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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 739 records · Page 41
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

The three-dimensional construction of leaves is coordinated with water use efficiency in conifers

Conifers prevail in the canopies of many terrestrial biomes, holding a great ecological and economic importance globally. Current increases in temperature and aridity are imposing high transpirational demands and resulting in conifer mortality. Therefore, identifying leaf structural determinants of water use efficiency is essential for predicting physiological impacts due to environmental variation. Here, using synchrotron-generated microtomography imaging, we extracted leaf volumetric anatomy and stomatal traits in 34 species across conifers with a special focus on Pinus , the richest conifer genus. We show that intrinsic water use efficiency (WUE i ) is positively driven by leaf vein volume. Needle-like leaves of Pinus , as opposed to flat leaves or flattened needles of other genera, showed lower mesophyll porosity, decreasing the relative mesophyll volume. This led to increased ratios of stomatal pore number per mesophyll or intercellular airspace volume, which emerged as powerful explanatory variables, predicting both stomatal conductance and WUE i . Our results clarify how the three-dimensional organisation of tissues within the leaf has a direct impact on plant water use and carbon uptake. By identifying a suite of structural traits that influence important physiological functions, our findings can help to understand how conifers may respond to the pressures exerted by climate change.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2020

Assessment of CO 2 trapping mechanisms in partially depleted oil-bearing sands

The goal of this work is to evaluate various CO 2 sequestration mechanisms occurring in the Morrow B Sandstone in the Farnsworth Unit. A history-matched numerical simulation model was created using extensive geological, petrophysical, and operational data collected from the field. The numerical model is competent to investigate the impact of residual, structural-stratigraphic, solubility, and mineral trapping mechanisms on the fluid transportation dynamics and petrophysical property variations. The model forecasts the field response of 20 years of WAG injections. Afterward, all wells were shut-in, and the reservoir was allowed to evolve for 1000 years to investigate the fate of injected CO 2 . In this paper, we assess the impacts of various trapping mechanisms on oil recovery and CO 2 storage efficacy. By analyzing the results reported from the numerical simulation model, the in-situ fluid composition and mineralogy changes are also investigated. More importantly, we seek to confirm the petrophysical property variations due to the CO 2 injection with observations from laboratory measurements. The experiences gained from this study provide valuable insights regarding physiochemical storage induced by the CO 2 injection activities and serve as a benchmark case for future CO 2 enhanced oil recovery (EOR) projects involving reactive solute transport.

42 ENGINEERING↗
DOE OSTI2020

Distribution of genetic variation underlying adult migration timing in steelhead of the Columbia River basin

Abstract Fish migrations are energetically costly, especially when moving between freshwater and saltwater, but are a viable strategy for Pacific salmon and trout ( Oncorhynchus spp.) due to the advantageous resources available at various life stages. Anadromous steelhead ( O. mykiss ) migrate vast distances and exhibit variation for adult migration phenotypes that have a genetic basis at candidate genes known as greb1L and rock1 . We examined the distribution of genetic variation at 13 candidate markers spanning greb1L , intergenic, and rock1 regions versus 226 neutral markers for 113 populations ( n = 9,471) of steelhead from inland and coastal lineages in the Columbia River. Patterns of population structure with neutral markers reflected genetic similarity by geographic region as demonstrated in previous studies, but candidate markers clustered populations by genetic variation associated with adult migration timing. Mature alleles for late migration had the highest frequency overall in steelhead populations throughout the Columbia River, with only 9 of 113 populations that had a higher frequency of premature alleles for early migration. While a single haplotype block was evident for the coastal lineage, we identified multiple haplotype blocks for the inland lineage. The inland lineage had one haplotype block that corresponded to candidate markers within the greb1L gene and immediately upstream in the intergenic region, and the second block only contained candidate markers from the intergenic region. Haplotype frequencies had similar patterns of geographic distribution as single markers, but there were distinct differences in frequency between the two haplotype blocks for the inland lineage. This may represent multiple recombination events that differed between lineages where phenotypic differences exist between freshwater entry versus arrival timing as indicated by Micheletti et al. (2018a). Redundancy analyses were used to model environmental effects on allelic frequencies of candidate markers, and significant variables were migration distance, temperature, isothermality, and annual precipitation. This study improves our understanding of the spatial distribution of genetic variation underlying adult migration timing in steelhead as well as associated environmental factors and has direct conservation and management implications.

