Polynomial Correction Research Articles

A Spectral Element Solution of the Poisson Equation with Shifted Boundary Polynomial Corrections: Influence of the Surrogate to True Boundary Mapping and an Asymptotically Preserving Robin Formulation

Abstract We present a new high-order spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order hp-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series, which traditionally is used within the shifted boundary method. Here, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small cut cells. Dirichlet, Neumann, and Robin boundary conditions are enforced weakly through a generalized: (i) Nitsche’s method and (ii) Aubin’s method. A consistent asymptotic preserving formulation of the embedded Robin formulations is presented. Several experiments and analyses of the numerical properties of the various weak forms are showcased. We include convergence studies under polynomial increase of the basis functions, p, mesh refinement, h, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. With this, we assess the influence of errors across variational forms, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

Investigation of the potential of in-line particle concentration measurements in gas particle separation processes by using low-cost particulate matter sensors

In addition to the costly laboratory-based particle measurement devices, low-cost particulate matter sensors for ambient air quality measurements can also be employed for the detection of PM in gas cleaning devices. The ability to spatially resolve the actual level of particle number concentration with high temporal resolution within a gas-particle separation process is of significant value for process monitoring and control. The quantitative reliability of the measurement data from low-cost particulate matter sensors remains uncertain when the particle concentrations in the apparatus exceed the upper measurement limit of the sensors, whether temporarily or permanently. This study examines the potential of low-cost particulate matter sensors for in-line particle concentration measurements in gas-particle separation processes. The measurement performance of the low-cost particulate matter sensors is initially examined under idealised conditions in a dust chamber. This is done in order to quantify the deviations of the low-cost particulate matter sensors in comparison to a state-of-the-art precision laboratory high-cost sensor. Subsequently, both sensor types are installed at a wet separator test facility, following the quantification of their respective measurement performances. The low-cost particulate matter sensors are capable of measuring particle number concentration above their specified limit (up to ≈6.1×104cm−3), depending on the measurement conditions. By applying a 3rd polynomial correction function, obtained from the initial quantification measurement, the low-cost particulate matter sensors are able to reproduce the particle number concentration measured by the high-cost sensor. Furthermore, the low-cost particulate matter sensors are capable of reproducing short-term fluctuations in the particle number concentration.

Signal drift in diffusion MRI of the brain: effects on intravoxel incoherent motion parameter estimates.

Signal drift has been put forward as one of the fundamental confounding factors in diffusion MRI (dMRI) of the brain. This study characterizes signal drift in dMRI of the brain, evaluates correction methods, and exemplifies its impact on parameter estimation for three intravoxel incoherent motion (IVIM) protocols. dMRI of the brain was acquired in ten healthy subjects using protocols designed to enable retrospective characterization and correction of signal drift. All scans were acquired twice for repeatability analysis. Three temporal polynomial correction methods were evaluated: (1) global, (2) voxelwise, and (3) spatiotemporal. Effects of acquisition order were simulated using estimated drift fields. Signal drift was around 2% per 5min in the brain as a whole, but reached above 5% per 5min in the frontal regions. Only correction methods taking spatially varying signal drift into account could achieve effective corrections. Altered acquisition order introduced both systematic changes and differences in repeatability in the presence of signal drift. Signal drift in dMRI of the brain was found to be spatially varying, calling for correction methods taking this into account. Without proper corrections, choice of protocol can affect dMRI parameter estimates and their repeatability.

The method to improve the accuracy of experiment using helium instead of hydrogen

Based on the hydrogen-helium similarity theory, the leakage experiments using helium instead of hydrogen can effectively reduce the risk of hydrogen deflagration, but the experimental results show that there is a large difference between the concentration of hydrogen and helium in initial period of leakage base on this theory. This paper focuses on leakage experiments under turbulent conditions in enclosed spaces. Based on the hydrogen-helium similarity theory, helium is used to replace hydrogen to find out the concentration of helium at multiple leakage speeds. A comparison is made between helium concentration values and hydrogen concentration values, and a polynomial correction algorithm is used to shrink the difference. The results show that this correction method can reduce the concentration average error between the helium and the hydrogen from 14.06% to 7.94% for the Reynolds number range of 3500–9500. Therefore, a polynomial fitting algorithm based on this theory can improve the accuracy of experiment using helium instead of hydrogen.

