Nonlinear Inversion Technique Research Articles

Earthquake focal mechanism analysis using data from seismic station SRKO

The Bulgarian University Seismic Network (BUSN), operated by the Faculty of Physics at Sofia University since 2023, is expanded by the newly established broadband seismic station SRKO. This study presents the first results from SRKO, in operation since August 2024, applying the INPAR method to investigate focal mechanisms of earthquakes in the surrounding regions of Bulgaria. The INPAR method, a nonlinear inversion technique, provides reliable solutions for moderate earthquakes with magnitudes between 4.0 and 5.0. Data from several data centers, including SAGE, GEOFON and EIDA, were used for this analysis. The results demonstrate the effectiveness of the INPAR method in determining earthquake mechanisms for moderate magnitudes in regions with limited seismic stations.

Nonlinear joint inversion of Rayleigh and Love wave dispersion curves based on Pearson correlation coefficient and thickness mean sharing

AbstractThe joint inversion of Rayleigh and Love waves plays a crucial role in mitigating the non‐uniqueness of surface wave inversion results and enhancing the stability of these inversions. Existing approaches to the joint inversion of Rayleigh and Love wave dispersion curves, which rely on conventional objective functions, often struggle with complex stratigraphic configurations and yield results of limited accuracy. This study introduces two novel nonlinear joint inversion techniques for Rayleigh and Love waves: Pearson correlation coefficient and thickness mean sharing. The Pearson correlation coefficient approach employs the Pearson correlation coefficient and alternating iterative objective functions to synchronize the shear wave velocity structures derived from Rayleigh and Love waves, thereby enhancing the accuracy of the joint inversion. Conversely, the thickness mean sharing method computes an average of the thickness values obtained in each iteration of the inversion, utilizing the traditional joint inversion objective function. Tests on three distinct stratigraphic structures—characterized by increasing velocity, high‐speed hard interlayers and low‐speed soft interlayers—as well as on measured data, demonstrate that the proposed methods significantly improve the stability and accuracy of nonlinear joint inversion.

Nonlinear dynamic inversion based full envelope robust flight control for coaxial compound helicopter

The flight control system of coaxial compound helicopters (CCHs) is a complex multi-variable system characterized by redundant control effectors, strong nonlinear characteristics, and multiple flight modes. This paper presents a unified robust control framework tailored for CCH full-envelope flight, utilizing nonlinear dynamic inversion and extended state observer techniques. The framework is specifically designed to tackle issues arising from a multi-mode nature, parametric uncertainties, and external disturbances. Within the unified framework, the study addresses inherent complexities such as nonlinearities and cross-coupling effects in CCH by leveraging nonlinear dynamic inversion to formulate flight control laws for both inner and outer-loops. To mitigate model uncertainties and external disturbances, the extended state observer is employed to estimate lumped disturbance effectively. Control allocation for both inner and outer-loops is devised to adapt to changes in control effector authorities and flight modes. Furthermore, an exponential control allocation strategy is proposed for the outer-loop to ensure a smooth pitch angle during transition flight. The effectiveness of the proposed control laws and control allocation strategies in achieving precise attitude tracking, transition flight and mission task elements (MTEs), even in the presence of notable model uncertainties and external disturbances, is demonstrated through numerical simulations.

