Statistical Inversion Method Research Articles

A virtual anti-scatter grid for multi-energy photon counting detector systems

Photon-counting (PC) systems are the next technological generation of medical computed tomography (CT) imaging and is being worked on by all major system providers. CT devices that are based on PC detectors enable multi-energy data collection. The information-content of this data can be used to obtain more detailed patient data, which improves the quality of reconstructions, compared to conventional detector systems. However, PC CT systems are subject to radiation scatter as just as any other imaging systems is. Conventionally anti-scatter grids (ASG) are used to reduce the scatter effect. These are however an imperfect solution, especially for PC detectors. In this work, a software-based scatter correction method, thus a virtual ASG is proposed. The method is tailoring a new statistical model in the measurement space and combining it with the statistical inversion method called Markov chain Monte Carlo (MCMC). The method can recover the measurement data from dense projections. We present the method on simulated data of a single photon emission computed tomography (SPECT) problem for which only under-sampled data is available. However, our approach can in principle be generalised to CT, PC-CT, Positron emission tomography (PET), radiotherapy, or even digital radiography problems. The results show that the proposed model performs similarly as physical ASGs and for cases where ASGs are not possible. The model further offers a significant improvement in the quality of the reconstruction image compared to the image reconstruction from original under-sampled data.

Finding the magnetopause location using soft X-ray observations and a statistical inverse method

Finding the magnetopause location using soft X-ray observations and a statistical inverse method

Statistical inversion of normal‐mode interference spectral parameter in ocean waveguide

AbstractA statistical inversion method of estimating normal mode interference spectral parameters (including waveguide invariance and interference frequency) from one‐dimensional broadband sound intensity spectrum (1D BSIS) is presented. First, the interference phase of 1D BSIS is compensated by a non‐linear factor for each pair of unknown parameters in a prior parameter space. Second, the singular value decomposition is performed for a data matrix, which is constructed by the phase‐compensated BSIS. Then, a misfit function is defined based on both the maximum singular value and the corresponding singular vector of the data matrix. Finally, the posterior probability distributions of unknown parameters are analysed based on statistical inversion theory. This method is suitable for the situation using a single receiver and without any environmental information. Numerical simulations illustrate the validity of the presented method.

Data driven statistical method for the multiscale characterization and modelling of fiber reinforced composites

Multiscale analysis of composite laminates allows for predicting the mechanical response of these materials avoiding cumbersome experimental campaigns. The matrix and fibre material properties and the size of the Representative Volume Element (RVE) are the main parameters affecting the accuracy of multiscale models. This paper proposes a statistical inverse method to calibrate micromechanical material parameters from macroscale experiments and 3D reconstruction. First, glass fiber reinforced epoxy laminates have been analysed with Computer Tomography (CT), then, the material 3D microstructure has been reconstructed and fibre, matrix, and voids were segmented. Tensile tests have been performed on the composite specimen, measuring the surface strains with a Digital Image Correlation (DIC) system. The reconstructed volume, converted to a voxel mesh, has been used to compute the homogenized response of composite by Fast Fourier Transform (FFT) analysis. By comparing the marginal distribution of homogenized material stiffness extracted from DIC data of tensile tests, with the conditioned distribution computed by varying the FFT model parameter, a Stochastic Volume Element (SVE) is finally calibrated. A probabilistic multiscale model based on the SVE that propagates the uncertainty from the microscale to the structure level is presented.

Development of the Sanya Incoherent Scatter Radar and Preliminary Results

AbstractLed by the Institute of Geology and Geophysics, Chinese Academy of Sciences, we have built a brand‐new modular active digital phased array, with all solid‐state transmission and digital receiving incoherent scatter radar (ISR) in Sanya (18.3°N, 109.6°E), a station in low latitude China called Sanya ISR (SYISR) since 2015. The development of SYISR involved the indoor design and development of key components, an outdoor prototype test, and the production and debugging of the entire array. The unique features of SYISR include a single‐channel directly connected T/R unit and antenna, a radar array monitoring and calibration network, environmental adaptability design and open architecture. The entire radar has 4,096 channels and 5,930 modules in total. All the technical indices, mainly including a >2 MW peak power, a 43 dBi antenna gain and a <120 K noise temperature, either meet or are superior to the designed value through an independent evaluation. Waveforms of single pulse, linear frequency modulation, Barker code, long pulse and alternating code have been implemented to meet multiple purposes. Four observational modes for ionospheric experiments, including zenith stare, perpendicular to geomagnetic field, meridian scan, and all sky scan, have been developed. We have implemented time domain decoding, frequency domain decoding, and statistical inversion methods in calculating the autocorrelation function and power spectra in signal processing. The preliminary experimental results on ionospheric parameters, plasma lines, irregularities and hard targets are reasonable and encouraging, which greatly enhances our confidence in achieving our scientific goals in the future.

