Statistical Moments Research Articles

The Use of Contrast and Gradient Features to Categorize Texture Images

Image texture is an important part of many types of images, for example medical images. Texture Analysis is the technique that uses measurable features to categorize complex textures. The main goal is to extract discriminative features that are used in different pattern recognition applications and texture categorization. This paper investigates the extraction of most discriminative features for different texture images from the “Colored Brodatz” dataset using two types of image contrast measures, as well as using the statistical moments on five bands (red, green, blue, grey, and black). The Euclidean distance measure is used in the matching step to check the similarity degree. The proposed method was tested on 112 classes of textures. The achieved results showed that the proposed method is accurate and fast concerning classification accuracy with low computational complexity. The achieved Recognition Rate (RR) was 100%.

Radial Basis Function Surrogates for Uncertainty Quantification and Aerodynamic Shape Optimization under Uncertainties

This paper investigates the adequacy of radial basis function (RBF)-based models as surrogates in uncertainty quantification (UQ) and CFD shape optimization; for the latter, problems with and without uncertainties are considered. In UQ, these are used to support the Monte Carlo, as well as, the non-intrusive, Gauss Quadrature and regression-based polynomial chaos expansion methods. They are applied to the flow around an isolated airfoil and a wing to quantify uncertainties associated with the constants of the γ−R˜eθt transition model and the surface roughness (in the 3D case); it is demonstrated that the use of the RBF-based surrogates leads to an up to 50% reduction in computational cost, compared with the same UQ method that uses CFD computations. In shape optimization under uncertainties, solved by stochastic search methods, RBF-based surrogates are used to compute statistical moments of the objective function. In applications with geometric uncertainties which are modeled through the Karhunen–Loève technique, the use on an RBF-based surrogate reduces the turnaround time of an evolutionary algorithm by orders of magnitude. In this type of applications, RBF networks are also used to perform mesh displacement for the perturbed geometries.

Efficient multi-fidelity computation of blood coagulation under flow.

Clot formation is a crucial process that prevents bleeding, but can lead to severe disorders when imbalanced. This process is regulated by the coagulation cascade, a biochemical network that controls the enzyme thrombin, which converts soluble fibrinogen into the fibrin fibers that constitute clots. Coagulation cascade models are typically complex and involve dozens of partial differential equations (PDEs) representing various chemical species' transport, reaction kinetics, and diffusion. Solving these PDE systems computationally is challenging, due to their large size and multi-scale nature. We propose a multi-fidelity strategy to increase the efficiency of coagulation cascade simulations. Leveraging the slower dynamics of molecular diffusion, we transform the governing PDEs into ordinary differential equations (ODEs) representing the evolution of species concentrations versus blood residence time. We then Taylor-expand the ODE solution around the zero-diffusivity limit to obtain spatiotemporal maps of species concentrations in terms of the statistical moments of residence time, [Formula: see text], and provide the governing PDEs for [Formula: see text]. This strategy replaces a high-fidelity system of N PDEs representing the coagulation cascade of N chemical species by N ODEs and p PDEs governing the residence time statistical moments. The multi-fidelity order (p) allows balancing accuracy and computational cost providing a speedup of over N/p compared to high-fidelity models. Moreover, this cost becomes independent of the number of chemical species in the large computational meshes typical of the arterial and cardiac chamber simulations. Using a coagulation network with N = 9 and an idealized aneurysm geometry with a pulsatile flow as a benchmark, we demonstrate favorable accuracy for low-order models of p = 1 and p = 2. The thrombin concentration in these models departs from the high-fidelity solution by under 20% (p = 1) and 2% (p = 2) after 20 cardiac cycles. These multi-fidelity models could enable new coagulation analyses in complex flow scenarios and extensive reaction networks. Furthermore, it could be generalized to advance our understanding of other reacting systems affected by flow.

