Stable Inversion Research Articles

Dynamical preparation and control of stable population inversion among semiconductor energy bands

Dynamical preparation and control of stable population inversion among semiconductor energy bands

Pathway-directed recyclable chirality inversion of coordinated supramolecular polymers

It remains challenging to elucidate the fundamental mechanisms behind the dynamic chirality inversion of supramolecular assemblies with pathway complexity. Herein, metal coordination driven assembly systems based on pyridyl-conjugated cholesterol (PVPCC) and metal ions (Ag+ or Al3+) are established to demonstrate pathway-directed, recyclable chirality inversion and assembly polymorphism. In the Ag(I)/PVPCC system, a competitive pathway leads Ag-Complex to form either kinetically controlled supramolecular polymer (Ag-SP I) or thermodynamically favored Ag-SP II, accompanied by reversible chiroptical inversion. Conversely, the Al(III)/PVPCC system displays a solvent-assisted consecutive pathway: the Al-Complex initially forms ethanol-containing Al-SP II, and subsequently converts into ethanol-free Al-SP I with opposite chiroptical performance upon thermal treatment. Moreover, stable chirality inversion in the solid state enables potential dynamic circularly polarized luminescence encryption when Ag(I)/PVPCC is co-assembled with thioflavin T. These findings provide the guidance for the dynamic modulation of chirality functionality in supramolecular materials for applications in information processing, data encryption, and chiral spintronics.

Hybrid-driven MRF seismic inversion for gas sand identification: A case study in the Yinggehai Basin

The Yinggehai Basin displays anomalies characterized by heightened levels of temperature and pressure. It represents a depositional model of a submarine fan with gravity-driven flow, demonstrating significant lateral reservoir heterogeneity and intricate spatial distribution of the gas reservoir. Identifying gas formations using elastic parameters as indicators relies on stable seismic inversion results. This requires regularization to alleviate ill-posedness in the inverse problem and to construct model features of the subsurface medium. Markov Random Field (MRF)is an effective soft-constrained regularization method. It enhances the marginal features of inversion results by penalizing the objective function with the gradient of neighborhood points. However, the standard MRF method relies only on the parameter model-driven and has poor applicability in areas with strong lateral inhomogeneity or complex depositional processes. In this research paper, we propose a novel approach to seismic inversion that integrates MRF neighborhood drive and incorporates both seismic data and a parametric model. Multi-order MRF neighborhoods are constructed by using seismic data in the horizontal direction (including horizontal diagonal) and parametric model data in the vertical direction (including vertical diagonal). The model-driven results are also utilized to couple seismic data and improve the stability of the hybrid-driven MRF inversion. In addition, we select the P-impedance as the parameter for inversion due to its heightened sensitivity towards gas formation within the study region. Consequently, we utilize the inversion results to delineate the presence of sandstone in the target layer and discern any indications of gas formation. The implementation of this method in the field has demonstrated its capability to enhance the stability of inversion outcomes, effectively integrating the lateral consistency of seismic data with the vertical precision of parametric model data. This approach significantly improves reservoir heterogeneity characterization and enhances accuracy in identifying sandstone and gas.

Integrated control‐structure co‐design for flexible manipulators

AbstractThis work aims to develop a methodology for the integrated control‐structure design of flexible multibody systems. Such an issue is linked to a generalization of the inverse dynamics problem, where both the feedforward control and mechanical parameters are designed to perform a desired trajectory optimally. The resulting constrained motion is aimed at improving the dynamic performance of the system, which is underactuated, with less power demand but possible elastic deformation. Stable inversion methods are required to deal with the internal dynamics related to underactuation. The proposed methodology is based on the optimal control theory. For illustrative purposes, we considered a fully actuated linear time‐invariant (LTI) system, establishing the integrated design results in a least‐squares homogeneous system and in an optimization problem. An additional formulation is proposed for the integrated design of the flexible multibody systems, where both a multi‐criterion optimization and a scalarized problem of a corresponding nonlinear programming problem are formulated to deal with the unstable zero‐dynamics related to non‐minimum phase systems and nonlinearity. The numerical simulation of an underactuated two‐masses‐spring‐damper system and a flexible robotic manipulator in planar motions are used for validation. Finally, the beneficial aspects of the integrated design are analyzed.