Statistical inverse estimation of source terms in the Helmholtz equation

In this paper, we study source identification for Helmholtz equation by using statistical inversion method. The approximate solutions are obtained from samples measured on the random boundary of the radiated electric field. Error estimates of exact solutions and their approximations are proven, and the validity of the methods is verified by numerical experiments.

Using statistical inverse methods for detecting defects in electronic components

The thermal modeling of electronic components is mandatory to optimize the cooling design versus reliability. Indeed most of failures are due to thermal phenomena [1]. Some of them are neglected or omitted by lack of data: ageing, manufacturing issues like voids in glue or solder joints, or material properties variability. Transient measurements of the junction-to-board temperature supply real thermal behavior of the component and PCB assembly to complete these missing data[2]. To complement and supplement the numerical model, inverse methods identification based on a statistical deconvolution approach, such as Bayesian one, is applied on these measurements to extract a Foster RC thermal network. The identification algorithm performances have been demonstrated on numerical as well as experimental dataset. Furthermore defects or voids can be detected using the extracted Foster RC networks.

Statistical Inversion Approach for Stress Estimation Based on Strain Monitoring in Continuously Pre-Stressed Concrete Beams

Stress is one of the most important physical indexes reflecting the mechanical behavior of concrete structures. In general, stress in structures cannot be directly monitored and can only be estimated through an established model of stress and strain. The accuracy of the estimated stress depends on the rationality of the established model for stress and strain. As the strain measured by sensors contains creep, shrinkage, and elastic strain, it is difficult to establish an analytical model for strain and stress. In this paper, a statistical inverse method was utilized to estimate the stress in continuously pre-stressed concrete beams based on the monitored strain. Stress in the beams and the model uncertainty factors were treated as model parameters. A linear-simplified method was adopted to determine the prior distribution of the stresses. The posterior distribution of the stresses at different locations during bridge construction can be obtained by the proposed method. A continuously pre-stressed concrete beam bridge was taken as the case study to verify the effectiveness of the proposed method. Additionally, the constitution of the total strain in the different construction stages was calculated. It was concluded that the creep strain is the dominant part of the total strain.

Statistical and deterministic inverse methods in the geosciences: introduction, review, and application to the nonlinear diffusion equation

Statistical and deterministic inverse methods in the geosciences: introduction, review, and application to the nonlinear diffusion equation

A synergic study on estimating surface downward shortwave radiation from satellite data

Surface downward shortwave radiation (DSR) is a fundamental variable in determining the Earth's radiation balance and is essential in many applications. Considerable efforts have been devoted to algorithm development, product generation, and validation. However, few studies have focused on comparing retrieval approaches, examining their strengths and weaknesses, and identifying the most suitable scenarios for each approach. In this study, we implemented and evaluated five representative DSR retrieval algorithms, including the forward parameterization approach, two physical inversion methods (look-up table (LUT) and optimization), and two statistical inversion methods (direct estimation and neural networks). We then proposed an algorithm-integration framework that combined the results of these DSR retrieval methods to further improve DSR estimation accuracy and consistency. To validate the DSR retrievals, we used in-situ data collected at 25 stations of the Baseline Surface Radiation Network (BSRN) over one year. Validation revealed that forward parameterization consistently performed best, with an overall root mean square error (RMSE) of 91.7 W/m2 or a relative RMSE of 16.9%, although it generated the fewest valid retrievals. For an identical data set, the LUT approach generated results comparable to those of parameterization. The neural network-based algorithm-integration approach reduced the RMSE by 11.0 W/m2 or the relative RMSE by 2.0%, compared to the best individual retrieval algorithm. Our analysis demonstrates that algorithm integration is a promising way to obtain DSR data that are superior to estimates from any individual retrieval algorithm.

Bayesian inversion for a fractional Lotka-Volterra model: An application of Canadian lynx vs. snowshoe hares

The Bayesian statistical inversion method is implemented to obtain the parameter estimations for fractional LotkaVolterra models, including the derivative orders, based on analytical solutions obtained by the multi-step homotopy method. For the posterior distributions of the parameter of interest, we used Markov Chain Monte Carlo method through the JAGS package within R software. The posterior predictive model–checking method is implemented to select the best model for a real data set.

