Function Interpolation Method Research Articles

Enhancing OpenEP: atrial conduction velocity and conduction velocity heterogeneity quantification through EP Workbench

Abstract Background Alterations in atrial conduction velocity have been implicated in the pathogenesis of atrial arrhythmias, with conduction velocity thought to be slower and more heterogeneous in areas promoting atrial fibrillation. However, clinical studies of conduction velocity are challenging since there are no available platforms that provide researchers and clinicians with the ability to quantify conduction velocity and analyse conduction heterogeneity from electroanatomic mapping data. Purpose In this paper we sought to address these limitations by enhancing the OpenEP Python library by (1) automating calculations using previously-published conduction velocity estimation algorithms; (2) providing an interactive graphical representing of conduction velocity and conduction heterogeneity and (3) implementing a histogram analysis tool to quantify conduction velocity and enable the identification of slow conduction regions. Method Two well-known and previously published methods for calculating cardiac conduction velocity, Triangulation and polynomial surface fitting, were implemented. Conduction velocity estimation was improved further by excluding regions with wave collisions and focal discharges. These regions were identified using the divergence of conduction velocity vector fields, with positive divergence representing focal sources and negative divergence representing areas of collision. Finally, calculated conduction velocities using electrogram data were interpolated over the surface mesh via Radial Basis Function (RBF) interpolation method. All algorithms were implemented in the OpenEP Python library (openep-py), with visualisation tools provided in the complementary graphical interface, EP Workbench. Results An extensible "Analysis" feature was created in openep-py, to provide conduction velocity and divergence calculations which are fully customisable through the graphical interface based on user preferences. The EP Workbench desktop application displays resulting cardiac surface maps to depict calculated conduction velocity and divergence fields, together with conventional activation and voltage maps (Figure 1). The histogram analysis tool can be used to identify regions of slow conduction velocity from electroanatomic mapping data. Using a threshold of <0.3m/s, two slow conducting channels are visualised on the posterior wall of a test case (Figure 2). The tool is not specific to conduction velocity and may also be used to quantify other data types in EP Workbench. Finally, computed conduction velocity and divergence values and surface maps are stored in the standard OpenEP data structure, allowing further analysis following data export. Conclusion We have extended OpenEP to provide a comprehensive set of tools for conduction velocity calculation and analysis which will facilitate investigation of the structural and functional basis of conduction velocity alterations in disease states.

A new contact and road model for multi-body dynamic simulation of wheeled vehicles on soft-soil terrain

AbstractIn this paper, a new high-performance and memory-efficient contact and road model is developed. Specifically, the road is modeled as a rectangular structured grid of deformable springs in the vertical direction, thus enabling fast execution. The new road model stands out due to its ability to handle large road scenarios by allocating computer memory dynamically for each spring, resulting in efficient memory utilization. Furthermore, each spring represents a small road patch that entails various information, such as the soil elevation, the soil properties, and the soil compaction, allowing for complicated simulations incorporating spatially varying soil properties and phenomena related to the multi-pass effect. In addition, using the new contact model, complex terrain geometries are handled in a computationally efficient way by approximating locally the irregular road profile with a suitable equivalent plane. For this, two different strategies are proposed, namely the radial basis function (RBF) interpolation method and the 3D enveloping contact model. Finally, the proposed techniques are implemented in Altair MotionSolve, a comprehensive multi-body simulation software for complex mechanical systems. In particular, a single-wheel test bed is initially examined followed by a four-wheeled rover model and the next-generation NATO reference mobility model (NG-NRMM). In all cases, the proposed model is validated by using available experimental data. Lastly, a case involving both wheeled and tracked vehicles is also examined by using a shared road model.