Evaluating acute nipple inversion, imaging findings and outcomes

PurposeAcute nipple inversion can be unsettling for patients and is sometimes associated with an underlying breast malignancy. It also poses a diagnostic challenge with lack of consensus management guidelines. This study reviewed institutional experience with new nipple inversion, including malignant association, imaging utilization, and outcomes, in an effort to improve management. MethodsA multisite institutional retrospective review was conducted of all breast imaging reports from 1/2010 to 6/2022 mentioning nipple inversion as an indication or finding. Patients with new nipple inversion, defined as arising since the time of last breast imaging exam or if reported as new by the patient/provider, were included for analysis. Retroareolar imaging findings, BI-RADS assessments/recommendations, pathology obtained from percutaneous or excisional biopsies, and follow-up imaging and clinical exams were collated. Cases of chronic or stable nipple inversion were excluded. Descriptive statistics were performed. ResultsA total of 414 patients had new nipple inversion, 387/414 (93.5 %) with benign or negative results at initial imaging and 27/414 (6.5 %) with malignant lesions. Diagnostic mammography/ultrasound detected 25/27 (92.6 %) cancers (sensitivity 92.6 %, specificity 75.5 %, PPV 20.8 %, NPV 99.3 %). Of 62 breast MRI exams performed in patients with negative mammogram/ultrasound, no cancers were detected in the retroareolar space with 2 incidental malignant lesions discovered distant from the nipple. ConclusionDiagnostic mammography/ultrasound is reliable in workups of acute nipple inversion, with a high sensitivity and NPV for excluding malignancy. Breast MRI and surgical referral should be reserved for patients with suspicious associated symptoms or clinical findings.

Analysis on stable imaging and inverse algorithm for artificial source EM data

Abstract The inversion of artificial source electromagnetic (EM) method data fundamentally involves constructing a mathematical relationship between observable data and geological structures. The aim of imaging and inversion is to construct a geophysical model that matches the observable results, thereby realizing the identification of subsurface targets. The results of EM data inversion, due to the simplicity of geophysical models, limit inversion computing efficiency. Moreover, complexity of actual geological structures, and lack of onsite observable data, are often hindered by non-uniqueness. The challenge in the interpretation of artificial source EM data is in enhancing both the precision and expeditiousness of the inversion process. It can be classified into three main types for EM data inversion: direct imaging inversion, deterministic inversion, and stochastic inversion. To enhance computational efficiency and reduce non-uniqueness in the results, effective inversion methods, prior geological information, geophysical data, and comprehensive analysis can help mitigate the issue of non-uniqueness in EM data inversion, thereby leading to more rational geophysical interpretation results. With the progress of technology such as computing centers and the development of artificial intelligence methods, future inversion techniques will become faster, more efficient, and more intelligent, and will be applied to the interpretation of artificial source EM data.