Collins, Erin E.↗
DOE OSTI2022

Controlling morphology in hybrid isotropic/patchy particle assemblies

Brownian dynamics is used to study self-assembly in a hybrid system of isotropic particles (IPs), combined with anisotropic building blocks that represent special “designer particles.” Those are modeled as spherical patchy particles (PPs) with binding only allowed between their patches and IPs. In this study, two types of PPs are considered: Octahedral PPs (Oh-PPs) and Square PPs (Sq-PPs), with octahedral and square arrangements of patches, respectively. The self-assembly is additionally facilitated by the simulated annealing procedure. Here, the resultant structures are characterized by a combination of local correlations in cubatic ordering and a symmetry-specific variation of bond orientation order parameters (SymBOPs). By varying the PP/IP size ratio, we detected a sharp crossover between two distinct morphologies in both types of systems. High symmetry phases, NaCl crystal for Oh-PP and square lattice for Sq-PP, are observed for larger size ratios. For the smaller ones, the dominant morphologies are significantly different, e.g., Oh-PPs form a compact amorphous structure with predominantly face-to-face orientation of neighboring PPs. Unusually, for a morphology without a long-range order, it is still possible to identify well organized coherent clusters of this structure, thanks to the adoption of our SymBOP-based characterization.

77 NANOSCIENCE AND NANOTECHNOLOGY↗
DOE OSTI2026

On the uncertainties in helium generation predictions for fission and fusion alloys

With ongoing advances in fusion and advanced fission reactors, quantifying irradiation effects in materials is critical. Transmutation-induced helium in cladding and structural materials can drive swelling and embrittlement, thereby reducing these components’ lifespans. Yet most studies ignore the considerable uncertainties in predicting helium generation rates. In this work, we created a code wrapper, F-SCATTER, that automatically performs simulations in FISPACT-II. We used this tool to investigate potential variance in helium generation rate, or He/dpa, calculations based on deviations in alloy composition, irradiating neutron flux spectrum, computational methodology, and nuclear data sources. We used 12 wt% Cr HT9 steel as the reference case and observed a 6.5%–98.3% He/dpa spread based on compositional variation within a single chemical specification, a 1.8%–11.5% He/dpa variation upon the incorporation of a 15% artificial uncertainty in flux at each energy, and a He/dpa difference as high as 231% when using ENDF/B-VIII.0 versus TENDL-2021 data libraries. Similar results were found for other prominent iron-based alloys, including Grade 91, castable nano-structured alloy, and 316H—where additional variations exist based on reactor type (e.g. thermal, fast, or fusion) and alloying elements such as carbon, nitrogen, and nickel. Based on the simulated results, we conclude that a significant part of the heat-to-heat variability in swelling responses of Fe-based alloys can be driven by impurity content in alloy compositions, and, therefore, chemical control should be a key element in supply chain design for advanced nuclear energy systems. Furthermore, we provide critical recommendations on best practices for evaluating and reporting helium production and lattice damage rates when computing predictions with multiphysics programs such as FISPACT-II.

FISPACT-II↗
DOE OSTI2021

Inherent stochasticity during insulator–metal transition in VO 2

Vanadium dioxide (VO 2 ), which exhibits a near-room-temperature insulator–metal transition, has great potential in applications of neuromorphic computing devices. Although its volatile switching property, which could emulate neuron spiking, has been studied widely, nanoscale studies of the structural stochasticity across the phase transition are still lacking. In this study, using in situ transmission electron microscopy and ex situ resistive switching measurement, we successfully characterized the structural phase transition between monoclinic and rutile VO 2 at local areas in planar VO 2 /TiO 2 device configuration under external biasing. After each resistive switching, different VO 2 monoclinic crystal orientations are observed, forming different equilibrium states. We have evaluated a statistical cycle-to-cycle variation, demonstrated a stochastic nature of the volatile resistive switching, and presented an approach to study in-plane structural anisotropy. Furthermore, our microscopic studies move a big step forward toward understanding the volatile switching mechanisms and the related applications of VO 2 as the key material of neuromorphic computing.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Discussion on molecular dynamics (MD) simulations of the asphalt materials