A comparative study of different radial basis function interpolation algorithms in the reconstruction and path planning of γ radiation fields

Accurate reconstruction of radiation field and path planning are very important for the safety of operators in the process of dismantling nuclear facilities. Based on radial basis function (RBF) interpolation algorithm, this paper discussed the application of inverse multiquadric radial basis Function (IMRBF) interpolation method to the reconstruction of gamma radiation field, and proved the feasibility of reconstructing a radiation field with multiple γ sources. The average relative errors of IMRBF interpolation results were 4.28% and 8.76%, respectively, for the experimental scenarios with single and double gamma sources. After comparing the consistency between the simulated scene and the experimental scene, IMRBF method and Cubic Spline method were respectively used to reconstruct the gamma radiation field by Geant4 simulation data. The results showed that the interpolation accuracy of IMRBF method was superior to that of Cubic Spline method. Further, more RBF interpolation algorithms were used to reconstruct the multi-γ source radiation field, and then the Probabilistic Roadmap (PRM) algorithm was used to optimize the human walking path in the radiation field reconstructed by different interpolation methods. The optimal paths in radiation fields generated by multiple interpolation methods were compared. The results herein contribute to a comprehensive understanding of RBF interpolation methods in reconstructing γ radiation fields and their application in optimizing paths in radiation environments. The insights may provide valuable information for decision-making in radiation protection during the decommissioning of nuclear facilities.

Research on the hydroelastic response of ice floes and wave scattering field

Abstract The marginal ice zone (MIZ) is the area between sea ice and open water, the structure of which is mainly determined by wave and ice interactions. Thus mastering the characteristics of MIZ is of great significance to the Arctic routes opening and the natural resources development. In this paper, the hydroelastic response of ice floes in waves is studied, a three-dimensional numerical wave tank is established based on the computational fluid dynamics (CFD) technology. The finite volume method (FVM) and finite element method (FEM) are respectively utilized for discrete fluid domain and ice domain. A mapping interface at the junction of the fluid and ice floes domains is created to perform data mapping by the shape function interpolation method and the least square method. This work presents a series of numerical simulations to study the fluid-solid interaction (FSI) of waves and ice floes. Under the given incident wave parameters, the vertical bending deformation of ice floes with different shapes under the excitation of waves, the effect of ice floes' deformation on the wave field are studied, and the effect of wave overwash on the transmitted wave field is emphasized. Results show that the shape of the ice floes significantly affects its elastic deformation and scattered wave field, and the wave overwash phenomenon attenuates the scattering wave.

A Stable and Efficient Interpolation Method for Two-dimensional Periodic Green's Functions

A Stable and Efficient Interpolation Method for Two-dimensional Periodic Green's Functions

A Comparison of Arterial Input Function Interpolation Methods for Patlak Plot Analysis of 68Ga-PSMA-11 PET Prostate Cancer Studies.

Positron emission tomography (PET) imaging enables quantitative assessment of tissue physiology. Dynamic pharmacokinetic analysis of PET images requires accurate estimation of the radiotracer plasma input function to derive meaningful parameter estimates, and small discrepancies in parameter estimation can mimic subtle physiologic tissue variation. This study evaluates the impact of input function interpolation method on the accuracy of Patlak kinetic parameter estimation through simulations modeling the pharmacokinetic properties of [68Ga]-PSMA-11. This study evaluated both trained and untrained methods. Although the mean kinetic parameter accuracy was similar across all interpolation models, the trained node weighting interpolation model estimated accurate kinetic parameters with reduced overall variability relative to standard linear interpolation. Trained node weighting interpolation reduced kinetic parameter estimation variance by a magnitude approximating the underlying physiologic differences between normal and diseased prostatic tissue. Overall, this analysis suggests that trained node weighting improves the reliability of Patlak kinetic parameter estimation for [68Ga]-PSMA-11 PET.

Study on the regional ASF prediction method based on the ordinary kriging interpolation

The eLoran system functions as a robust backup to the Global Navigation Satellite System (GNSS), providing substantial signal power, robust anti-jamming capabilities, and an easily maintainable ground system. Nonetheless, the propagation time delay of the system, primarily driven by the Additional Secondary Phase Factor (ASF), significantly influences the positioning accuracy. Compensating for ASF can effectively enhance the positioning performance. Traditional propagation delay theories frequently yield substantial discrepancies between the predicted and measured values in regions characterized by extended propagation distances and varying topographic features. To address this issue, we conducted ASF measurements within a selected test area using a limited number of measurement points. We employed the ordinary Kriging interpolation method to predict ASF values across the entire test area, and used cross-validation to validate our predictions. The results confirmed the accuracy and effectiveness of the Kriging interpolation algorithm in predicting ASF values within specific regions. The cross-validation demonstrated that the errors remained within acceptable ranges. Furthermore, we applied Ordinary Kriging, Inverse Distance Weighting, and Radial Basis Function Interpolation methods to evaluate the positioning accuracy of the test area before and after ASF correction. Compared to other methods, using the Ordinary Kriging interpolation algorithm for predicting ASF values resulted in a corrected positioning accuracy of up to 68.8 m at various test locations. This approach effectively resolves the challenge of low accuracy in theoretical calculations in complex environments. By utilizing ordinary Kriging interpolation, we required measurements of only a few ASF values within a specific region to create an ASF correction map, addressing the challenges related to inaccurate theoretical calculations in complex pathways and avoiding time-consuming and labor-intensive large-scale measurements. The results of this study offer valuable theoretical support for improving the accuracy of land-based navigation systems.