Automotive Electronic Control Unit Ground Line Health Monitoring Method

Electronic Control Units (ECUs) have been used in the automotive industry for decades to control one or more of the vehicle subsystems. The ECUs communicate primarily using the in-vehicle Controller Area Network (CAN) communication protocol. The recent rapid development of connected, electric, and autonomous vehicles expands the number of ECUs and complexity of the CAN network required to integrate vehicle systems and deliver the desired functionalities. This demands increased reliability of the ECUs to ensure for robust vehicle performance. One of the most common ECU failure modes is the ECU ground fault. A ground fault occurs when the ground path in the ECU circuit is corroded, which is usually developed slowly over time. Such failure usually results in various symptoms including ECU incapable of functioning and further impacts the vehicle functionalities negatively. This type of fault can be difficult to detect prior to vehicle functionality loss. It usually involves routinely testing the resistance of the ground circuit, visually inspecting the connectors and wirings, and checking the voltage drop across the ground circuit. Therefore, it is highly desirable to continuously monitor the ECU ground line health status to predict any degradation and thus prevent vehicle functionality losses.
 This paper presents a novel method to monitor the health status of ECU ground line. The method leverages measured CAN voltage data to estimate the ECU ground state of health. The CAN voltage measurements are preprocessed and fed into a real-time data buffer of predefined size. Statistical moments are calculated from the buffered data to generate health indicators, which are then combined to form a fused health indicator. The fused health indicator is used to determine the health stage of ECU ground line. The health stage is classified based on the relationship between ground line degradation level and the ECU communication loss status. The method was developed and validated using actual vehicle data.

Exploring the parameter space of an endohedral atom in a cylindrical cavity.

Endohedral fullerenes, or endofullerenes, are chemical systems of fullerene cages encapsulating single atoms or small molecules. These species provide an interesting challenge of Potential Energy Surface determination as examples of non-covalently bonded, bound systems. While the majority of studies focus on C60 as the encapsulating cage, introducing some anisotropy by using a different fullerene, e.g., C70 can unveil a double well potential along the unique axis. By approximating the potential as a pairwise Lennard-Jones (LJ) summation over the fixed C cage atoms, the parameter space of the Hamiltonian includes three tunable variables: (M, ɛ, σ) representing the mass of the trapped species, the LJ energy, and length scales respectively. Fixing the mass and allowing the others to vary can imitate the potentials of endohedral species trapped in more elongated fullerenes. We choose to explore the LJ parameter space of an endohedral atom in C70 with ɛ ∈ [20, 150 cm-1], and σ ∈ [2.85, 3.05 Å]. As the barrier height and positions of these wells vary between [1, 264 cm-1] and [0.35, 0.85 Å] respectively, using a 3D direct product basis of 1D harmonic oscillator (HO) wavefunctions centred at the origin where there is a local maximum is unphysical. Instead we propose the use of a non-orthogonal basis set, using 1D HO wavefunctions centred in each minimum and compare this to other choices. The ground state energy of the X@C70 is tracked across the LJ parameter space, along with its corresponding nuclear translational wavefunctions. A classification of the wavefunction characteristics, namely the prolateness and "peanut-likeness" based on its statistical moments is also proposed. Excited states of longer fullerenes are assigned quantum numbers, and the fundamental transitions of Ne@C70 are tracked across the parameter space.

Multispectral polarization microscopy of different stages of human oral tissue: A polarization study.

Many optical techniques have been used in various diagnostics and biomedical applications since a decade and polarization imaging is one of the non-invasive and label free optical technique to investigate biological samples making it an important tool in diagnostics, biomedical applications. We report a multispectral polarization-based imaging of oral tissue by utilizing a polarization microscope system with a broadband-light source. Experiments were performed on oral tissue samples and multispectral Stokes mapping was done by recording a set of intensity images. Polarization-based parameters like degree of polarization, angle of fast axis, retardation and linear birefringence have been retrieved. The statistical moments of these polarization components have also been reported at multiples wavelengths. The polarimetric properties of oral tissue at different stages of cancer have been analyzed and significant changes from normal to pre-cancerous lesions to the cancerous are observed in linear birefringence quantification as (1.7 ± 0.1) × 10-3 , (2.5 ± 0.2) × 10-3 and (3.3 ± 0.2) × 10-3 respectively.