Reconstruction of Temperature, Accumulation Rate, and Layer Thinning From an Ice Core at South Pole, Using a Statistical Inverse Method

AbstractData from the South Pole ice core (SPC14) are used to constrain climate conditions and ice‐flow‐induced layer thinning for the last 54,000 years. Empirical constraints are obtained from the SPC14 ice and gas timescales, used to calculate annual‐layer thickness and the gas‐ice age difference (Δage), and from high‐resolution measurements of water isotopes, used to calculate the water‐isotope diffusion length. Both Δage and diffusion length depend on firn properties and therefore contain information about past temperature and snow‐accumulation rate. A statistical inverse approach is used to obtain an ensemble of reconstructions of temperature, accumulation‐rate, and thinning of annual layers in the ice sheet at the SPC14 site. The traditional water‐isotope/temperature relationship is not used as a constraint; the results therefore provide an independent calibration of that relationship. The temperature reconstruction yields a glacial‐interglacial temperature change of 6.7 ± 1.0°C at the South Pole. The sensitivity of δ18O to temperature is 0.99 ± 0.03 ‰°C−1, significantly greater than the spatial slope of 0.8‰°C−1 that has been used previously to determine temperature changes from East Antarctic ice core records. The reconstructions of accumulation rate and ice thinning show millennial‐scale variations in the thinning function as well as decreased thinning at depth compared to the results of a 1‐D ice flow model, suggesting influence of bedrock topography on ice flow.

Uncertainty quantification of in situ horizontal stress with pressuremeter using a statistical inverse analysis method

Knowledge of in situ stress magnitude and orientation plays a very important role in geological/geotechnical engineering and in the development of energy resources, such as caprock integrity, waste fluid disposal, geological storage of CO2, and geothermal energy extraction. The uncertainty of estimated parameters, especially horizontal stress, from in situ tests such as pressuremeter tests is a long-standing challenge owing to the existence of uncertainties from geomaterial spatial variability, measurement errors, limited information, and modelling methods. Therefore, non-unique solutions are often encountered in pressuremeter interpretation. In this study, a statistical inverse analysis method is proposed to solve this issue by combining a closed-form solution, finite-difference model, and selected optimization algorithms. The objective of the statistical inverse analysis is to determine the optimal parameters by minimizing the sum of squared errors while providing the confidence intervals of inversed parameters. Random variables generated in the optimization process reproduce the potential parameter uncertainties. The Jacobian matrix and confidence intervals are derived from the optimization process to evaluate variability of the predicted horizontal stress and ground properties. A workflow is presented that demonstrates a statistical inverse method for analyzing pressuremeter results and helps quantify uncertainties of the ground properties and in situ stress magnitudes and orientations derived from a pressuremeter test.

Investigation of precision, accuracy and confidence of X-ray diffraction for determining crystallite size in nanopowders

X-ray diffraction (XRD) is often utilized as a method of determining bulk sample crystallite size in powder characterization. While it is generally accepted that XRD peak broadening allows for qualitative crystallite size comparisons, its use for quantitative information is still debated. This study investigates the quantitative capability of XRD for determining the crystallite sizes of magnesium oxide nanocrystals by examining the precision, accuracy and uncertainty using the whole pattern (WP) weighted least-squares and Williamson–Hall (WH) methods. The precision of the methods was investigated by re-preparing, re-running and re-analysing identical samples. Both methods were found to be precise within 2 nm. The accuracy of the methods was investigated by comparing them against independent crystallite size analyses using visual particle identification from scanning electron microscopy micrographs and from indirect calculations using Brunauer–Emmett–Teller (BET) adsorption-determined surface areas. The WP method was found to be more accurate than the WH method, which consistently underpredicted the crystallite size. Finally, the confidence of the methods was investigated using a Bayesian inference statistical inversion method. The WP method was found to have a narrower confidence distribution in its crystallite size determination than the WH method. The broad WH confidence indicates that reliable quantitative single-measurement crystallite size determinations are not feasible using the WH technique. However, the WP method demonstrated precision, accuracy and confidence, allowing quantitative crystallite size determinations to be made.