Converted wave tomography based on inverse level set and adjoint formulation

SUMMARY Shear wave velocity (Vs) is a fundamental property of elastic media whose estimation from PS converted waves is challenging and requires modelling the boundary where P to S conversion occurs. This paper presents a PS tomography where seismic wave conversion/reflection points correspond to reflectors modelled with the level-set function set to zero [ϕ(x, z) = 0]. The proposed method aims for stable Vs inversion in a seismic acquisition setting using multicomponent receivers. Synthetic models simulating true Vs, Vp and the location of the geological reflector are used in the study. The inversion starts by locating a flat reflector, ϕ(x, z) = 0, which defines the zone Ω1 between the surface and the reflector, where the initial Vs and Vp fields are also set. To calculate the traveltimes of incident PT (P wave that propagates in Ω1 from source to the reflector), converted PS and reflected PP waves, for both observed and modelled data (forward problem), the methodology proposed by Rawlinson and Sambridge is adopted. This method uses the arrival times of the P waves, Tpt, from the seismic source at each reflector point as secondary sources generating the times Tps and Tpp. These times are calculated as a solution to the eikonal equation by using the Fast Marching method. The PS and PP residual times are minimized by updating Vs, Vp and ϕ(x, z) = 0 through adjoint variables designed from a formulation using Lagrange Multipliers in a variational context. The performance of the algorithm is evaluated for models with synclinal, sinusoidal and monoclinal reflector geometries using numerical tests considering the inversion of: (1) ϕ, given the true values of Vs and Vp; (2) ϕ and Vs, given the true value of Vp; (3) ϕ and Vp, given the true value of Vs and (4) the three parameters ϕ, Vs and Vp, simultaneously. Good results are obtained by inverting Vs and ϕ, given the true value of Vp. The simultaneous inversion of the three parameters exhibits promising results, despite the illumination problems caused by the different distribution of the PS, PP and PT time gradients due to the geometry of the reflectors and the acquisition setting (sources–receivers in the same plane). The proposed tomography estimates Vs and reflector positions which could help in statics corrections and improve the lithological characterization of near surface.

Role of Bay of Bengal low‐pressure systems in the formation of mid‐tropospheric cyclones over the Arabian Sea and western India

AbstractArabian Sea mid‐tropospheric cyclones (MTCs), responsible for extreme rainfall events in Western India, often coincide with monsoon low‐pressure systems (LPSs) over the Bay of Bengal. However, the influence of Bay of Bengal LPSs on the formation of Arabian Sea MTCs remains unclear. This study utilizes the Weather Research and Forecasting Model (WRF) to investigate the atmospheric connection between these two basins. By introducing a balanced bogus vortex over the Bay of Bengal, cyclonic systems are induced over the Arabian Sea in the majority of ensemble members, exhibiting characteristics consistent with observations. In particular, as the Bay of Bengal vortex moves westward, the middle tropospheric trough deepens, horizontal wind shear increases, the low‐level Arabian Sea stable inversion layer weakens, and the middle troposphere moisture content over Western India and the northeast Arabian Sea rises. Subsequently, MTC genesis occurs over the northeast Arabian Sea along the western edge of the trough within 2–4 days of model integration. A vorticity budget analysis highlights the critical role of vorticity advection and tilting during the initial 24 h of MTC genesis, while vortex stretching becomes the dominant vorticity source during rapid intensification. To substantiate these findings further, a mechanism denial experiment is conducted using a real‐world instance of a coexistent Arabian Sea MTC and Bay of Bengal LPS, replicated in the model. In this experiment, conditions unfavorable for LPS genesis are created by cooling and drying the Bay of Bengal. The results demonstrate that the absence or reduced intensity of the Bay of Bengal LPS inhibits formation of the Arabian Sea MTC. In all, this study presents compelling evidence for the significant influence of Bay of Bengal low‐pressure systems on the formation of severe weather‐inducing MTCs over the Arabian Sea and Western India.

High‐resolution reservoir prediction method based on data‐driven and model‐based approaches

AbstractThe Jiyang depression in the southeastern part of the Bohai Bay Basin has a relatively large scale set of shale oil in the Paleogene Shahejie Formation, but the complex internal components lead to narrow frequency bands, low resolution and difficulty in reservoir information extraction. Impedance is important information for reservoir characterization, and how to predict high‐resolution impedance using available information is particularly important. Deep learning, known for its effectiveness in addressing non‐linear problems, has found extensive applications in various fields of oil and gas exploration. However, the challenges of overfitting and poor generalization persist due to the limited availability of training datasets. Besides, existing methods often use networks to solve a single problem in fact, deep learning can deal with a series of problems intelligently. In order to partially solve the above problems, an intelligent storage prediction network framework is proposed in this paper. Physical information is introduced to realize data‐driven and model‐based approaches, thus solving the problem of difficult construction of training datasets. The processing part accomplishes the high‐resolution processing of seismic records, thus solving the problems of narrow bandwidth and low resolution. Initial model constraints are introduced so as to obtain more stable inversion results. Finally, the well data is compared and analysed to identify and predict the lithology and complete the intelligent prediction of unconventional reservoirs. The results are compared with the traditional model‐driven inversion method, revealing that the approach presented in this paper exhibits higher resolution in predicting dolomite. This contributes to the establishment of a robust data foundation for reservoir evaluation.