Full waveform inversion of cross-hole radio frequency electromagnetic data

Summary We consider application of full waveform inversion (FWI) to radio-frequency electromagnetic (EM) data. Radio-frequency imaging (RIM) is a cross-borehole technique to image electromagnetic subsurface properties from measurements of transmitted radio-frequency waves. It is used in coal seam imaging, ore exploration and various engineering and civil engineering applications. RIM operates at frequencies from 50 kHz to several tens of MHz. It differs from other geophysical EM methods, because the frequency band includes the transition between the wave propagation and diffusion regimes. RIM data are acquired in two-dimensional cross-hole sections in a reciprocal manner. Traditionally, radio-frequency data are inverted by straight ray tomography because it is inexpensive and easy to implement. It is argued that due to attenuation, the sensitivity of the transmitted electric field is the strongest within the first Fresnel zone of the ray connecting the transmitter and receiver. While straight ray tomography is a simple to implement and fast method, the non-linearity in the relationship between model parameters and data is often strong enough to warrant non-linear inversion techniques. FWI is an iterative high resolution technique, in which the physical properties are updated to minimize the misfit between the measured and modelled wavefields. Full waveform techniques have been used and extensively studied for the inversion of seismic data, and more recently, they have been applied to the inversion of GPR data. Non-linear inversion methods for RIM data are less advanced. Their use has been hindered by the high cost of full wave modelling and the high conductivity contrasts of many RIM targets, and, to some extent, by the limitations of the measuring instruments. We present the first application of this methodology to simultaneous conductivity and permittivity inversion of RIM data. We implement the inversion in the frequency domain in two dimensions using L-BFGS optimization. We analyze the sensitivity of the data to the model parameters and the parameter trade-off and validate the proposed methodology on a synthetic example with moderate conductivity variations and localized highly conductive targets. We then apply the FWI methodology to a field data set from Sudbury, Canada. For the field data set, we determine the most appropriate preprocessing steps that take into account specific peculiarities of RIM: the insufficient prior information about the subsurface and the limitations of the measuring equipment. We show that FWI is applicable under the conditions of RIM and is robust to imperfect prior knowledge: we obtain satisfactory model recoveries starting from homogeneous initial models in all of our examples. Just as other methods, FWI underestimates large conductivity contrasts due to the loss of sensitivity of the transmitted electric field to the conductivity variations as the conductivity increases above a certain level. The permittivity inside high conductors can not be recovered, however, recovering permittivity variations in the resistive zones helps obtain better focused conductivity images with fewer artifacts. Overall, FWI produces cleaner, less noisy and higher resolution reconstructions than the methods currently used in practice.

Hamiltonian Monte Carlo based elastic full-waveform inversion of wide-angle seismic data

SUMMARY Full-waveform inversion (FWI) of seismic data provides quantitative constraints on subsurface structures. Despite its widespread success, FWI of data around the critical angle is challenging because of the abrupt change in amplitude and phase at the critical angle and the complex waveforms, especially in the presence of a sharp velocity contrast, such as at the Moho transition zone (MTZ). Furthermore, the interference of refracted lower crustal (Pg) and upper mantle (Pn) arrivals with the critically reflected Moho (PmP) arrivals in crustal and mantle studies makes the application of conventional FWI based on linearized model updates difficult. To address such a complex relationship between the model and data, one should use an inversion method based on a Bayesian formulation. Here, we propose to use a Hamiltonian Monte Carlo (HMC) method for FWI of wide-angle seismic data. HMC is a non-linear inversion technique where model updates follow the Hamiltonian mechanics while using the gradient information present in the probability distribution, making it similar to iterative gradient techniques like FWI. It also involves procedures for generating distant models for sampling the posterior distribution, making it a Bayesian method. We test the performance and applicability of HMC based elastic FWI by inverting the non-linear part of the synthetic seismic data from a three-layer and a complex velocity model, followed by the inversion of wide-angle seismic data recorded by two ocean bottom seismometers over a 70 Ma old oceanic crustal segment in the equatorial Atlantic Ocean. The inversion results from both synthetic and real data suggest that HMC based FWI is an appropriate method for inverting the non-linear part of seismic data for crustal studies.

2D boundary-condition-free nonlinear inversion technique applied to optical shear vibration induced microelastography

Optical microelastography (OME) has emerged as a new technique for quantifying cellular mechanical properties. However, accurately reconstructing viscoelastic properties at the microscale level from noisy 2D displacement fields remains a challenge. This study introduces a 2D boundary-condition-free nonlinear inversion (2D-NoBC-NLI) approach, addressing challenges of interpreting noisy data and deducing full-field 3D displacements from 2D measurements. OME requires vibrating the cell and mapping the shear modulus based on wave-induced displacements within the cell. The shear modulus distribution is recovered via a coupled adjoint field NLI reconstruction to allow 2D-NoBC-NLI. Validation was conducted through numerical simulations at 36 kHz on a homogeneous sphere of 75 μm diameter and an assigned viscoelastic modulus, G*, of 800 + i150 Pa. The same reconstruction approach was also applied to experimental data obtained from polyacrylamide (PAAm) microbeads of the same diameter. Results demonstrated relative differences from true simulated values of 0.7% and 45% for storage and loss moduli, respectively, with a coefficient of variation under 1% for homogeneous regions. When applying this method to PAAm microbeads, viscoelastic reconstructions showed the potential of OME under experimental conditions. These findings highlight the accuracy of 2D-No BC-NLI reconstruction in OME for precise microscale characterization and mapping of the viscoelastic cell structure.