Retrieval of Oceanic Total Precipitable Water Vapor and Cloud Liquid Water from Fengyun-3D Microwave Sounding Instruments

Fengyun-3D (FY-3D) satellite is the latest polar-orbiting meteorological satellite launched by China and carries 10 instruments onboard. Its microwave temperature sounder (MWTS) and microwave humidity sounder (MWHS) can acquire a total of 28 channels of brightness temperatures, providing rich information for profiling atmospheric temperature and moisture. However, due to a lack of two important frequencies at 23.8 and 31.4 GHz, it is difficult to retrieve the total precipitable water vapor (TPW) and cloud liquid water path (CLW) from FY-3D microwave sounder data as commonly done for other microwave sounding instruments. Using the channel similarity between Suomi National Polar-orbiting Partnership (NPP) advanced technology microwave sounder (ATMS) and FY-3D microwave sounding instruments, a machine learning (ML) technique is used to generate the two missing low-frequency channels of MWTS and MWHS. Then, a new dataset named as combined microwave sounder (CMWS) is obtained, which has the same channel setting as ATMS but the spatial resolution is consistent with MWTS. A statistical inversion method is adopted to retrieve TPW and CLW over oceans from the FY-3D CMWS. The intercomparison between different satellites shows that the inversion products of FY-3D CMWS and Suomi NPP ATMS have good consistency in magnitude and distribution. The correlation coefficients of retrieved TPW and CLW between CMWS and ATMS can reach 0.95 and 0.85, respectively.

Retrieval of forest vertical leaf area index and clumping index through field measurement and remote sensing techniques: A review

Forest vertical stratification directly affects the transfer of solar radiation and water evapotranspiration in the canopy. Consequently, leaf photosynthesis varies in different parts of the canopy, affecting the growth of trees and the succession process of the plant community. The vertical distribution of forest structural parameters, such as the leaf area index (LAI) and clumping index (CI), is critical not only for understanding of the forest carbon, nitrogen, and water cycles and remote sensing radiative transfer processes, but also for forest monitoring and management practices. Forest vertical LAI and CI can be obtained through field measurement and remote sensing inversion methods. For field measurements, LAI-2200, digital hemispherical photography, and photosynthetically active radiation sensors are frequently used instruments. Vertical structural measurements are usually carried out through destructive sampling, tower-base measurement, and mobile elevators. In general, field measurements are more accurate; however, they are time-consuming and laborious, making them unsuitable for large-scale applications. Remote sensing technology uses the passive optical method, the light detection and ranging (LiDAR) technology, and synthetic aperture radar (SAR) methods to estimate forest vertical LAI and CI. The classical passive optical method is based on either the empirical vegetation index estimation method or physical model inversion method. The vegetation index method establishes an empirical relationship between vegetation structural parameters and vegetation indices. Subsequently, the relationship is used to estimate the structural parameters for different layers. The physical model inversion method is based on a physical radiative transfer model. In general, the application of both passive optical and SAR methods in forest vertical parameter retrieval is limited. As an active remote sensing technology, the LiDAR technology has shown its unique advantages in the inversion of vertical structural parameters. Based on the platform used, LiDAR can be classified into three categories: Terrestrial laser scanning (TLS), airborne laser scanning (ALS), and spaceborne laser scanning. The basic rationale of the LiDAR technology is based on the Beer-Lambert Law, which estimates canopy LAI and CI from the canopy gap fraction; both empirical statistical and physical inversion methods are used during the process. The use of LiDAR to estimate LAI and CI is gaining popularity worldwide, but most of the research is based on TLS and ALS. The development of new space-based LiDAR technologies, such as the Global Ecosystem Dynamics Investigation on the International Space Station, will greatly advance the acquisition of forest vertical structural information on a global scale. However, the relatively complex operation and data processing requirements have limited its widespread application. The vertical distribution of forest LAI can be represented by the normal, Weibull, beta, chi-square, and Johnson’s S-B distributions. The CI values usually decrease with the increase in canopy height because the upper part of the canopy usually has larger gaps. However, mature global vertical LAI and CI products that can meet the application requirements are currently unavailable. Due to the limits of ground measurements, forest vertical LAI and CI retrieved from remote sensing lack sufficient ground validation. In field measurements, the LAI values of trees, shrubs, and herbs should be acquired simultaneously in order to validate the remote sensing LAI. The development of wireless sensor networks and unmanned aerial vehicles provides convenient and effective methods to obtain a large amount of vertical structural information useful for validation purposes. Forest vertical LAI and CI have been applied in land surface and radiative transfer models and forest management studies. In future research, new field measurement and remote sensing inversion methods should be explored. Various LiDAR data sources can be used to generate high-quality vertical structural products, and these vertical products should be fully validated to better meet the requirements proposed by the research and management communities.