Long-term ice mass changes in Greenland and Antarctica derived from satellite laser ranging

Time-variable gravity field models obtained from satellite gravimetric techniques allow for the assessment of ice sheet mass changes in remote polar regions, such as Greenland and Antarctica. So far, GRACE has been the primary mission for obtaining the global time-variable gravity field models. However, GRACE was launched in 2002, thus very little is known about the global mass changes before this data, as well as between GRACE and its successor – GRACE Follow-On. We derive a method of gravity field recovery based on Satellite Laser Ranging (SLR) data to geodetic satellites that allows for obtaining direct ice mass change estimates for a period longer by 10 years than that provided by the GRACE missions. The developed method is based on splitting normal equation systems and re-stacking the solutions which allow for stable inversion, reduces the correlations between obtained parameters, stabilizes the ice mass estimates in polar regions, and reduces the noise over oceans by a factor of four. The secular trends obtained from SLR are equal to −113.5 and −82.8 Gt/year, whereas these are −119.1 and −83.3 Gt/year from GRACE and GRACE-FO to degree and order 10 for Greenland and West Antarctica, respectively, for the common period of 2002–2021 and after removing the post-glacial rebound effect. Despite the conformity of the trend and patterns, an underestimation is observed in the solutions expanded to degree and order 10. Therefore, scaling factors between GRACE/GRACE-FO expanded up to a degree and order 10 × 10 and 60 × 60 were derived and applied to SLR solutions to account for the differences in mass estimates due to the truncation of the models. SLR data revealed that in Greenland the smallest ice mass trends are for 1995–2000, 2000–2005, and 2015–2020 which are equal to +54.3, −15.5, and −75.9 Gt/year. The largest ice mass depletion periods took place in 2005–2010, 2010–2015, and recently in 2019–2021 with trends of −213.9, −287.2, and −276.1 Gt/year, respectively. For Greenland and West Antarctica, the period 2010–2015 is characterized by the most enormous ice depletion events, whereas the later 5-year period of 2015–2020 provided a near mass-equilibrium for Antarctica reducing the negative trend and returning to the situation from the ‘90s when no significant ice mass changes were observed.

Stable and efficient seismic impedance inversion using quantum annealing with L1 norm regularization

Abstract Seismic impedance inversion makes a significant contribution to locating hydrocarbons and interpreting seismic data. However, it suffers from non-unique solutions, and a direct linear inversion produces large errors. Global optimization methods, such as simulated annealing, have been applied in seismic impedance inversion and achieved promising inversion results. Over the last decades, there has been an increasing interest in quantum computing. Owing to its natural parallelism, quantum computing has a powerful computational capability and certain advantages in solving complex inverse problems. In this article, we present a stable and efficient impedance inversion using quantum annealing with L1 norm regularization, which significantly improves the inversion accuracy compared to the traditional simulated annealing method. Tests on a one-dimensional 10-layer model with noisy data demonstrate that the new method exhibits significantly improved accuracy and stability. Additionally, we perform seismic impedance inversion for synthetic data from the overthrust model and field data using two methods. These results demonstrate that the quantum annealing impedance inversion with L1 norm regularization dramatically enhances the accuracy and anti-noise ability.

Asymptotical left inversion for single-input linear systems

A constructive design of a stable left inverse for single-input linear systems is proposed. Based on structural properties relating left inversion and feasibility of input reconstruction, the input observer is realized within the state-space approach, for arbitrary initial condition. While previous design is concerned with a nominal plant, that is with known parameters, we propose in a second step to apply this inversion methodology to a simplified identification problem. Reformulating the unknown parameters as a fictitious unknown input, convergent parameter estimates are retrieved from measurement. The overall methodology is illustrated by simulations.