Magnetic data modelling of salt domes in Eastern Mediterranean, offshore Egypt

We analyzed magnetic data of a deep-water area in the Eastern Mediterranean, offshore Egypt where there are numerous salt domes. Regional-residual separation of the reduced to pole (RTP) magnetic anomalies was performed with the discrete wavelet transform, and the residual magnetic anomalies were interpreted as mainly due to the susceptibility contrast between sediments and salt bodies. A multiscale boundary analysis of the anomalies yielded meaningful lineaments at both large and short scales. Salt bodies lineaments are very well defined in the short-scale map. Then, we inverted the data by a nonlinear and non-iterative 3D inversion technique, to delineate the salt bodies and estimate their magnetization contrast. We used the depths obtained from the Euler deconvolution to form a set of maxima and minima constraints in the inversion. Our procedure yielded an interesting map of the top of salt in the deep-water region, entirely based on the magnetic data, which agrees well with the seismic top of salt. This is a somewhat surprising result, in some respects not obvious, demonstrating that magnetic prospecting can be an important and low-cost tool for the exploration of salt diapirs, especially when seismic data are partially or completely inaccessible or of poor quality.

Imaging the subcellular viscoelastic properties of mouse oocytes

In recent years, cellular biomechanical properties have been investigated as an alternative to morphological assessments for oocyte selection in reproductive science. Despite the high relevance of cell viscoelasticity characterization, the reconstruction of spatially distributed viscoelastic parameter images in such materials remains a major challenge. Here, a framework for mapping viscoelasticity at the subcellular scale is proposed and applied to live mouse oocytes. The strategy relies on the principles of optical microelastography for imaging in combination with the overlapping subzone nonlinear inversion technique for complex-valued shear modulus reconstruction. The three-dimensional nature of the viscoelasticity equations was accommodated by applying an oocyte geometry-based 3D mechanical motion model to the measured wave field. Five domains-nucleolus, nucleus, cytoplasm, perivitelline space, and zona pellucida-could be visually differentiated in both oocyte storage and loss modulus maps, and statistically significant differences were observed between most of these domains in either property reconstruction. The method proposed herein presents excellent potential for biomechanical-based monitoring of oocyte health and complex transformations across lifespan. It also shows appreciable latitude for generalization to cells of arbitrary shape using conventional microscopy equipment.

Consensus Tracking of Nonlinear Agents Using Distributed Nonlinear Dynamic Inversion with Switching Leader-Follower Connection

In this paper, a consensus tracking protocol for nonlinear agents is presented, which is based on the Nonlinear Dynamic Inversion (NDI) technique. Implementation of such a technique is new in the context of the consensus tracking problem. The tracking capability of nonlinear dynamic inversion (NDI) is exploited for a leader-follower multi-agent scenario. We have provided all the mathematical details to establish its theoretical foundation. Additionally, a convergence study is provided to show the efficiency of the proposed controller. The performance of the proposed controller is evaluated in the presence of both (a) random switching topology among the agents and (b) random switching of leader-follower connections, which is realistic and not reported in the literature. The follower agents track various trajectories generated by a dynamic leader, which describes the tracking capability of the proposed controller. The results obtained from the simulation study show how efficiently this controller can handle the switching topology and switching leader-follower connections.