The application of asphalt materials in pavement engineering has been increasingly widespread and sophisticated over the past several decades. Variations in the properties of asphalt binder during mixing, transportation, and paving can affect the performance of asphalt pavement. However, the asphalt material is a non-homogeneous and complex organic substance, consisting of various molecules with widely various molecular weights, elemental compositions, and structures. This complexity leads to difficulties for researchers to clearly and immediately understand the properties of asphalt materials and their variations. The multi-scale research approach combines macroscopic experimental data and microscopic simulation results from a practical engineering perspective. It helps to improve the understanding of asphalt materials. The molecular dynamics (MD) simulation proposes a corresponding molecular model of asphalt material based on experimental data, and the simulation algorithm is able to derive properties similar to those of real asphalt. Here, this paper provides a comprehensive review of the current studies on MD simulation of asphalt materials, including modeling, properties, and multi-scale analysis. As a key part of the computational simulation, this paper discusses the typical asphalt binder and asphalt-aggregate interface models constructed by different groups, and also presents their differences from real samples and their feasibility based on fundamental properties. After the introduction of molecular models, the extensive work made by researchers based on molecular models is categorically reviewed and discussed. The strengths and weaknesses of MD simulation methods in the study of asphalt materials are also summarized in order to provide the reader with a more comprehensive understanding of the relevant contents and to guide subsequent research.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Dynamic Structure Discovery Applied to the Ion Transport in the Ubiquitous Lithium-ion Battery Electrolyte LP30

The electrolytes of the today omnipresent lithium-ion batteries (LIBs) have for more than 25 years been based upon 1 M LiPF 6 in a 50:50 EC:DMC mixture—commonly known as LP30. The success of the basic design of the LP30 electrolyte, with many variations and additions made over the years, is unchallenged. Yet, some molecular level fundamentals of LP30 are surprisingly elusive: the structure of the first solvation shell of the Li + cation is still a topic of current debate; the details of the dynamics are not fully understood; the interpretation of structural and dynamic properties is highly dependent on the analysis methods used; the contributions by different species to the ion transport and the energetics involved are not established. We here apply dynamic structure discovery analysis as implemented in CHAMPION to molecular dynamics simulation trajectories to bring new light on the structure and dynamics within LP30 and especially the (Li + ) ion transport to rationalize further development of LIB electrolytes.

25 ENERGY STORAGE↗
DOE OSTI2021

Magnetism in metastable and annealed compositionally complex alloys

Compositionally complex materials (CCMs) present a potential paradigm shift in the design of magnetic materials. These alloys exhibit long-range structural order coupled with limited or no chemical order. As a result, extreme local environments exist with a large variations in the magnetic energy terms, which can manifest large changes in the magnetic behavior. In the current work, the magnetic properties of (Cr, Mn, Fe, Ni) alloys are presented. These materials were prepared by room-temperature combinatorial sputtering, resulting in a range of compositions with a single bcc structural phase and no chemical ordering. The combinatorial growth technique allows CCMs to be prepared outside of their thermodynamically stable phase, enabling the exploration of otherwise inaccessible order. The mixed ferromagnetic and antiferromagnetic interactions in these alloys causes frustrated magnetic behavior, which results in an extremely low coercivity (<1mT), which increases rapidly at 50 K. At low temperatures, the coercivity achieves values of nearly 500 mT, which is comparable to some high-anisotropy magnetic materials. Further, commensurate with the divergent coercivity is an atypical drop in the temperature dependent magnetization. These effects are explained by a mixed magnetic phase model, consisting of ferro-, antiferro-, and frustrated magnetic regions, and are rationalized by simulations. A machine-learning algorithm is employed to visualize the parameter space and inform the development of subsequent compositions. Annealing the samples at 600 °C orders the sample, more-than doubling the Curie temperature and increasing the saturation magnetization by as much as 5×. Simultaneously, the large coercivities are suppressed, resulting in magnetic behavior that is largely temperature independent over a range of 350 K. The ability to transform from a hard magnet to a soft magnet over a narrow temperature range makes these materials promising for heat-assisted recording technologies.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Structural polymorphism of amyloid fibrils in ATTR amyloidosis revealed by cryo-electron microscopy