Baseline-Free Detection Method for Change of Lateral Stiffness of High-Rise Building Based on Statistical Moment Curvature

In recent times, there has been a notable increase in the quantity of high-rise buildings, attributed to the swift advancements in both economic growth and construction technology. Assessing the structural integrity of high-rise buildings is important to ensure their safe operation. However, existing structural health monitoring methods typically require a baseline, involving either the measured dynamic and static responses from an intact structure or the finite element (FE) model corresponding to an undamaged state. These prerequisites are often challenging to acquire in practical scenarios. This study introduces a novel baseline-free method for detecting reduction in the lateral stiffness of high-rise buildings. The method is based on the statistical moment curvature (SMC) concept, determined through applying central difference to the second-order statistical moment of displacement. Initially, theoretical formulas were derived to demonstrate the viability of utilizing SMC for identifying reduction in the lateral stiffness of high-rise buildings. Subsequently, a FE model of a representative high-rise building was constructed to validate the proposed approach and assess its sensitivity, where different structural types and noise levels were considered. Lastly, a field test was conducted on a 33-story shear wall structure to provide additional validation for the proposed method. The results confirmed its effectiveness in accurately detecting reduction in the lateral stiffness of high-rise building. It offers two primary benefits: firstly, it obviates the need for a baseline, rendering it more convenient and applicable in real-world scenarios; secondly, its heightened sensitivity to sudden drops in lateral stiffness allows for early-stage detection of structural damage.

Reconstructing the phase diagram of iron in the terapascal region via the statistical moment method

Reconstructing the phase diagram of iron in the terapascal region via the statistical moment method

How Should Enterprises Quantify and Analyze (Multi-Party) APT Cyber-Risk Exposure in their Industrial IoT Network?

Industrial Internet of Things (IIoT) networks (e.g., a smart grid industrial control system) are increasingly on the rise, especially in smart cities around the globe. They contribute to meeting the day-to-day needs (e.g., power, water, manufacturing, transportation) of the civilian society, alongside making societal businesses more efficient, productive, and profitable. However, it is also well known that IoT devices often operate on poorly configured security settings. This increases the chances of occurrence of (nation-sponsored) stealthy spread-based APT malware attacks in IIoT networks that might go undetected over a considerable period of time. Such attacks usually generate a negative first-party QoS impact with financial consequences for companies owning such IIoT network infrastructures. This impact spans (i.e., aggregates) space (i.e., the entire IIoT network or a sub-network) and time (i.e., duration of business disruption), and is a measure of significant interest to managers running their businesses atop such networks. It is of little use to network resilience boosting managers if they have to wait for a cyber-attack to happen to gauge this impact. Consequently, one of the questions that intrigues us is: can managers estimate this first-party impact prior to APT cyber-attack(s) causing financial damage to companies? In this paper, we propose the first computationally efficient and quantitative network theory framework to (a) characterize this first-party impact apriori as a statistical distribution over multiple attack configurations in a family of malware-driven APT cyber-attacks specifically launched on businesses running atop IIoT networks, (b) accurately compute the statistical moments (e.g., mean) of the resulting impact distribution, and (c) tightly bound the accuracy of worst-case risk estimate of such a distribution - captured through the tail of the distribution, using the Conditional Value at Risk (CVaR) metric. In relation to (a) above, our methodology extends the seminal Factor Analysis of Information Risk (FAIR) cyber-risk quantification methodology that does not explicitly account for network interconnections among system-risk contributing variables. We validate the effectiveness of our theory using trace-driven Monte Carlo simulations based upon test-bed experiments conducted in the FIT IoT-Lab. We further illustrate quantitatively that even if spread-based APT cyber-attacks induce a statistically light-tailed first-party cyber-loss distribution on an IIoT networked enterprise in the worst case, the aggregate multi-party cyber-risk distribution incurred by the same enterprise in supply-chain ecosystems could be heavy-tailed. This will pose significant market scale-up challenges to cyber-security improving commercial cyber (re-)insurance businesses. We subsequently propose managerial action items to mitigate the first-party cyber-risk exposure emanating from any given IIoT driven enterprise.