Nonlinear amplitude inversion using a hybrid quantum genetic algorithm and the exact zoeppritz equation

The amplitude versus offset/angle (AVO/AVA) inversion which recovers elastic properties of subsurface media is an essential tool in oil and gas exploration. In general, the exact Zoeppritz equation has a relatively high accuracy in modelling the reflection coefficients. However, amplitude inversion based on it is highly nonlinear, thus, requires nonlinear inversion techniques like the genetic algorithm (GA) which has been widely applied in seismology. The quantum genetic algorithm (QGA) is a variant of the GA that enjoys the advantages of quantum computing, such as qubits and superposition of states. It, however, suffers from limitations in the areas of convergence rate and escaping local minima. To address these shortcomings, in this study, we propose a hybrid quantum genetic algorithm (HQGA) that combines a self-adaptive rotating strategy, and operations of quantum mutation and catastrophe. While the self-adaptive rotating strategy improves the flexibility and efficiency of a quantum rotating gate, the operations of quantum mutation and catastrophe enhance the local and global search abilities, respectively. Using the exact Zoeppritz equation, the HQGA was applied to both synthetic and field seismic data inversion and the results were compared to those of the GA and QGA. A number of the synthetic tests show that the HQGA requires fewer searches to converge to the global solution and the inversion results have generally higher accuracy. The application to field data reveals a good agreement between the inverted parameters and real logs.

Dynamic modeling and robust nonlinear control of a laboratory gas turbine engine

In this paper, a physics-based mathematical model of a mini SR-30 gas turbine engine (GTE) is presented using a state variable modeling method. This model is developed by utilizing approximated maps of the engine components and generic governing equations that describe the engine's behavior. The mathematical model captures both the steady-state and transient behavior of the engine. Using the developed model, a modern control system design approach is proposed to satisfy the stability and performance of the GTE. The objective of this control design is to adjust the required fuel flow rate to the combustion chamber such that the shaft speed of the engine tracks its command signal as closely as possible. The proposed robust nonlinear fuel flow controller is developed based on the nonlinear dynamic inversion (NDI) technique, augmented with a dual extended Kalman filter (DEKF) estimation design. DEKF-based state and parameter estimation are used here to ensure robustness against model inaccuracies due to not accurately known component maps, as well as process and measurement noises. To validate the developed mathematical model, the realistic data extracted from the SR-30 GTE are utilized. The simulation results of the proposed control design approach indicate that it is effective in achieving satisfactory performance of the engine. Also, the proposed NDI controller is compared with the well-established proportional-integral-derivative (PID) controller to illustrate its superiority.

Angular acceleration estimation-based incremental nonlinear dynamic inversion for robust flight control

The robustness of the incremental nonlinear dynamic inversion (INDI) technique depends on the accuracy of the feedback angular accelerations. However, angular accelerations are usually not readily available. They are obtained from differentiation and are sensitive to time delays and noise. These undesired effects decrease the robustness of the system to disturbances and model uncertainties. A novel INDI-based flight control strategy, named angular acceleration estimation-based INDI (EINDI), is proposed in this paper to solve the problem of acquiring accurate angular accelerations. The EINDI method combines the control surface deflections and an adaptive technique to estimate angular accelerations, which can reduce the effects of noise and time delays on angular accelerations, thereby ensuring the robustness of the system. Furthermore, stability analysis based on the Lyapunov theory demonstrated that the EINDI was robust to model uncertainties. Simulation results showed that EINDI was effective in reducing the influence of noise and time delays and overcoming the influence of the center of gravity (CG) changes and a series of model uncertainties.

The consensus of non-linear agents under switching topology using dynamic inversion in the presence of communication noise and delay

In this study, a consensus protocol for non-linear multi-agent systems using the non-linear dynamic inversion (NDI) technique is presented. It is named as distributed NDI or DNDI. The agents are considered to be working in a randomly changing environment which is realistic. The randomness in the operating environment is introduced by random switching communication topology, delays, and noise. The consensus protocol is obtained as a closed-form expression, which is a critical requirement for real-time implementation. Also, various cases regarding the communication issues have been considered to study the performance of the DNDI controller. The simulation results are found to be satisfactory.