ATTR amyloidosis is caused by the deposition of transthyretin in the form of amyloid fibrils in virtually every organ of the body, including the heart. This systemic deposition leads to a phenotypic variability that has not been molecularly explained yet. In brain amyloid conditions, previous studies suggest an association between clinical phenotype and the molecular structures of their amyloid fibrils. Here we investigate whether there is such an association in ATTRv amyloidosis patients carrying the mutation I84S. Using cryo-electron microscopy, we determined the structures of cardiac fibrils extracted from three ATTR amyloidosis patients carrying the ATTRv-I84S mutation, associated with a consistent clinical phenotype. We found that in each ATTRv-I84S patient, the cardiac fibrils exhibited different local conformations, and these variations can co-exist within the same fibril. Our finding suggests that one amyloid disease may associate with multiple fibril structures in systemic amyloidoses, calling for further studies.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2023

Structuring Nutrient Yields throughout Mississippi/Atchafalaya River Basin Using Machine Learning Approaches

To minimize the eutrophication pressure along the Gulf of Mexico or reduce the size of the hypoxic zone in the Gulf of Mexico, it is important to understand the underlying temporal and spatial variations and correlations in excess nutrient loads, which are strongly associated with the formation of hypoxia. This study’s objective was to reveal and visualize structures in high-dimensional datasets of nutrient yield distributions throughout the Mississippi/Atchafalaya River Basin (MARB). For this purpose, the annual mean nutrient concentrations were collected from thirty-three US Geological Survey (USGS) water stations scattered in the upper and lower MARB from 1996 to 2020. Eight surface water quality indicators were selected to make comparisons among water stations along the MARB over the past two decades. Principal component analysis (PCA) was used to comprehensively evaluate the nutrient yields across thirty-three USGS monitoring stations and identify the major contributing nutrient loads. The results showed that all samples could be analyzed using two main components, which accounted for 81.6% of the total variance. The PCA results showed that yields of orthophosphate (OP), silica (SI), nitrate–nitrites (NO 3 -NO 2 ), and total suspended sediment (TSS) are major contributors to nutrient yields. It also showed that land-planted crops, density of population, domestic and industrial discharges, and precipitation are fundamental causes of excess nutrient loads in MARB. These factors are of great significance for the excess nutrient load management and pollution control of the Mississippi River. It was found that the average nutrient yields were stable within the sub-MARB area, but the large nitrogen yields in the upper MARB and the large phosphorus yields in the lower MARB were of great concern. t-distributed stochastic neighbor embedding (t-SNE) revealed interesting nonlinear and local structures in nutrient yield distributions. Clustering analysis (CA) showed the detailed development of similarities in the nutrient yield distribution. Moreover, PCA, t-SNE, and CA showed consistent clustering results. This study demonstrated that the integration of dimension reduction techniques, PCA, and t-SNE with CA techniques in machine learning are effective tools for the visualization of the structures of the correlations in high-dimensional datasets of nutrient yields and provide a comprehensive understanding of the correlations in the distributions of nutrient loads across the MARB.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Data for Clumping Index Estimation With 30°-tilted Cameras in Row Crops: Evaluation of Methods and Segment Size Effects