A line loss reduction optimization for renewable energy‐based distribution networks using a probabilistic approach

SummaryThe large integration of distributed generation (DG) and electric vehicles (EVs) facilitates daily life and reduces carbon emissions. However, it brings problems such as increased uncertainty and power electronic permeability to the distribution network. In this context, it is important to solve the line loss optimization problem as closely as possible to the real situation. Therefore, this paper solves the line loss optimization problem based on a probabilistic model of three‐phase unbalanced modern power electrical distribution network. Firstly, the paper proposes a frequency domain model of the distribution network and the optimization problem model. Then, to deal with the uncertainty of the system, the point estimation method (PEM) and Monte Carlo Simulation (MCS) are used. Finally, this paper proposes two optimization schemes considering the power quality when single and multiple DGs are integrated. The simulation results demonstrate that the PEM using the Gram–Charlier series expansion of the fourth‐order statistical moments to estimate the probability density function (PDF) is significantly less time consuming than the traditional MCS method while maintaining accuracy. The comparative analysis reveals that under the optimization scheme with single DG integration, the expected value of system line loss and expected line loss rate are reduced from 283.95 kW to 218.96 kW (22.89% reduction) and from 13.44% to 10.68%, respectively. Under the dual DG integrated optimization scheme, the expected value of line losses and expected line loss rate are reduced from 283.95 kW to 208.88 kW (26.4% reduction) and from 13.44% to 10.25%, respectively. However, after taking the uncertainty into account, the operational reliability of the system has been enhanced under different scenarios. In addition, it is found that the optimization approach incorporating multiple DG units can effectively mitigate the system line loss under diverse operational scenarios while enhancing the system's DG integration capability.

POLYNOMIAL ESTIMATION OF DATA MODEL PARAMETERS WITH NEGATIVE KURTOSIS

Context. The paper focuses on the problem of estimating the center of distribution of the random component of experimental data for density models with a negative kurtosis.
 Objective. The goal of this research is to develop methods to improve the efficiency of polynomial estimation of parameters of experimental data with a negative kurtosis coefficient.
 Method. The study applies a relatively new approach to obtaining estimates for the center of the probability distribution from the results of experimental data with a stochastic component. This approach is based on polynomial estimation methods that rely on the mathematical apparatus of Kunchenko's stochastic polynomials and the description of random variables by higher-order statistics (moments or cumulants). A number of probability density distributions with a negative kurtosis coefficient are used as models of the random component.
 As a measure of efficiency, the ratio of variance of the estimates for the center of the distribution found using polynomial and classical methods based on the parameter of amount of information obtained is used.
 The relative accuracy of polynomial estimates in comparison with the estimates of the mean, median and quantile estimates (center of curvature) is researched using the Monte Carlo method for multiple tests.
 Results. Polynomial methods for estimating the distribution center parameter for data models of probability distribution density with a negative kurtosis coefficient have been constructed.
 Conclusions. The research carried out in this paper confirms the potentially high efficiency of polynomial estimates of the coordinates of the center of the experimental data, which are adequately described by model distributions with a negative kurtosis. Statistical modeling has confirmed the effectiveness of the obtained estimates in comparison with the known non-parametric estimates based on the statistics of the mean, median, and quantile, even with small sample sizes.

Non-Gaussian buffeting analysis of large structures by means of a Proper Orthogonal Decomposition

Proper Orthogonal Decomposition is implemented in a Bispectral Stochastic Analysis of large MDOF structures. Although Higher Order Stochastic Analysis has been introduced few decades ago, its establishment has found place only in a theoretical way, or applied to very small systems. Its heavy computational cost as well as resource consumption burden has let it aside for practical civil engineering applications. Nevertheless, in wind engineering, the community is increasingly considering the importance of non-Gaussian nature of wind induced vibrations. This is of interest for the extreme value analysis, for which many theoretical and empirical models exist for non-Gaussian random processes. In this context, a (bi-)spectral approach is realized in a 2-D frequency space where the bispectrum is computed then the 3rd statistical moment is obtained by means of a twofold integration. These operations are quite heavy as soon as the dimension of the problem increases. A novel algorithm implementation is proposed. It hinges on (i) the use of Proper Orthogonal Decomposition and the (ii) development of an optimized numerical method. The proposed algorithmic arrangement minimizes the number of operations, and consequently saves time and memory, while conserving precision in estimating 3rd order statistical moments of structural responses.