Stable Inversion of Piecewise Affine Systems with Application to Feedforward and Iterative Learning Control

Stable Inversion of Piecewise Affine Systems with Application to Feedforward and Iterative Learning Control

Unmeasurable flexible dynamics monitoring and tracking controller design for guidance and control system of hypersonic flight vehicle

The guidance and control system (GCS) is an important and indispensable component of hypersonic flight vehicle (HFV), and it can be viewed as a special cyber–physical system (CPS). However, the monitoring of the HFV dynamics is difficult since the inertial navigation equipment can only provides the information of acceleration and angular rate. In this case, the flexible states of HFV are difficult to be measured. This paper proposes a sliding mode observer (SMO) based GCS design method for HFV. With considering the measurement difficulties, the flexible states of HFV are assumed to be unmeasurable. Then a SMO is designed to estimated them. Combined with the actual flight environment of HFV, the measuring devices are assumed to be suffering from disturbance, then the SMO should not only guarantee the estimation error stable, but also reduce the influence of external disturbances. What is worse, the input of HFV is suffering from input delay, which brings more difficulties to the controller design. By state transformation,input delay model of HFV is transformed into a delay free model,then a SMO based robust controller is proposed for HFV based on the stable inversion controller design strategy. By the proposed SMO based controller, the GCS can be constructed for HFV. Numeral simulation are carried on HFV, and simulation results show shat, the designed SMO can realize the monitoring of flexible states of HFV and the designed GCS can get a good tracking performance.

Combined non-convex second-order total variation with overlapping group sparsity for full waveform inversion

Full waveform inversion (FWI) can provide an accurate velocity model by matching observed and simulated seismograms. Mathematically, FWI is a highly ill-posed inverse problem that the observed data are independent of the model and cannot be inverted accurately. For a stable and reasonable inversion, proper regularization methods have to be taken into account. We propose a novel composite regularization for frequency-domain FWI problem, which uses a non-convex second-order total variation (TV) term and an overlapping group sparse TV (OGS-TV) regularization term. Compared with the conventional TV regularization, our method has better accuracy and robustness, and could effectively make use of the sparsity in the velocity model. Furthermore, the alternating direction multiplier algorithm with the adaptive selection of the penalty parameters is developed to solve this composite constraint problem, which can improve the stability of the FWI process. To illustrate the superior method both visually and quantitatively, we experimentally compare the proposed method with the conventional TV regularized FWI and the second-order total generalized variation regularized FWI and overlapping group sparsity regularized FWI on the well-known geological models.

ON NUMERICAL BEAMFORMING FOR CORRELATED DIPOLE TYPE SOURCES

A numerical beamforming method for correlated dipole-type sources in the frequency domain is developed. The typical configuration of the location of acoustic field sources and microphones is considered to analyze the noise generated under an aircraft flying in turbulent flow regimes. Three problems of numerical beamforming are studied, allowing one to take into account a priori information about connections between the components of the dipole function at a point. They differ in the number of real functions to be found: six (no connection), four (components are in phase) and two (the direction of the dipole moment is known). The discretization parameters of source function grids and microphones are estimated to ensure stable matrix inversion in numerical beamforming algorithms. The issues of nonuniqueness of solutions for the tangent components of the dipole function in the first and second problems are identified and discussed; for the third problem such an issue does not arise.