The clumping index (CI) quantifies the spatial distribution of foliage elements and is essential for accurately estimating the plant area index (PAI), canopy radiative transfer, and photosynthesis. Traditionally, the finite-length averaging method (LX), the gap size distribution method (CC), and a combined approach of CC and LX (CLX) have been applied to instruments like TRAC and digital hemispherical photography to estimate CI. However, a comprehensive evaluation of these methods in row crops remains limited, especially regarding the influence of segment size on CI. Meanwhile, digital cameras offer a cost-effective and user-friendly solution for canopy measurements in row crops, yet their application in this context remains underexplored. In this study, we employed a new approach using a 30°-tilted digital camera to estimate CI in corn and soybean fields, applying the LX, CC, and CLX methods. We systematically assessed the performance of these three methods by combining field measurements in real-world fields with simulations using the LESS 3D radiative transfer model. Our results showed that CLX applied to the whole image and 45° segment offered accurate estimation of CI (bias within ±0.1, RMSE < 0.2) and PAI (bias within ±0.4, RMSE < 1) in real-world fields and LESS simulations. The accuracy of the LX method was highly sensitive to segment size, with the best performance observed at the 15° segment (PAI bias within ±0.4). In contrast, the CC method remained stable across different segment sizes, and its performance was generally comparable to that of LX, except at the 15° segment. Across view zenith angles, CI derived from CC generally showed a continuous increase, while those from LX and CLX followed a rising trend at small zenith angles but began to decline at 68°, likely due to an increasing proportion of no-gap segments. Seasonally, LX tended to show decreasing CI during early growth stages but increased as the canopy matured, whereas CC and CLX showed gradually increasing CI before plateauing at peak PAI. The 30°-tilted camera effectively captured CI variations across different angles and growth stages, making it a practical and robust instrument for row crop canopy structure analysis. Applying these CI methods to digital cameras offers a low-cost and accessible CI estimation alternative, improving canopy structure monitoring accuracy in row crops.

Modeling↗
DOE OSTI2022

Ig-VAE: Generative modeling of protein structure by direct 3D coordinate generation

While deep learning models have seen increasing applications in protein science, few have been implemented for protein backbone generation—an important task in structure-based problems such as active site and interface design. We present a new approach to building class-specific backbones, using a variational auto-encoder to directly generate the 3D coordinates of immunoglobulins. Our model is torsion- and distance-aware, learns a high-resolution embedding of the dataset, and generates novel, high-quality structures compatible with existing design tools. We show that the Ig-VAE can be used with Rosetta to create a computational model of a SARS-CoV2-RBD binder via latent space sampling. We further demonstrate that the model’s generative prior is a powerful tool for guiding computational protein design, motivating a new paradigm under which backbone design is solved as constrained optimization problem in the latent space of a generative model.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

The Structure of Liquid and Glassy Carbamazepine

To enhance the solubility of orally administered pharmaceuticals, liquid capsules or amorphous tablets are often preferred over crystalline drug products. However, little is known regarding the variation in bonding mechanisms between pharmaceutical molecules in their different disordered forms. In this study, liquid and melt-quenched glassy carbamazepine have been studied using high energy X-ray diffraction and modeled using Empirical Potential Structure Refinement. The results show significant structural differences between the liquid and glassy states. The liquid shows a wide range of structures; from isolated molecules, to aromatic ring correlations and NH-O hydrogen bonding. Upon quenching from the liquid to the glass the number of hydrogen bonds per molecule increases by ~50% at the expense of a ~30% decrease in the close contact (non-bonded) carbon-carbon interactions between aromatic rings. During the cooling process, there is an increase in both singly and doubly hydrogen-bonded adjacent molecules. Although hydrogen-bonded dimers found in the crystalline states persist in the glassy state, the absence of a crystalline lattice also allows small, hydrogen-bonded NH-O trimers and tetramers to form. This proposed model for the structure of glassy carbamazepine is consistent with the results from vibrational spectroscopy and nuclear magnetic resonance.

Benmore, Chris J. (ORCID:0000000170077749)↗
DOE OSTI2023

Finding simplicity: unsupervised discovery of features, patterns, and order parameters via shift-invariant variational autoencoders *