Dynamical system analysis of scalar field cosmology in coincident f(Q) gravity

In this article, we investigate scalar field cosmology in the coincident f(Q) gravity formalism. We calculate the motion equations of f(Q) gravity under the flat Friedmann-Lemaître-Robertson-Walker background in the presence of a scalar field. We consider a non-linear f(Q) model, particularly f(Q) = − Q + α Q n , which is nothing but a polynomial correction to the Symmetric Teleparallel Equivalent to General Relativity (STEGR) case. Further, we assumed two well-known specific forms of the potential function, specifically the exponential from V(ϕ) = V 0 e −β ϕ and the power-law form V(ϕ) = V 0 ϕ −k . We employ some phase-space variables and transform the cosmological field equations into an autonomous system. We calculate the critical points of the corresponding autonomous systems and examine their stability behaviors. We discuss the physical significance corresponding to the exponential case for parameter values n = 2 and n = − 1 with β = 1, and n = − 1 with β=3 . Moreover, we discuss the same corresponding to the power-law case for the parameter value n = − 2 and k = 0.16. We also analyze the behavior of corresponding cosmological parameters such as scalar field and dark energy density, deceleration, and the effective equation of state parameter. Corresponding to the exponential case, we find that the results obtained for the parameter constraints in Case III is better among all three cases, and that represents the evolution of the Universe from a decelerated stiff era to an accelerated de-Sitter era via matter-dominated epoch. Further, in the power-law case, we find that all trajectories exhibit identical behavior, representing the evolution of the Universe from a decelerated stiff era to an accelerated de-Sitter era. Lastly, we conclude that the exponential case shows better evolution as compared to the power-law case.

A correction method for mitigating absorbance discrepancies between near-infrared spectrometers through the incorporation of blended carbon-titanium dioxide powder

In near-infrared spectroscopy analysis, ensuring the accurate transfer of models between different instruments relies on maintaining the accuracy of instrument wavelengths and absorbance. To mitigate absorbance drift at different wavelength points, this paper proposes a near-infrared spectroscopy point-by-point quadratic polynomial correction method based on carbon-titanium dioxide powder samples. The method establishes a quadratic polynomial relationship model for the absorbance of each wavelength point between the main instrument and slave instruments. The study utilized two S450 grating-based diffuse reflection near-infrared spectroscopy instruments, with one serving as the main instrument and the other as the slave instrument. The point-by-point quadratic polynomial was employed to correct wheat spectra collected by the slave instruments, and a crude protein content prediction model for wheat was established, comparing it with linear regression correction. After correction, the average Euclidean distance of wheat spectra decreased by 66.71%, from 0.0937 to 0.0321, and the average peak-valley Euclidean distance decreased by 72.28%, from 0.0203 to 0.0056. The standard deviation of the predicted results decreased by 90.69%, from 1.4372 to 0.1338. The correction effect of the method combined with traditional preprocessing methods was superior to using preprocessing methods alone. Overall, the near-infrared spectroscopy point-by-point quadratic polynomial correction method based on carbon-titanium dioxide powder samples significantly reduces spectral differences between different instruments, enhances spectral consistency, and diminishes prediction errors, achieving improved model sharing between instruments.