Real-space tight-binding model for twisted bilayer graphene based on mapped Wannier functions

Twisted multi-layer heterostructures have been considered a platform for studying highly correlated many-particle systems, hosting the emerging phenomena of correlation physics. Electronic structure calculations of such materials which mainly focused on Twisted Bilayer Graphene (TBG) in the framework of the independent-electron approximation, despite the complexity, accurately match the experimental results around the Fermi level. Here, we present a convenient real space π-bands tight-binding model to calculate the band structure of TBG based on the Wannier interlayer hopping parameters obtained from non-twisted bilayer graphene. Our approach is based on mapping hopping parameters onto the real space TBG interlayer couplings, using Radial Basis Function (RBF) interpolation method. This accurate mapping, naturally, transfers all the symmetries and orbital shape features of the bilayer graphene lattice to the TBG lattice and preserves coupling orientation dependencies in addition to distance dependencies. The importance of this fact is explained by showing the effect of the threefold rotation symmetry of AB′ interlayer coupling on the flat band of the TBG band structure. In addition, due to real space study, the model gives us a comprehensive and intuitive view of the role of each interlayer orbital coupling in the TBG band structure and the structural variation relative to twisting.

Artificial Intelligence Technology Boosts the Construction of English Intelligent Teaching Mode in Colleges and Universities

Abstract With the goal of improving the teaching effect of English in colleges and universities, this paper combines artificial intelligence technology with teaching and constructs an English-intelligent tutoring teaching model. By analyzing the main forms of artificial intelligence in the field of teaching, the advantages of intelligent teaching mode in college English teaching are explored from personalized teaching as well as teaching effect. Combining the RBF neural network to analyze the students’ learning data, the function approximation principle and interpolation method are used to improve the accuracy of the analysis of the students’ data. Using the form of network topology, the information transfer process in English intelligent teaching is explored. To improve the classification of students’ abilities and the prediction of their grades, judgment trees are added to the network. The English Intelligent Assisted Teaching model was applied to teaching to explore its feasibility and effectiveness. The results show that students’ satisfaction with personalized teaching is 0.85, and their satisfaction with personalized evaluation is 0.8. The students’ translation ability under the English Intelligent Assisted Teaching Model has improved more, from 75 to 90.

Reconstruction of incomplete flow fields based on unsupervised learning

The loss of flow field data is a common challenge in experimental techniques and data transmission processes. Traditional methods often rely on another complete or high-fidelity dataset to compensate for missing information regarding fluid characteristics. However, obtaining such complete data is not always feasible. In this study, we propose an unsupervised learning approach that efficiently, swiftly, and cost-effectively reconstructs the complete flow field by leveraging the inherent properties of the flow field itself, without the need for high-fidelity data integration. Our proposed methods primarily encompass three techniques: Fourier basis function interpolation, the Autoregressive Integrated Moving Average model (ARIMA), and AMF (ARIMA-Median Filter). The first two methods are founded on the temporal evolution characteristics of the flow field and reconstruct the missing fluid information by extrapolating from the available data. The AMF method, building upon the ARIMA reconstruction outcomes, employs a median filtering approach to eliminate noise and outliers introduced by the lack of spatial correlation. All three methods perform well, with the AMF method exhibiting superior performance, particularly in cases involving scatter missing data with higher absence rates. While ARIMA and AMF entail longer computational times, they offer stability and are less prone to overfitting during the reconstruction process, rendering them more suitable for reconstructing flow fields with varying degrees of missing data. The Fourier basis function interpolation method, with its shorter runtime, is better suited for swiftly reconstructing flow fields with lower rates of missing data. However, it is worth noting that improper parameter selection for this method can lead to overfitting. Additionally, the non-stationarity of the flow field has an impact on reconstruction accuracy, with regions exhibiting stronger non-stationarity resulting in more noticeable reconstruction errors. For the ARIMA method, which relies on temporal correlation for reconstruction, the missing structures have a relatively minor effect. However, in the case of flow fields with block missing data, the improvement in reconstruction accuracy when using median filtering compared to ARIMA is not substantial.

Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n+1 “failed to produce acceptable results”.In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).

A framework for simulating snow accumulation and ice accretion on high-speed trains

A simulation framework for three-dimensional particle trajectory tracking, snow accumulation and ice accretion modelling on solid bodies in high-speed air flow is presented. The framework, named HADICE, solves the aerodynamic flow field as Reynolds averaged Navier-Stokes (RANS), with optional hybrid RANS/LES capabilities for modelling complex turbulent flows. Particle trajectory tracking is performed in an Eulerian-Lagrangian approach and wall collision is modelled using a hard collision model with reflect condition and a momentum based trap condition. A novel accurate and robust iterative evaluation method for the local collection efficiency is proposed for arbitrary 3D geometry and flow. Ice accretion is modelled in a multi-step approach and iced geometry is updated through mesh deformation using a radial basis function (RBF) interpolation method. Snow accumulation and ice accretion predictions based on the framework are validated against climatic wind tunnel experimental measurements. A full 3D simulation is demonstrated for snow accumulation and ice accretion on a 1:8 scaled high-speed train model. Using the presented framework, snow accumulation and icing simulation for high-speed trains can be conducted in an accurate and efficient manner, which is of great importance for physical investigations and the design of anti-snow and ice protection systems.

Design of large-angle distortion-free and spot adjustable LED spot array projector based on microlens array and eyepiece

The spot array projector has long been a challenging and intriguing research focus in the fields of projection and lighting applications. In the current literature, there is a lack of a comprehensive and detailed presentation on the design and analysis method for a spot array generator based on the structure of imaging eyepiece and MLA. We present a novel design and optimization method for a large-angle, distortion-free and spot adjustable LED spot array projector that is composed of an eyepiece, two microlens arrays (MLAs), and a micro-LED array (MLEDA). The eyepiece system is optimized using imaging optical methods to project sub-beams to the target plane with a large angle. The sub-lens of condenser MLA is also optimized using imaging optical methods to refocus the collimating beam and match the numerical aperture (NA) with the eyepiece, and the sub-lens of the collimating MLA is acquired by using simulated annealing (SA) global illumination optics optimization method to achieve collimation and far-field homogenization. The predistortion MLEDA and the MLAs are proposed and implemented by the radial basis function (RBF) interpolation method, which correct the large-angle distortion introduced by the eyepiece. Both near-field and far-field applications can be realized by the proposed system. In the near-field applications, different spot geometries at the near-field target plane can be achieved. In the far-field applications, the power matching of the MLEDA is used to improve far-field uniformity of spot array. Moreover, the predefined-geometry arrangement spot array can be realized in both near and far fields. Two design examples with full field of view (FOV) projection of 80° and 100° are provided to validate the proposed method. Overall, the proposed system offers a promising solution for various applications requiring target identification or 3D calibration.

Net-HDMR Metamodeling Method for High-Dimensional Problems

Abstract Metamodel technology provides an efficient method to approximate complex engineering design problems. However, the approximation for high-dimensional problems usually requires a large number of samples for most traditional metamodeling methods, which leads to the difficulty of “curse of dimensionality.” To address the aforementioned issue, this paper presents the Net-high dimension model representation (HDMR) method based on the Cut-HDMR framework. Compared with traditional HDMR modeling, the Net-HDMR method incorporates two novel modeling approaches that improve the modeling efficiency of high-dimensional problems. The first approach enhances the modeling accuracy of HDMR by using the net function interpolation method to decompose the component functions into a series of one-dimensional net functions. The second approach adopts the CV-Voronoi sequence sampling method to effectively represent one-dimensional net functions with limited samples. Overall, the proposed method transforms complex high-dimensional problems into fitting finite one-dimensional splines, thereby increasing the modeling efficiency while ensuring approximate accuracy. Six numerical benchmark examples with different dimensions are examined to demonstrate the accuracy and efficiency of the proposed Net-HDMR. An engineering problem of thermal stress and deformation analysis for a jet engine turbine blade was introduced to verify the engineering feasibility of the proposed Net-HDMR.