A New Robust Weak Supervision Deep Learning Approach for Reservoir Properties Prediction in Malaysian Basin Field

The conventional seismic inversion approach is practical for operational work, as it only uses simple linearized algorithms and assumptions, but may be less applicable when dealing with a complex geological setting, especially in the Malay basin fields, as it may introduce non-linear noises and non-unique solutions. In the Malay basin, we also frequently struggle with a scarcity of reliable well data when performing seismic inversions. This makes finding an accurate prior model for inversion challenging and contributes to high uncertainty in properties’ estimation. Implementation of deep learning for seismic inversion has become routine and has shown increasing capability in addressing nonlinearity in inverse problems. In this work, we develop a robust approach to deep learning-based seismic inversion to predict elastic properties from seismic data. The approach incorporates synthetic well and seismic data generation from a set of rock physics knowledge called the rock physics library, which plays a significant role in dataset input for network training, validation, and testing to improve elastic properties in this field. The deep learning network architecture comprising UNET and RESNET-18 with weak supervision networks has proven to be useful to enhance computational work efficiency and prediction accuracy while handling the non-linearity of the data and the non-uniqueness of the solutions. We successfully validated the proposed method on actual field data from a clastic fluvial-dominated field in the Malay basin. Upon comparative analysis with the conventional method, both inversion results are comparable and capable of identifying the reservoir occurrence and distribution. The conventional method exposed the presence of scattered amplitude noises and prominent seismic imprints masking the reservoir. Meanwhile, the proposed method showed more stable, clearer definition and fewer noise inversion results but with a faster turn-around time and a more efficient workflow. There are substantial improvements of up to 31% in correlation accuracy achieved upon implementing the proposed method for elastic properties prediction compared to the conventional. The result implies that the proposed method can provide a good elastic properties prediction framework while addressing data limitations and sparsity issues in typical deep learning-based inversions.

Constrained one-step material decomposition reconstruction of head CT data from a silicon photon-counting prototype.

Spectral CT material decomposition provides quantitative information but is challenged by the instability of the inversion into basis materials. We have previously proposed the constrained One-Step Spectral CT Image Reconstruction (cOSSCIR) algorithm to stabilize the material decomposition inversion by directly estimating basis material images from spectral CT data. cOSSCIR was previously investigated on phantomdata. This study investigates the performance of cOSSCIR using head CT datasets acquired on a clinical photon-counting CT (PCCT) prototype. This is the first investigation of cOSSCIR for large-scale, anatomically complex, clinical PCCT data. The cOSSCIR decomposition is preceded by a spectrum estimation and nonlinear counts correction calibration step to address nonideal detectoreffects. Head CT data were acquired on an early prototype clinical PCCT system using an edge-on silicon detector with eight energy bins. Calibration data of a step wedge phantom were also acquired and used to train a spectral model to account for the source spectrum and detector spectral response, and also to train a nonlinear counts correction model to account for pulse pileup effects. The cOSSCIR algorithm optimized the bone and adipose basis images directly from the photon counts data, while placing a grouped total variation (TV) constraint on the basis images. For comparison, basis images were also reconstructed by a two-step projection-domain approach of Maximum Likelihood Estimation (MLE) for decomposing basis sinograms, followed by filtered backprojection (MLE + FBP) or a TV minimization algorithm (MLE + TVmin ) to reconstruct basis images. We hypothesize that the cOSSCIR approach will provide a more stable inversion into basis images compared to two-step approaches. To investigate this hypothesis, the noise standard deviation in bone and soft-tissue regions of interest (ROIs) in the reconstructed images were compared between cOSSCIR and the two-step methods for a range of regularization constraintsettings. cOSSCIR reduced the noise standard deviation in the basis images by a factor of two to six compared to that of MLE + TVmin , when both algorithms were constrained to produce images with the same TV. The cOSSCIR images demonstrated qualitatively improved spatial resolution and depiction of fine anatomical detail. The MLE + TVmin algorithm resulted in lower noise standard deviation than cOSSCIR for the virtual monoenergetic images (VMIs) at higher energy levels and constraint settings, while the cOSSCIR VMIs resulted in lower noise standard deviation at lower energy levels and overall higher qualitative spatial resolution. There were no statistically significant differences in the mean values within the bone region of images reconstructed by the studied algorithms. There were statistically significant differences in the mean values within the soft-tissue region of the reconstructed images, with cOSSCIR producing mean values closer to the expectedvalues. The cOSSCIR algorithm, combined with our previously proposed spectral model estimation and nonlinear counts correction method, successfully estimated bone and adipose basis images from high resolution, large-scale patient data from a clinical PCCT prototype. The cOSSCIR basis images were able to depict fine anatomical details with a factor of two to six reduction in noise standard deviation compared to that of the MLE + TVmin two-stepapproach.