Abstract Recent advances in scanning tunneling and transmission electron microscopies (STM and STEM) have allowed routine generation of large volumes of imaging data containing information on the structure and functionality of materials. The experimental data sets contain signatures of long-range phenomena such as physical order parameter fields, polarization, and strain gradients in STEM, or standing electronic waves and carrier-mediated exchange interactions in STM, all superimposed onto scanning system distortions and gradual changes of contrast due to drift and/or mis-tilt effects. Correspondingly, while the human eye can readily identify certain patterns in the images such as lattice periodicities, repeating structural elements, or microstructures, their automatic extraction and classification are highly non-trivial and universal pathways to accomplish such analyses are absent. We pose that the most distinctive elements of the patterns observed in STM and (S)TEM images are similarity and (almost-) periodicity, behaviors stemming directly from the parsimony of elementary atomic structures, superimposed on the gradual changes reflective of order parameter distributions. However, the discovery of these elements via global Fourier methods is non-trivial due to variability and lack of ideal discrete translation symmetry. To address this problem, we explore the shift-invariant variational autoencoders (shift-VAEs) that allow disentangling characteristic repeating features in the images, their variations, and shifts that inevitably occur when randomly sampling the image space. Shift-VAEs balance the uncertainty in the position of the object of interest with the uncertainty in shape reconstruction. This approach is illustrated for model 1D data, and further extended to synthetic and experimental STM and STEM 2D data. We further introduce an approach for training shift-VAEs that allows finding the latent variables that comport to known physical behavior. In this specific case, the condition is that the latent variable maps should be smooth on the length scale of the atomic lattice (as expected for physical order parameters), but other conditions can be imposed. The opportunities and limitations of the shift VAE analysis for pattern discovery are elucidated.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Real-time gravitational replicas: formalism and a variational principle

This work is the first step in a two-part investigation of real-time replica wormholes. Here we study the associated real-time gravitational path integral and construct the variational principle that will define its saddle-points. We also describe the general structure of the resulting real-time replica wormhole saddles, setting the stage for construction of explicit examples. These saddles necessarily involve complex metrics, and thus are accessed by deforming the original real contour of integration. However, the construction of these saddles need not rely on analytic continuation, and our formulation can be used even in the presence of non-analytic boundary-sources. Furthermore, at least for replica- and CPT-symmetric saddles we show that the metrics may be taken to be real in regions spacelike separated from a so-called ‘splitting surface’. This feature is an important hallmark of unitarity in a field theory dual.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Multimodal and multiscale strengthening mechanisms in Al-Ni-Zr-Ti-Mn alloy processed by laser powder bed fusion additive manufacturing

The unique thermokinetics of laser-powder bed fusion additive manufacturing (L-PBFAM) has been exploited for development of a novel high-strength Al-Ni-Ti-Zr-Mn alloy. The addition of 0.5 wt% Mn leads to extraordinary improvement in ultimate tensile strength (502 MPa) and work hardening due to the activation of two Mn-induced strengthening mechanisms. First, by a bimodal particle strengthening effect due to Al 31 Mn 6 Ni 12 nano-quasi-crystal particles rejected in inter-dendritic spaces and fibrous Al 3 Ni eutectic dendritic channels, which predominately contributes to the strength improvement, and second by solid solution strengthening from remaining Mn entrapped in Al. These mechanisms supplement the particle strengthening effect imparted by coherent and incoherent Al 3 (Ti,Zr) co-precipitates present at melt pool boundaries, dislocation strengthening due to solidification induced strain, and Hall-Petch and backstress strengthening effect due to heterogenous grain size distribution occurring at various length scales. In conclusion, the solidification dynamics and hierarchical heat distribution that are associated with L-PBFAM resulted in complex spatial variations in these strengthening phenomena and were investigated via a high-throughput multiscale structure–property correlation that involved thermokinetic simulation, X-ray diffraction, high-resolution nanoindentation mapping, and site-specific transmission electron microscopy of the alloy.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Probabilistic Context Neighborhood model for lattices

Here we present the Probabilistic Context Neighborhood model designed for two-dimensional lattices as a variation of a Markov random field assuming discrete values. In this model, the neighborhood structure has a fixed geometry but a variable order, depending on the neighbors’ values. Our model extends the Probabilistic Context Tree model, originally applicable to one-dimensional space. It retains advantageous properties, such as representing the dependence neighborhood structure as a graph in a tree format, facilitating an understanding of model complexity. Furthermore, we adapt the algorithm used to estimate the Probabilistic Context Tree to estimate the parameters of the proposed model. We illustrate the accuracy of our estimation methodology through simulation studies. Additionally, we apply the Probabilistic Context Neighborhood model to spatial real-world data, showcasing its practical utility.

97 MATHEMATICS AND COMPUTING↗