Study of groundwater temperature spatio-temporal variation in the city of Wroclaw. Preliminary results.

In Wrocław over the past 20 years, the recorded increase in the average ambient air temperature is approx. 0.5 °C. In addition, urbanisation has created a phenomenon called the Urban Heat Island (UHI). The combined effect of the UHI and climate change can influence groundwater temperature by penetrating underground. The phenomenon of elevated groundwater temperatures in urban areas is called the Subsurface Urban Heat Island (SSUHI). This study focuses on the spatial distribution of temperature in shallow aquifers in the city of Wroclaw. The spatial distribution was analysed based on groundwater temperature maps developed with various interpolation techniques. The results of measurements in a network of 65 piezometers conducted in the 2004-2005 hydrological year have been used in this study. The temperatures recorded show seasonal variation with the lowest recorded in February (1.3 °C) and the highest in August (24.5 °C) with the annual average of 13.2 °C. The data has been processed with the Inverse Distance Weighted (IDW), Radial Basis Function (RBF) and Kriging interpolation methods. Furthermore, the cokriging method based on multivariate linear regression models has been used to assess if variable such as distance from the city centre improve the interpolation performance expressed as the lowest RMS error value. It has been established that the Kriging univariate regression model based interpolation method produced the best results (RMS error equal to 1.33 °C) and that the distance from the city centre has improved the accuracy of interpolation.

Fast Numerical Solution of a Kind of Nonlinear Integral Equations—Dyson-Schwinger Equations for Quark Propagator in Hadron Physics

The nonlinear integral equation has been widely studied and has become the heart of the matter in many scientific and engineering fields, such as seismology, optical fiber evolution, radio astronomy, and hadron physics with Quantum Chromodynamics. The Dyson-Schwinger Equations (DSEs) approach provides an essential nonperturbative approach to investigating the properties of hadrons and hot/dense quark matter. Mathematically, the Dyson-Schwinger Equations are a group of coupled nonlinear integral equations of quark propagators, gluon propagators, ghost propagators, and various vertices. On account of the non-linearity and singularity of the coupled equations, it is almost impossible to solve the DSEs analytically. One has to resort to the numerical solution of the equations, in which efficient fast algorithms are key points in practice. In this work, two improvements for numerically solving the nonlinear and singular integral equation for quark propagator in a vacuum are proposed. One is a modified interpolation method for unknown functions in the integral with high degrees of freedom. The other is the parallelization on CPUs with OpenMP in GCC Comparing the CPU times with different algorithms, our results indicate that our proposed methods can greatly improve the efficiency and reduce the computation time of the CPU.

3D Near-surface Velocity Model Building by Integrating Surface, Borehole Seismic, and Well-logging Data in a Small-scale Testbed

This study aims to build a 3D velocity model for investigating and evaluating underground contaminants flowing through groundwater pathways. To accomplish this goal, we performed seismic exploration at a test site with fractured rock layers within a depth of 100 m. The acquired seismic data was then used to construct a three-dimensional (3D) P-wave velocity model of the test site. Although comprehensive geophysical exploration may be useful for investigating the near-surface structure, this paper focuses only on constructing a P-wave velocity model using the seismic method. The primary information of the model consists of extracted velocities from a two-dimensional surface, borehole first-arrival traveltime tomography results, and full waveform sonic log data. Since the test site has the spatial restriction of the survey line, we intended to improve the geological structure analysis results using various quantitative and qualitative analysis methods. First, to increase the reliability of velocity information, we performed a traveltime analysis on the zero vertical interval (ZVI) gather from the borehole seismic data. Then, we identified the qualitative information of the fracture zone's location by analyzing the amplitude variation of the ZVI gather. Moreover, we extracted structural information using the common reflection point gather from the borehole seismic data to supplement the obtained lithological information. However, the information for constructing a 3D P-wave velocity model was still insufficient due to the spatial constraints of the survey line and the limited depth of the borehole seismic survey. We filled this gap using the radial basis function interpolation method. We could verify the completed 3D P-wave velocity model by comparing it with core log data. Overall, the integrated interpretation of the final velocity model and the analysis results could provide a probable pathway that indicates hydraulic connectivity.