A borehole trajectory inversion scheme to adjust the measurement geometry for 3D travel-time tomography on glaciers

Abstract. Cross-borehole seismic tomography is a powerful tool to investigate the subsurface with a very high spatial resolution. In a set of boreholes, comprehensive three-dimensional investigations at different depths can be conducted to analyse velocity anisotropy effects due to local changes within the medium. Especially in glaciological applications, the drilling of boreholes with hot water is cost-efficient and provides rapid access to the internal structure of the ice. In turn, movements of the subsurface such as the continuous flow of ice masses cause deformations of the boreholes and complicate a precise determination of the source and receiver positions along the borehole trajectories. Here, we present a three-dimensional inversion scheme that considers the deviations of the boreholes as additional model parameters next to the common velocity inversion parameters. Instead of introducing individual parameters for each source and receiver position, we describe the borehole trajectory with two orthogonal polynomials and only invert for the polynomial coefficients. This significantly reduces the number of additional model parameters and leads to much more stable inversion results. In addition, we also discuss whether the inversion of the borehole parameters can be separated from the velocity inversion, which would enhance the flexibility of our inversion scheme. In that case, updates of the borehole trajectories are only performed if this further reduces the overall error in the data sets. We apply this sequential inversion scheme to a synthetic data set and a field data set from a temperate Alpine glacier. With the sequential inversion, the number of artefacts in the velocity model decreases compared to a velocity inversion without borehole adjustments. In combination with a rough approximation of the borehole trajectories, for example, from additional a priori information, heterogeneities in the velocity model can be imaged similarly to an inversion with fully correct borehole coordinates. Furthermore, we discuss the advantages and limitations of our approach in the context of an inherent seismic anisotropy of the medium and extend our algorithm to consider an elliptic velocity anisotropy. With this extended version of the algorithm, we analyse the interference between a seismic anisotropy in the medium and the borehole coordinate adjustment. Our analysis indicates that the borehole inversion interferes with seismic velocity anisotropy. The inversion can compensate for such a velocity anisotropy. Based on the modelling results, we propose considering polynomials up to degree 3. For such a borehole trajectory inversion, third-order polynomials are a good compromise between a good representation of the true borehole trajectories and minimising compensation for velocity anisotropy.

Robust Flight-Path Angle Consensus Tracking Control for Non-Minimum Phase Unmanned Fixed-Wing Aircraft Formation in the Presence of Measurement Errors

The robust flight-path angle consensus tracking control problem for multiple unmanned fixed-wing aircrafts is investigated in this paper, where the non-minimum phase properties and the presence of measurement errors are systematically addressed. A three-module control scheme is proposed for each aircraft: a Distributed Observer that obtains the available information from the reference system and the neighbor aircraft to provide the estimates of the reference states; a Casual Stable Inversion that calculates the bounded estimates of the desired input, desired external states, and most importantly, desired internal states to resolve the divergence issues caused by the non-minimum phase properties; and a Local Measurement Error Rejection Controller that includes a measurement error estimator (MEE) to actively compensate for the adverse effect of measurement errors to achieve robust consensus tracking control. Stability, convergence, and robustness of the proposed control are analyzed, showing that (1) the non-minimum phase issue can be systematically resolved by the designed Casual Stable Inversion to ensure aircraft internal stability and flight safety, and (2) the consensus tracking accuracy can be improved by tuning a single MEE parameter, which is favorable in practical applications to large-scale unmanned aircraft formations. Comparative simulation results with classic PID-based consensus control demonstrate the advantage of the proposed control in transient oscillations, steady-state tracking accuracy, and robustness against measurement errors.