Joint multizonal transdimensional Bayesian inversion of surface wave dispersion and ellipticity curves for local near-surface imaging

SummaryPhysical properties of near-surface soil and rock layers play a fundamental role in the seismic site effects analysis, being an essential element of seismic hazard assessment. Site-specific mechanical properties (i.e. shear- and compressional-wave velocities and mass density) can be inferred from surface wave dispersion and horizontal-to-vertical or ellipticity data by non-linear inversion techniques. Nevertheless, results typically exhibit significant inherent non-uniqueness as different models may fit the data equally well. Standard optimization inversion techniques minimize data misfit, resulting in a single representative model, rejecting other models providing similar misfit values. An alternative inversion technique can be formulated in the Bayesian framework, where the posterior probability density on the model space is inferred. This paper introduces an inversion approach of surface wave dispersion and ellipticity data based on a novel multizonal transdimensional Bayesian formulation. In particular, we parametrize 1-D layered velocity models by the varying number of Voronoi nuclei, allowing us to treat the number of layers as an unknown parameter of the inverse problem. The chosen parametrization leads to the transdimensional formulation of the model space, sampled by a reversible jump Markov chain Monte Carlo algorithm to provide an ensemble of random samples following the posterior probability density of model parameters. The used type of the sampling algorithm controls a model complexity (i.e. the number of layers) self-adaptively based on the measured data's information content. The method novelty lies in the parsimonious selection of sampling models and in the multizonal formulation of prior assumptions on model parameters, the latter allows including additional site-specific constraints in the inversion. These assumptions may be based on, e.g. stratigraphic logs, standard penetration tests, known water table, and bedrock depth. The multizonal formulation fully preserves the validity of the transdimensional one, as demonstrated analytically. The resultant ensemble of model samples is a discrete approximation of the posterior probability density function of model parameters and associated properties (e.g. VS30, quarter-wavelength average velocity profile and theoretical SH-wave amplification function). Although the ultimate result is the posterior probability density function, some representative models are selected according to data fit and maximum of the posterior probability density function. We first validate our inversion approach based on synthetic tests and then apply it to field data acquired from the active seismic survey and single-station measurements of ambient vibrations at the SENGL seismic station site in central Switzerland.

Analysis of a Nonlinear Technique for Microwave Imaging of Targets Inside Conducting Cylinders

Microwave imaging of targets enclosed in circular metallic cylinders represents an interesting scenario, whose applications range from biomedical diagnostics to nondestructive testing. In this paper, the theoretical bases of microwave tomographic imaging inside circular metallic pipes are reviewed and discussed. A nonlinear quantitative inversion technique in non-Hilbertian Lebesgue spaces is then applied to this kind of problem for the first time. The accuracy of the obtained dielectric reconstructions is assessed by numerical simulations in canonical cases, aimed at verifying the dependence of the result on the size of the conducting enclosure and comparing results with the conventional free space case. Numerical results show benefits in lossy environments, although the presence and the type of resonances should be carefully taken into account.

Adaptive non‐linear integral sliding mode fault‐tolerant control allocation scheme for octorotor UAV system

This study proposes a new active fault-tolerant control (FTC) and fault estimation scheme for a non-linear octorotor system. The proposed method utilises the idea of an online control allocation (CA) scheme to fully engage the rotors redundancy based on the information from the fault estimation unit. The nominal performance is first achieved using non-linear dynamic inversion (NDI) technique and then to incorporate the robustness, an adaptive non-linear sliding mode control is united with a baseline NDI controller. The proposed method is used to attain the desired altitude and attitude tracking control of an octorotor system. Furthermore, to control the un-actuated states (called internal dynamics) of octorotor system, a separate integral sliding mode-based NDI controller is designed that provides the translational axes control by generating the desired roll and pitch commands. Simulations on the non-linear model of octorotor system validate the dominant performance of the proposed scheme compared to the existing methods in the literature.

Three‐dimensional electrical structure of the northwestern margin of the Karamay region, China, revealed by magnetotelluric data