A gradient-based Markov chain Monte Carlo method for full-waveform inversion and uncertainty analysis

Traditional full-waveform inversion (FWI) methods only render a “best-fit” model that cannot account for uncertainties of the ill-posed inverse problem. Additionally, local optimization-based FWI methods cannot always converge to a geologically meaningful solution unless the inversion starts with an accurate background model. We seek the solution for FWI in the Bayesian inference framework to address those two issues. In Bayesian inference, the model space is directly probed by sampling methods such that we obtain a reliable uncertainty appraisal, determine optimal models, and avoid entrapment in a small local region of the model space. The solution of such a statistical inverse method is completely described by the posterior distribution, which quantifies the distributions for parameters and inversion uncertainties. To efficiently sample the posterior distribution, we introduce a sampling algorithm in which the proposal distribution is constructed by the local gradient and the diagonal approximate Hessian of the local log posterior. Our algorithm is called the gradient-based Markov chain Monte Carlo (GMCMC) method. The GMCMC FWI method can quantify inversion uncertainties with estimated posterior distribution given sufficiently long Markov chains. By directly sampling the posterior distribution, we obtain a global view of the model space. Theoretically speaking, statistical assessments do not depend on starting models. Our method is applied to the 2D Marmousi model with the frequency-domain FWI setting. Numerical results suggest that our method can be readily applied to 2D cases with affordable computational efforts.

Diagnosing the Scale- and Space-Dependent Horizontal Eddy Diffusivity at the Global Surface Ocean

AbstractOceanic tracers are transported by oceanic motions of all scales, but only the large-scale motions are resolved by the present-day Earth system models. In these models, the unresolved lateral sub-gridscale tracer transport is generally parameterized through diffusive closures with a scale-independent diffusion coefficient. However, evidence from observations and theory suggests that diffusivity varies spatially and is length-scale dependent. Here we provide new scale-dependent quantification of the global surface diffusivities. To this end we use a recently developed statistical inversion method, MicroInverse, to diagnose horizontal surface diffusivities from observed sea surface temperature and idealized model simulation. We compare the results to theoretical estimates of mixing by the large-scale shear and by the sub-gridscale velocity fluctuations. The diagnosed diffusivity magnitude peaks in the tropics and western boundary currents with minima in the subtropical gyres (~3000 and ~100 m2 s−1) at ~40-km scale, respectively. Focusing on the 40–200-km length scale range, we find that the diffusivity magnitude scales with the length scale to a power n that is between 1.22 and 1.54 (90% confidence) in the tropics and also peaks at values above 1 in the boundary currents. In the midlatitudes we find that 0.58 < n < 0.87 (90% confidence). Comparison to the theory suggests that in regions with n > 1 the horizontal mixing is dominated by large-scale shear, whereas in regions where n < 1 the horizontal mixing is due to processes that are small compared to the 40–200-km length scale range considered in this study.

Seismic AVO statistical inversion incorporating poroelasticity

Seismic amplitude variation with offset (AVO) inversion is an important approach for quantitative prediction of rock elasticity, lithology and fluid properties. With Biot–Gassmann’s poroelasticity, an improved statistical AVO inversion approach is proposed. To distinguish the influence of rock porosity and pore fluid modulus on AVO reflection coefficients, the AVO equation of reflection coefficients parameterized by porosity, rock-matrix moduli, density and fluid modulus is initially derived from Gassmann equation and critical porosity model. From the analysis of the influences of model parameters on the proposed AVO equation, rock porosity has the greatest influences, followed by rock-matrix moduli and density, and fluid modulus has the least influences among these model parameters. Furthermore, a statistical AVO stepwise inversion method is implemented to the simultaneous estimation of rock porosity, rock-matrix modulus, density and fluid modulus. Besides, the Laplace probability model and differential evolution, Markov chain Monte Carlo algorithm is utilized for the stochastic simulation within Bayesian framework. Models and field data examples demonstrate that the simultaneous optimizations of multiple Markov chains can achieve the efficient simulation of the posterior probability density distribution of model parameters, which is helpful for the uncertainty analysis of the inversion and sets a theoretical fundament for reservoir characterization and fluid discrimination.

Thermospectroscopic infrared imaging of a confined drying process

We present an infrared (IR) imaging technique that allows us to retrieve quantitative concentration and thermal maps with relatively fast acquisition times for samples that are evolving in time and have micron-scale spatial resolution. As a proof-of-concept, we image the transient drying kinetics of a μL drop of colloidal suspension in a confined geometry. Quantitative concentration maps inside the drying droplet are retrieved. Transport phenomena such as colloid redistribution inside the droplet due to inhomogeneous drying can be highlighted by this means. A numerical inverse method based on the acquired images that allows one to estimate intrinsic properties of the studied material, such as the collective diffusion coefficient of the mixture, is presented. Such a technique combined with statistical inverse methods provides a useful, non-invasive means of visualizing and estimating parameters of materials evolving in time.