical parameters of chlorophyll α in photosynthetic organisms using the method of nonlinear laser fluorimetry

We study the possibility of solving the multiparameter inverse problem of nonlinear laser fluorimetry of molecular systems with high local concentration of fluorophores (by the example of chlorophyll α molecules in photosynthetic organisms). The algorithms are proposed that allow determination of up to four photophysical parameters of chlorophyll α from the experimental fluorescence saturation curves. The uniqueness and stability of the inverse problem solution obtained using the proposed algorithms were assessed numerically. The laser spectrometer, designed in the course of carrying out the work and aimed at nonlinear laser fluorimetry in the quasi-stationary and nonstationary excitation regimes is described. The algorithms, proposed in this paper, are tested on pure cultures of microalgae Chlorella pyrenoidosa and Chlamydomonas reinhardtii under different functional conditions.

Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

Multiparameter inverse problem of approximation by functions with given supports

Let Lp(S), 0 < p < +∞, be a Lebesgue space of measurable functions on S with ordinary quasinorm ∥·∥p. For a system of sets {Bt|t ∈ [0, +∞)n} and a given function ψ: [0, +∞)n ↦ [ 0, +∞), we establish necessary and sufficient conditions for the existence of a function f ∈ Lp(S) such that inf {∥f − g∥ppg ∈ Lp(S), g = 0 almost everywhere on S\Bt} = ψ (t), t ∈ [0, +∞)n. As a consequence, we obtain a generalization and improvement of the Dzhrbashyan theorem on the inverse problem of approximation by functions of the exponential type in L2.