A Multi-Fidelity Integration Method for Reliability Analysis of Industrial Robots

Abstract A multi-fidelity integration method is proposed to analyze the reliability of multiple performance indicators (MPI) for industrial robots. In order to high-fidelity mapping the performance of industrial robots, a unified multi-domain model (UMDM) is first established. The contribution-degree analysis is then used to classify the input random variables into interacting and non-interacting ones. Thus, the high-dimensional integration of reliability analysis is separated into a low-dimensional integration and multiple one-dimensional integrations in an additive form. Here, the low-dimensional integration consisting of the interacting variables is calculated using the high-precision mixed-degree cubature formula (MDCF), and the computational results are treated as high-fidelity data. The one-dimensional integration consisting of non-interacting variables is then computed by the highly efficient five-point Gaussian Hermite quadrature (FGHQ), and the computational results are named low-fidelity data. A multi-fidelity integration method is constructed by fusing the high-fidelity data and the low-fidelity data to obtain the statistical moments of the MPI. Subsequently, the probability density function and the failure probability of the MPI are estimated using the saddlepoint approximation method. Finally, some representative methods are performed to verify the superiority of the proposed method.

Reliability-based topology optimization of fractionally-damped structures under nonstationary random excitation

This contribution proposes an effective reliability-based topology optimization (RBTO) framework for the layout of viscoelastic dampers of energy-dissipating structures under nonstationary seismic excitation, in which the constitutive behavior of viscoelastic damper is described by the novel fractional derivative models. To formulate the RBTO problem, the installation number of fractional viscoelastic dampers is minimized under the constraints of dynamic reliability, and the design variables are chosen as the damper parameters and the existence variables of the potential fractional viscoelastic dampers. The explicit responses are first established for the fractionally-damped structure, and the explicit response sensitivities with respect to the design variables are further derived through the adjoint variable method (AVM). On this basis, an explicit time-domain method (ETDM) is developed for explicit formulation of the statistical moments of responses and the moment sensitivities of the fractionally-damped structure, which are further employed to analytically derive the first-passage dynamic reliabilities and the reliability sensitivities of the structure based on the classical level-crossing process theory. Finally, the RBTO problem for the layout of fractional viscoelastic dampers is solved with the gradient-based method of moving asymptotes (MMA), in which the existence variables of fractional viscoelastic dampers can be driven to binary solutions by the solid isotropic material with penalization (SIMP) technique. An engineering application involving a building frame structure installed with fractional viscoelastic dampers is presented to validate the feasibility of the proposed ETDM-based RBTO framework.

Underwater image quality assessment method based on color space multi-feature fusion

The complexity and challenging underwater environment leading to degradation in underwater image. Measuring the quality of underwater image is a significant step for the subsequent image processing step. Existing Image Quality Assessment (IQA) methods do not fully consider the characteristics of degradation in underwater images, which limits their performance in underwater image assessment. To address this problem, an Underwater IQA (UIQA) method based on color space multi-feature fusion is proposed to focus on underwater image. The proposed method converts underwater images from RGB color space to CIELab color space, which has a higher correlation to human subjective perception of underwater visual quality. The proposed method extract histogram features, morphological features, and moment statistics from luminance and color components and concatenate the features to obtain fusion features to better quantify the degradation in underwater image quality. After features extraction, support vector regression(SVR) is employed to learn the relationship between fusion features and image quality scores, and gain the quality prediction model. Experimental results on the SAUD dataset and UIED dataset show that our proposed method can perform well in underwater image quality assessment. The performance comparisons on LIVE dataset, TID2013 dataset,LIVEMD dataset,LIVEC dataset and SIQAD dataset demonstrate the applicability of the proposed method.