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

<p style='text-indent:20px;'>We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ L\geq1 $\end{document}</tex-math></inline-formula>) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level <inline-formula><tex-math id="M3">\begin{document}$ L+1 $\end{document}</tex-math></inline-formula>, which includes the sparse grids from approximation level <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula>. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.</p>

The reconstruction of 3D radiation field based on sparse measurement data

With the development of nuclear technology, an increasing number of occupational workers are exposed to radioactive environments. The three-dimensional(3D) dose rate field is of great significance in the radiation protection field. Many researchers focus on the reconstruction of 2D radiation fields but few of them paid attention to the reconstruction of the 3D dose rate field based on sparse measurement data. In this work, the net function interpolation method is introduced to reconstruct 3D radiation field using sparse sampling nodes. In contrast to traditional dose rate interpolation methods, we interpolated the value of the inverse of the dose rate. By using the inverse, the field to be interpolated follows a power function, which fits the net function much better than the direct interpolation. The proposed method includes interpolation on sampling grid lines, interpolation on sampling grid plane and interpolation in the three-dimensional space. Additionally, four simulation experiments are carried out to prove the effectiveness and accuracy of the proposed method. The first two cases are radiation fields formed by a single radioactive source without and with shielding. Radioactive environments caused by multiple radioactive sources without and with shielding are conducted in Case 3 and Case 4. The reconstruction radiation field is compared to the radiation field simulated by the Monte Carlo method. The average relative errors of the four cases are 0.82%, 2.13%, 1.71% and 4.27% respectively using approximately 1% measurement data. The experimental data provided by other researchers are also used to validate the effectiveness of the proposed method and the minimum relative error of Case 5 is 4.67%. Therefore, the net function is an interpolation method suitable for the radiation field and it can reconstruct the 3D radiation field accurately with sparse data.

Characteristic Analysis of the Spatio-Temporal Distribution of Key Variables of the Soil Freeze–Thaw Processes over Heilongjiang Province, China

The soil freeze–thaw phenomenon is one of the most outstanding characteristics of the soil in Heilongjiang Province. Quantitative analysis of the characteristics of changes in key variables of the soil freeze–thaw processes is of great scientific importance for understanding climate change, as well as ecological and hydrological processes. Based on the daily surface temperature and air temperature data in Heilongjiang Province for the past 50 years, the spatial–temporal distribution characteristics of key variables and their correlations with air temperature and latitude in the freeze–thaw process of soil were analyzed using linear regression, the Mann–Kendall test, the local thin disk smooth spline function interpolation method, and correlation analysis; additionally, the spatial–temporal distribution of key variables and the changes in the surface temperature during the freeze–thaw process are discussed under different vegetation types. The results show that there is a trend of delayed freezing and early melting of key variables of the soil freeze–thaw process from north to south. From 1971 to 2019 a, the freezing start date (FSD) was delayed at a rate of 1.66 d/10 a, the freezing end date (FED) advanced at a rate of 3.17 d/10 a, and the freezing days (FD) were shortened at a rate of 4.79 d/10 a; with each 1 °C increase in temperature, the FSD was delayed by about 1.6 d, the FED was advanced by about 3 d, and the FD was shortened by about 4.6 d; with each 1° increase in latitude, the FSD was delayed by about 2.6 d, the FED was advanced by about 2.8 d, and the FD was shortened by about 5.6 d. The spatial variation in key variables of the soil freeze–thaw process under the same vegetation cover was closely related to latitude and altitude, where the lower the latitude and altitude, the more obvious the variation trend; among them, the interannual variation trend of key variables of soil freeze–thaw under meadow cover was the most obvious, which varied by 9.65, 16.86, and 26.51 d, respectively. In addition, the trends of ground temperature under different vegetation types were generally consistent, with the longest period of unstable freeze–thaw and the shortest period of stable freeze in coniferous forests, compared to the shortest period of unstable freeze–thaw and the longest period of stable freeze in meadows. The results of the study are important for our understanding of soil freeze–thaw processes and changes in Heilongjiang Province, as well as the evolution of high-latitude permafrost; they also promote further exploration of the impact of soil freeze–thaw on agricultural production and climate change.