ПОБУДОВА ТА ПЕРШІ РЕЗУЛЬТАТИ ІНТЕРПРЕТАЦІЇ ТРИВИМІРНОЇ МОДЕЛІ ЩІЛЬНОСТІ МАНТІЇ ПІД УКРАЇНСЬКИМ ЩИТОМ

Background. Mantle density models are key tools for understanding fundamental geological and physical processes occurring within the Earth. Many parameters used in mantle density models remain poorly understood and undefined. Among others, these include data on the composition and rheology of the mantle, which can vary significantly. Methods. The method of creating density models (density) significantly influences the final result. Modeling with one-dimensional models simplifies the calculation process but generalizes the distribution of mantle density, assuming it is homogeneous in the horizontal direction. This limitation does not allow for the consideration of lateral variations in mantle density, which can be important at the regional level. In this study, we present a quasi-three-dimensional model of mantle density beneath the Ukrainian Shield, obtained on the basis of a set of one-dimensional density curves. Polynomial corrections for heterogeneity were applied during the calculations, compensating for the shortcomings of one-dimensional models. This three-dimensional model was derived by recalculating one-dimensional velocity curves obtained by seismic tomography for 21 mantle domains in the depth range of 50 to 2600 km. The process of transforming P-wave velocity curves into a density model includes the following steps: determining seismic boundaries in the mantle as points of inflection of the first derivative of P-wave velocity curves for each mantle domain; creating a synthetic S-wave mantle model beneath the Ukrainian Shield by recalculating P-wave velocity curves; solving the Adams-Williamson equation using seismic velocities (P, S) for each domain with subsequent polynomial correction to account for heterogeneity; selecting a reference mantle model that would serve as the basis for converting velocity curves into density through the comparison of gravitational potential on the Earth's surface and calculated values from existing reference mantle models (PREM, PREMA, PREMC, IASP91 AK135). The AK135 model was chosen as the reference model based on the comparison of calculated and observed gravitational potential at the central point of the Ukrainian Shield. This study focuses on the final stages of constructing the mantle density model, taking into account mass balancing of the upper and lower mantle for each domain when determining density using the Adams-Williamson equation and introducing polynomial corrections relative to the AK135 reference model; calculating densities for each of the 21 mantle domains and their three-dimensional integration. Results. In this study, we present a quasi-three-dimensional model of mantle density beneath the Ukrainian Shield, obtained on the basis of a set of one-dimensional density curves, with polynomial corrections for inhomogeneity incorporated into the calculations, compensating for the shortcomings of one-dimensional model calculations. This three-dimensional model was obtained by recalculating one-dimensional velocity curves obtained by the seismic tomography method for P-waves, calculated for 21 mantle domains in the depth range from 50 to 2600 km. Conclusions. This study focuses on the final stages of constructing the mantle density model, considering balancing the mass of the upper and lower mantle for each domain in determining density using the Adams-Williamson equation and introducing polynomial corrections relative to the AK135 reference model; calculating densities for each of the 21 mantle domains and their three-dimensional integration. The obtained mantle density model of the Ukrainian Shield is well aligned with the division of the mantle into three main layers: lithosphere, upper mantle, and lower mantle. Each of the structural layers has its own visual pattern of density heterogeneity.

Simple ePDF: A Pair Distribution Function Method Based on Electron Diffraction Patterns to Reveal the Local Structure of Amorphous and Nanocrystalline Materials.

Amorphous, glassy or disordered materials play important roles in developing structural materials from metals or ceramics, devices from semiconductors or medicines from organic compounds. Their local structure is frequently similar to crystalline ones. A computer program is presented here that runs under the Windows operating system on a PC to extract pair distribution function (PDF) from electron diffraction in a transmission electron microscope (TEM). A polynomial correction reduces small systematic deviations from the expected average Q-dependence of scattering. Neighbor distance and coordination number measurements are supplemented by either measurement or enforcement of number density. Quantification of similarity is supported by calculation of Pearson's correlation coefficient and fingerprinting. A rough estimate of fractions in a mixture is computed by multiple least-square fitting using the PDFs from components of the mixture. PDF is also simulated from crystalline structural models (in addition to measured ones) to be used in libraries for fingerprinting or fraction estimation. Crystalline structure models for simulations are obtained from CIF files or str files of ProcessDiffraction. Data from inorganic samples exemplify usage. In contrast to previous free ePDF programs, our stand-alone program does not need a special software environment, which is a novelty. The program is available from the author upon request.

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.

Application and Improvement of the Particle Swarm Optimization Algorithm in Source-Term Estimations for Hazardous Release

Hazardous gas release can pose severe hazards to the ecological environment and public safety. The source-term estimation of hazardous gas leakage serves a crucial role in emergency response and safety management practices. Nevertheless, the precision of a forward diffusion model and atmospheric diffusion conditions have a significant impact on the performance of the method for estimating source terms. This work proposes the particle swarm optimization (PSO) algorithm coupled with the Gaussian dispersion model for estimating leakage source parameters. The method is validated using experimental cases of the prairie grass field dispersion experiment with various atmospheric stability classes. The results prove the effectiveness of this method. The effects of atmospheric diffusion conditions on estimation outcomes are also investigated. The estimated effect in extreme atmospheric diffusion conditions is not as good as in other diffusion conditions. Accordingly, the Gaussian dispersion model is improved by adding linear and polynomial correction coefficients to it for its inapplicability under extreme diffusion conditions. Finally, the PSO method coupled with improved models is adapted for the source-term parameter estimation. The findings demonstrate that the estimation performance of the PSO method coupled with improved models is significantly improved. It was also found that estimated performances of source parameters of two correction models were significantly distinct under various atmospheric stability classes. There is no single optimal model; however, the model can be selected according to practical diffusion conditions to enhance the estimated precision of source-term parameters.

A simple method of bias correction for GCM derived streamflow at catchment scale

ABSTRACT In this study, a simple polynomial bias correction method is developed to correct the bias in the forecasted streamflow (runoff) derived from a global circulation model (GCM). First, a set of polynomial correction factors was derived comparing observed and GCM-derived runoff for a hindcast period (1961–2000) for each of the 11 selected GCMs. The correction factors are used to correct the GCM-derived streamflow for projected periods (2046–2065 and 2081–2100) for the Intergovernmental Panel on Climate Change scenarios A2 and B1 (CMIP3) for the Murray-Hotham catchment of Western Australia. The assumption is that the correction factors derived for each GCM for the observed period (1961–2000) are valid for the projected periods. Results show the method reduces biases considerably for the projected runoff at a catchment scale. The method developed here uses CMIP3 data but it may be applicable to any GCM data, such as CMIP5/CMIP6.

Enumeration of Corner Polyhedra and 3-Connected Schnyder Labelings

We show that corner polyhedra and 3-connected Schnyder labelings join the growing list of planar structures that can be set in exact correspondence with (weighted) models of quadrant walks via a bijection due to Kenyon, Miller, Sheffield and Wilson.
 Our approach leads to a first polynomial time algorithm to count these structures, and to the determination of their exact asymptotic growth constants : the number $p_n$ of corner polyhedra and $s_n$ of 3-connected Schnyder woods of size $n$ respectively satisfy $(p_n)^{1/n}\to 9/2$ and $(s_n)^{1/n}\to 16/3$ as $n$ goes to infinity.
 While the growth rates are rational, like in the case of previously known instances of such correspondences, the exponent of the asymptotic polynomial correction to the exponential growth does not appear to follow from the now standard Denisov-Wachtel approach, due to a bimodal behavior of the step set of the underlying tandem walk. However a heuristic argument suggests that these exponents are $-1-\pi/\arccos(9/16)\approx -4.23$ for $p_n$ and $-1-\pi/\arccos(22/27)\approx -6.08$ for $s_n$, which would imply that the associated series are not D-finite.

Research on Two-stage Estimation of Partially Linear Single-index Model with Longitudinal Data

Partial linear single-index model is a kind of semi-parametric model with wide application. In this paper, we deal with the partial linear single-index model under longitudinal data. A "two-stage estimation method" without iteration by using local polynomial and bias correction generalized estimation equation is proposed. under some regularity conditions, the asymptotic properties of the connection function and unknown parameter estimator are investigated. Numerical simulation shows that the proposed method is robust.

Probabilistic error assessment and correction of design code-based shear strength prediction models for reliability analysis of prestressed concrete girders

Aiming at probabilistic error assessment of design code models for shear strength prediction of prestressed concrete (PC) girders, this study compiled an experimental database containing 369 PC girders that failed in shear. Using the experimental database, this paper first assessed seven well-received shear strength models from five concrete structure and bridge design codes, including ACI 318-19, AASHTO LRFD 2017, CSA A23.3:19, CSA S6:19 and fib MC 2010. In view of the fact that systematic error exists in those models, polynomial correction terms were calibrated for each model together with the remaining error quantified based on the compiled experimental database and Bayesian updating. The resulted models can be used for shear strength predictions with better accuracy and, more importantly, with the model uncertainty quantified probabilistically. In the end, a case study of fragility analysis was conducted to show the benefits of the developed probabilistic models.

A New GNSS TEC Neural Network Prediction Algorithm With the Data Fusion of Physical Observation

This work models and predicts anomalous jumps in GNSS total electron content (TEC) caused by geomagnetic disturbance or solar radiation by introducing F10.7 and regional Dst indices, which are used to characterize solar radiation and geomagnetic disturbance status, respectively. We then introduced long-distance constraint stations to restrain ionospheric variations during high solar activity, which reduced the influence of unmodelled factors on prediction accuracy. In our experimental work, Global Ionosphere Map (GIM) TEC data from 11 stations located at different latitudes were used for comparison during maximum (2014) and minimum (2018) of solar cycle 24. Sample data for >27 days were used for learning, and then ionospheric changes for the whole year were predicted using the sliding window method. The results showed that, the standard deviation for the global ionospheric prediction products could be guaranteed to be within 2 TECU. The proximity station constraint method proposed in this paper and the Dst high-order polynomial correction method during geomagnetic disturbance control the prediction error within -4.4 TECU, while the change of adjacent days is -51.4 TECU in 2014. The new algorithm reduced the TEC jump effect in predictions, and the prediction of ionospheric products provided by this algorithm can meet the basic requirements of ionospheric delay correction in high-precision positioning.

On a Hypothetical Model with Second Kind Chebyshev’s Polynomial–Correction: Type of Limit Cycles, Simulations, and Possible Applications

In this article, we explore a new extended Lienard-type planar system with “corrections” of the second kind Chebyshev’s polynomial Un. The number and type of limit cycles are also studied. The discussion on the y(t)—component of the solution of the Lienard system is connected to searching for the solution of the synthesis of filters and electrical circuits. Numerical experiments, depicting our outcomes using CAS MATHEMATICA, are presented.

Shifted boundary polynomial corrections for compressible flows: high order on curved domains using linear meshes

In this work we propose a simple but effective high order polynomial correction allowing to enhance the consistency of all kind of boundary conditions for the Euler equations (Dirichlet, characteristic far-field and slip-wall), both in 2D and 3D, preserving a high order of accuracy without the need of curved meshes. The method proposed is a simplified reformulation of the Shifted Boundary Method (SBM) and relies on a correction based on the extrapolated value of the in cell polynomial to the true geometry, thus not requiring the explicit evaluation of high order Taylor series. Moreover, this strategy could be easily implemented into any already existing finite element and finite volume code. Several validation tests are presented to prove the convergence properties up to order four for 2D and 3D simulations with curved boundaries, as well as an effective extension to flows with shocks.

ACCESS: Confirmation of a Clear Atmosphere for WASP-96b and a Comparison of Light Curve Detrending Techniques

One of the strongest Na i features was observed in WASP-96b. To confirm this novel detection, we provide a new 475–825 nm transmission spectrum obtained with Magellan/IMACS, which indeed confirms the presence of a broad sodium absorption feature. We find the same result when reanalyzing the 400–825 nm VLT/FORS2 data. We also utilize synthetic data to test the effectiveness of two common detrending techniques: (1) a Gaussian processes (GP) routine, and (2) common-mode correction followed by polynomial correction (CMC+Poly). We find that both methods poorly reproduce the absolute transit depths but maintain their true spectral shape. This emphasizes the importance of fitting for offsets when combining spectra from different sources or epochs. Additionally, we find that, for our data sets, both methods give consistent results, but CMC+Poly is more accurate and precise. We combine the Magellan/IMACS and VLT/FORS2 spectra with literature 800–1644 nm HST/WFC3 spectra, yielding a global spectrum from 400 to 1644 nm. We used the PLATON and Exoretrievals retrieval codes to interpret this spectrum, and find that both yield relatively deeper pressures where the atmosphere is optically thick at log-pressures between and bars, respectively. Exoretrievals finds solar to supersolar Na i and H2O log-mixing ratios of and , respectively, while PLATON finds an overall metallicity of dex. Therefore, our findings are in agreement with the literature and support the inference that the terminator of WASP-96b has few aerosols obscuring prominent features in the optical to near-infrared (near-IR) spectrum.

Thermistor string corrections in data from very weakly stratified deep-ocean waters

An important parameter for observational studies on dynamical variations of the ocean is temperature (T) which determines density variations together with salinity. While the achievement of high absolute accuracy of T-sensors is a formidable task, even the maintaining of high precision is difficult for any long-term deep-ocean observations. This is because the deep ocean is very weakly stably stratified in density, in which all the dynamics is covered by a range of about 0.001 °C. One requires high-resolution T-sensors with low noise levels and precisely calibrated. However, common Negative Temperature Coefficient (NTC) thermistor T-sensors have electronic drift causing a bias of temperature with time. The drift rate varies with temperature (change) and time (age) between 0.0002 and 0.001 °C wk−1 for T-sensors used here. A string of such T-sensors moored in the open-ocean for more than a month after calibration requires a drift-correction. When moored in the deep-ocean, a secondary correction is necessary even immediately after calibration. Secondary correction uses reference layers that are homogeneous to within 0.00005 °C over 100 m vertically plus an additional polynomial correction to remove remaining bias prior or after the reference date. Secondary correction may also involve pressure information on surface tides and mooring deflection. In this paper, secondary corrections are described for two cases of data. For one case from the West-Mediterranean varying temperature-density relationship and near-homogeneous conditions exist. The other is from hadal-deep instruments on a long mooring line in the Challenger Deep, Mariana Trench, affected by tidal flow. Nearby shipborne CTD-data are always needed to establish the temperature-density relationship and the background vertical temperature slope to which the moored T-sensor data are referenced. The stability of moored T-sensors is compared with that of temperature sensors of CTD.

Fully automated background phase correction using M-estimate SAmple consensus (MSAC)-Application to2Dand 4D flow.

Flow quantification by phase-contrast MRI is hampered by spatially varying background phase offsets. Correction performance by polynomial regression on stationary tissue may be affected by outliers such as wrap-around or constant flow. Therefore, we propose an alternative, M-estimate SAmple Consensus (MSAC) to reject outliers, and improve and fully automate background phase correction. The MSAC technique fits polynomials to randomly drawn small samples from the image. Over several trials, it aims to find the best consensus set of valid pixels by rejecting outliers to the fit and minimizing the residuals of the remaining pixels. The robustness of MSAC to its few parameters was investigated and verified using third-order polynomial correction fits on a total of 118 2D flow (97 with wrap-around) and 18 4D flow data sets (14 with wrap-around), acquired at 1.5 T and 3T. Background phase was compared with standard stationary correction and phantom correction. Pulmonary/systemic flow ratios in 2D flow were derived, and exemplary 4D flow analysis was performed. The MSAC technique is robust over a range of parameter choices, and a unique set of parameters is suitable for both 2D and 4D flow. In 2D flow, phase errors were significantly reduced by MSAC compared with stationary correction (p=0.005), and stationary correction shows larger errors in pulmonary/systemic flow ratios compared with MSAC. In 4D flow, MSAC shows similar performance as stationary correction. The MSAC method provides fully automated background phase correction to 2D and 4D flow data and shows improved robustness over stationary correction, especially with outliers present.