ABSTRACTIn order to investigate the three‐dimensional structures of intrusive granite and the deep structure of the Darbut fault in the northwestern margin of the Karamay region, western Junggar Basin, China, new magnetotelluric data were collected along six profiles across the Darbut fault. The magnetotelluric data were processed using a robust estimation technique to obtain the magnetotelluric impedance. Then the off‐diagonal components of the impedance tensor were inverted using a three‐dimensional nonlinear conjugate gradient inversion technique, which was performed using open‐source three‐dimensional electromagnetic inversion software. The final three‐dimensional model includes two major low‐resistivity anomalies and two major high‐resistivity anomalies. The first low‐resistivity anomaly corresponds to the location of the Darbut fault, which indicates that metallic elements are abnormally enriched there. The second one is located beneath the Darbut fault, and it is most likely a magma channel in the middle crust. The two major high‐resistivity anomalies are distributed on either side of the Darbut fault. We interpret them as Karamay rocks and Akebasitao rocks and suggest that they were formed in an extensional setting. The cross sections of three‐dimensional magnetotelluric inversion result reveal that the Darbut fault has been reformed by the later magmatism, leading to the variation of its downward extent along its strike. Moreover, our inversion result also indicates that a magma channel exists in the middle crust of the region.

A Neural Adaptive Approach for Active Fault-Tolerant Control Design in UAV

Faults in aircraft actuators can cause serious issues in safety. Due to the autonomous nature of the unmanned aerial vehicles (UAVs), faults can lead to more serious problems in these systems. In this paper, a new active fault tolerant control (FTC) system design for an UAV is presented. The proposed design uses a neural network adaptive structure for fault detection and isolation (FDI), then, the FDI signal combined with a nonlinear dynamic inversion technique is used to compensate for the faults in the actuators. The proposed scheme detects and isolates faults in the actuators of the system in real-time without the need of controller reconfiguration in presence of faults in the actuators. The simulation results of the designed FTC system when it is applied to WVU YF-22 UAV show that the proposed design can successfully detect and isolate the faults and compensate for their effect.

Reducing seismic reflector distortion beneath gas clouds in the Malay Basin using full-wavefield imaging approaches

A seismic example from the Malay Basin is presented, demonstrating improved seismic imaging beneath gas clouds using full-wavefield imaging approaches. Overall imaging concepts, synthetic examples, and field implementation strategies are discussed, and results that tie with well information are presented. Seismic imaging beneath gas clouds using the full-wavefield redatuming technique improves the image by estimating the waveform transmission operators via equivalent-medium representation of the overburden from the gas cloud reflection response for use in a form of multidimensional deconvolution of the wavefield. The other example shown uses the full-wavefield migration seismic imaging technique, which utilizes primaries and higher-order multiples as signals to improve the reflectivity estimation in imaging. The demonstrated full-wavefield imaging approach uses information carried by the gas cloud reflection response to correct seismic image distortion. Removing the internal multiple using conventional demultiple processing in the gas cloud area will also remove the valuable information of the subsurface that it carries. Such multiples must be preserved for this method to be successful. The information is translated into transmission operators that are estimated by simulating the reflection response through an effective medium of the gas cloud overburden. The effective medium is obtained via nonlinear full-waveform inversion techniques from the reflections of the gas cloud overburden area. Finally, a deconvolutional process removes the transmission operators from the gas cloud reflections and recovers the underlying reflectors. Full-wavefield imaging can reconstruct the amplitudes of the reflection response below a gas cloud overburden zone so that the complex transmission imprint on the area underneath is removed properly. The Malay Basin field case study shows that implementation of this approach can provide a reliable amplitude image of the subsurface affected by gas clouds, calibrated and verified by the well information.

Flight Control Design using Incremental Nonlinear Dynamic Inversion with Fixed-lag Smoothing Estimation

In this study, a flight control design strategy based on incremental nonlinear dynamic inversion (INDI) and smoothing algorithm is presented. The INDI is an enhanced version of the nonlinear dynamic inversion technique with a better robust performance. It reduces the aircraft model dependence via the feedback information of the state derivative, that is, angular acceleration for the aircraft attitude motion (or angular velocity related to the wind axis parameters). However, the state derivatives cannot always be obtained by direct measurement, thus they need to be estimated on-line. Moreover, taking into account that INDI requires one-step delay of state derivative and the fact that the inertial measurement unit (IMU) operates faster than the flight control computer (FCC), there will be additional information that can be used to improve the estimation. Therefore, a fixed-lag smoothing algorithm based on a discrete Kalman filter is proposed for angular velocity and angular acceleration estimation. The smoother utilizes the state variable from the sensor measurement and thus dealing with noise and delay.