Robust topology optimization of truss-like continuum structures under uncertain loads based on cloud model

The traditional structural performance may be seriously degraded due to applied indeterministic cases. A novel framework for robust topology optimization of truss-like continuum considering uncertain loads is presented. The robust optimization formula is constructed as minimizing the weighted sum of expectancy and standard variance of structural compliance with volume constraints. In this framework, both the magnitude and direction of applied loads are uncertain and independent. Truss-like continuum provides the topology optimal structure theoretically. Based on the classic truss-like optimization method, the statistical moments of compliance are accurately evaluated utilizing a bivariate cloud model tool. An effective strategy based on linear decoupling is reported to largely reduce the total number of finite element analysis. The optimization is achieved with gradient-based mathematic programming. Several numerical examples demonstrate a better structural robust performance obtained by the proposed algorithm than the traditional one.

Reliability-based analysis of seismic bearing capacity of shallow strip footings resting on soils with randomly varying geotechnical and earthquake parameters

Seismic bearing capacity of strip footings is a challenging task for geotechnical engineers due to its stochastic framework instigated by the natural uncertainties incorporated into geotechnical properties and earthquake parameters. Consequently, the introduction of the random field theory into reliability analysis may provide power tools to succor designers check how reliable their designs. This paper aims to assess the seismic bearing capacity of shallow strip footings resting on soils with randomly varying parameters. Bearing capacity formulas for purely cohesive and cohesive-frictional soils are considered. The influence of the type of the autocorrelation functions (ACFs), the scale of fluctuations (SOFs) and the coefficient of variation (COV) of the random parameters are investigated. Statistical moments, probability density function (PDF) and failure probability (Pf) of the seismic bearing capacity are computed. It is shown that the Single Exponential (SNE) ACF is the most appropriate function to characterize the spatial variability of the soil properties since it provides conservative results. On other hand, the results indicate that the increase in the coefficients of variation (COV) of the cohesion or the friction angle increases the variability of the seismic bearing capacity while this variability remains unaffected when the COV of the seismic coefficient increases. The results also highlight that the effect of the vertical SOF on the PDF and the failure probability is much more significant than that of the horizontal SOF. In addition, the mean seismic bearing capacity fluctuates slightly as the horizontal or vertical SOF increases so that the increment of variation is between 0.4% and 2% for the both two soil types.

Isotopic effect in thermodynamic properties of solid neon

Isotopic effects in mean-square displacement, specific heat at constant pressure and linear thermal expansion coefficient of solid neon have been considered using the statistical moment method in statistical mechanics. This approach allows us to study these effects including the anharmonicity of lattice vibrations. We implement calculations for 20Ne and 22Ne isotopes in temperature range from 0 K to the melting temperature (24.6 K) using Lennard-Jones potential to describe the interaction between Ne quantum particles. Our numerical results are compared with those of path-integral Monte Carlo simulations and previous experimental data showing the good agreement. This research presents an effective statistical approach to study isotopic effects in thermodynamic quantities of materials.

Higher-order statistical moments of predicted sonic boom waveforms through turbulence

Sonic booms generated by supersonic aircraft are affected by turbulence in the atmospheric boundary layer through which they propagate. Turbulence effects lead to random variability of the sonic boom waveforms measured on the ground, complicating the prediction of such waveforms. As an initial effort to predict the waveform variability, the solution of the coherent or mean sonic boom waveform has previously been formulated by the author. The current paper extends the formulation to derive an expression of the second-order statistical moment necessary to calculate the variance of the spectral amplitudes. Since the derivation uses a full wave equation, results using the derived expression will be compared with those obtained using a parabolic approximation. In order to fully quantify the uncertainty of our predictions, the higher order statistical moments are also formulated and are shown to be approximately zero. Consequently, the probability density function (pdf) of the spectral amplitudes is predicted to be Gaussian. The formulation is further extended to determine the pdf of the loudness of sonic booms and quantify the uncertainties associated with the loudness prediction. Results from the formulation are compared with available flight test data.

High-order moment closure models with random batch method for efficient computation of multiscale turbulent systems.

We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz '96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation.