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Sparsifying priors for Bayesian uncertainty quantification in model discovery

We propose a probabilistic model discovery method for identifying ordinary differential equations governing the dynamics of observed multivariate data. Our method is based on the sparse identification of nonlinear dynamics (SINDy) framework, where models are expressed as sparse linear combinations of pre-specified candidate functions. Promoting parsimony through sparsity leads to interpretable models that generalize to unknown data. Instead of targeting point estimates of the SINDy coefficients, we estimate these coefficients via sparse Bayesian inference. The resulting method, uncertainty quantification SINDy (UQ-SINDy), quantifies not only the uncertainty in the values of the SINDy coefficients due to observation errors and limited data, but also the probability of inclusion of each candidate function in the linear combination. UQ-SINDy promotes robustness against observation noise and limited data, interpretability (in terms of model selection and inclusion probabilities) and generalization capacity for out-of-sample forecast. Sparse inference for UQ-SINDy employs Markov chain Monte Carlo, and we explore two sparsifying priors: the spike and slab prior, and the regularized horseshoe prior. UQ-SINDy is shown to discover accurate models in the presence of noise and with orders-of-magnitude less data than current model discovery methods, thus providing a transformative method for real-world applications which have limited data.

97 MATHEMATICS AND COMPUTING↗

Machine learning assisted bayesian inference of mix and hot-spot conditions in NIF implosions

Experiments on the National Ignition Facility (NIF) have provided clear evidence of ablator material mixing into the Hot-Spot, leading to degraded performance. However, inferring the amount of mix and Hot-Spot conditions from typical experimental observations (e.g. x-ray spectra and images) is highly challenging. Here, we have developed an analysis method that utilizes machine learning assisted Bayesian inference to find the probability distributions of the Hot-Spot and mix conditions. This approach uses a neural network, trained on an idealized 2-dimensional representation of the Hot-Spot and mix distribution, and Bayesian inference to find the statistical distributions of Hot-Spot conditions that provide a match with observations. We have tested this method with synthetic data from simulations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Applicability study of Bayesian optimization in core neutronic design using a toy model

At the Japan Atomic Energy Agency (JAEA), an innovative design approach named ARKADIA (Advanced Reactor Knowledge- and AI-aided Design Integration Approach through the whole plant life cycle) for advanced nuclear reactors is currently under development. One task in ARKADIA is to build a system that automatically optimizes core and fuel designs by conducting core neutronic and thermal-hydraulic calculations, fuel integrity evaluations, and plant dynamic analyses. This system will be implemented to automatically find an optimal design that minimizes (or maximizes) objective function defined by core performance while varying the core and fuel design parameters such as fuel pin diameter, core height and diameter. In this study, as the first step of system development, we focused only on core neutronic design and conducted a study of automatic optimization. As the optimization algorithm, Bayesian optimization (BO), an effective method for optimization problems with expensive computational cost of objective function, was utilized. The applicability of BO was studied based on single- and two-objective optimization examples of core neutronic design in a toy model. As a result, in the former, it was shown that BO can give the optimal solution, which matches the reference solution calculated by a brute force calculation well, with a small number of required calculations. Usability on core neutronic designs, where the computational cost per case is high, was confirmed. In the latter, it was found that BO can give a Pareto solutions-set that shows good agreement with the reference solution. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Recent Developments in Hydrogeologic Applications for Strain Tensor Analyses

Changes in fluid pressure deform porous media and this effect occurs in a variety of hydrogeologic processes, from the change in storage during pumping or injection to fluctuations in water levels caused by barometric pressure. We have developed instruments for measuring small strains in porous media, and we have used the resulting strain data to evaluate well testing, hydraulic fracturing, manual loading at the ground surface, and ambient hydrologic processes, like rainfall and evaporation. A particularly important application is the use of strain tensor data measured at shallow depths to analyze well tests or hydraulic fractures conducted at much greater depths. An early demonstration of this technique was conducted at the North Avant Field north of Tulsa, Oklahoma, where Pennsylvanian sandstone creates a confined aquifer and oil reservoir at a depth of 530m. We have showed that the strains caused by injecting into the aquifer could be measured at a depth of 30m and used to evaluate the properties of the aquifer. We recently expanded the array of strainmeters at the North Avant Field by deploying three more instruments at shallow depth (30m) along with a deep instrument at 520m depth in the winter, 2021. The deep instrument is deployed in shale caprock slightly above the aquifer. To our knowledge, the deep strainmeter at the North Avant Field is the deepest strainmeter ever deployed and this required refining methods originally developed for shallow deployments. The instrument was lowered to depth on oil field tubing and cemented in place using techniques and materials developed for use in oil wells. Optical fiber used to communicate with the instrument was cemented in the annulus on the outside of tubing. This is significant because the techniques we used could readily be extended to greater depth, suggesting that strainmeters can be deployed over a wide range of depths for monitoring critical subsurface processes. For example, it suggests that strainmeters could be deployed through the caprock to monitor for leaks from underlying CO2 storage reservoirs. The strainmeter array at the North Avant Field has been used to characterize deformation during a series of injection tests in the spring and summer, 2021. All the new strainmeters respond to pumping, and the strainmeters we installed earlier also responded. To our knowledge, this is the first application of well testing in a deep aquifer that was monitored by an array of strainmeters—our earlier work used strainmeters at a single location. We are currently analyzing the strain data using an analytical solution, a proxy-based Bayesian inversion algorithm, and other methods. Strainmeter data has also been used to characterize periodic pumping tests by us and Riley Blais. A periodically varying pumping rate causes both hydraulic head and strain signals that vary with the same period as the pumping. The peaks and troughs of the head in monitoring wells lag behind the peaks and troughs of the head in the pumping well, and this lag time increases with distance from the pumping well. The lag time of the pressure and the distance to the monitoring well can be used in a simple analysis to estimate the hydraulic diffusivity of the aquifer. The lag time determined from strain data can be used to estimate aquifer properties using the same analysis that works for the pressure only for strain data measured at particular locations. That is because the strain field in a confining unit advances upward, laterally and then downward even though the pressure in the underlying aquifer only advances laterally, according to our recent simulations. We have field data showing that a small periodic signal superimposed on an injection rate at the North Avant Field will create a periodic strain signal at shallow strainmeters. The field data and the recent simulations suggest that including a periodic component to injection or pumping and then monitoring the resulting strain signal could be a way to monitor the subsurface.

Murdoch, Larry↗

Recent Developments in Hydrogeologic Applications for Strain Tensor Analyses

Changes in fluid pressure deform porous media and this effect occurs in a variety of hydrogeologic processes, from the change in storage during pumping or injection to fluctuations in water levels caused by barometric pressure. We have developed instruments for measuring small strains in porous media, and we have used the resulting strain data to evaluate well testing, hydraulic fracturing, manual loading at the ground surface, and ambient hydrologic processes, like rainfall and evaporation. A particularly important application is the use of strain tensor data measured at shallow depths to analyze well tests or hydraulic fractures conducted at much greater depths. An early demonstration of this technique was conducted at the North Avant Field north of Tulsa, Oklahoma, where Pennsylvanian sandstone creates a confined aquifer and oil reservoir at a depth of 530m. We have showed that the strains caused by injecting into the aquifer could be measured at a depth of 30m and used to evaluate the properties of the aquifer. We recently expanded the array of strainmeters at the North Avant Field by deploying three more instruments at shallow depth (30m) along with a deep instrument at 520m depth in the winter, 2021. The deep instrument is deployed in shale caprock slightly above the aquifer. To our knowledge, the deep strainmeter at the North Avant Field is the deepest strainmeter ever deployed and this required refining methods originally developed for shallow deployments. The instrument was lowered to depth on oil field tubing and cemented in place using techniques and materials developed for use in oil wells. Optical fiber used to communicate with the instrument was cemented in the annulus on the outside of tubing. This is significant because the techniques we used could readily be extended to greater depth, suggesting that strainmeters can be deployed over a wide range of depths for monitoring critical subsurface processes. For example, it suggests that strainmeters could be deployed through the caprock to monitor for leaks from underlying CO2 storage reservoirs. The strainmeter array at the North Avant Field has been used to characterize deformation during a series of injection tests in the spring and summer, 2021. All the new strainmeters respond to pumping, and the strainmeters we installed earlier also responded. To our knowledge, this is the first application of well testing in a deep aquifer that was monitored by an array of strainmeters—our earlier work used strainmeters at a single location. We are currently analyzing the strain data using an analytical solution, a proxy-based Bayesian inversion algorithm, and other methods. Strainmeter data has also been used to characterize periodic pumping tests by us and Riley Blais. A periodically varying pumping rate causes both hydraulic head and strain signals that vary with the same period as the pumping. The peaks and troughs of the head in monitoring wells lag behind the peaks and troughs of the head in the pumping well, and this lag time increases with distance from the pumping well. The lag time of the pressure and the distance to the monitoring well can be used in a simple analysis to estimate the hydraulic diffusivity of the aquifer. The lag time determined from strain data can be used to estimate aquifer properties using the same analysis that works for the pressure only for strain data measured at particular locations. That is because the strain field in a confining unit advances upward, laterally and then downward even though the pressure in the underlying aquifer only advances laterally, according to our recent simulations. We have field data showing that a small periodic signal superimposed on an injection rate at the North Avant Field will create a periodic strain signal at shallow strainmeters. The field data and the recent simulations suggest that including a periodic component to injection or pumping and then monitoring the resulting strain signal could be a way to monitor the subsurface.

Murdoch, Larry↗

Precipitation and Latent Heating Distributions from Satellite Passive Microwave Radiometry: Method and Uncertainties - Part 1

A revised Bayesian algorithm for estimating surface rain rate, convective rain proportion, and latent heating/drying profiles from satellite-borne passive microwave radiometer observations over ocean backgrounds is described. The algorithm searches a large database of cloud-radiative model simulations to find cloud profiles that are radiatively consistent with a given set of microwave radiance measurements. The properties of these radiatively consistent profiles are then composited to obtain best estimates of the observed properties. The revised algorithm is supported by an expanded and more physically consistent database of cloud-radiative model simulations. The algorithm also features a better quantification of the convective and non-convective contributions to total rainfall, a new geographic database, and an improved representation of background radiances in rain-free regions. Bias and random error estimates are derived from applications of the algorithm to synthetic radiance data, based upon a subset of cloud resolving model simulations, and from the Bayesian formulation itself. Synthetic rain rate and latent heating estimates exhibit a trend of high (low) bias for low (high) retrieved values. The Bayesian estimates of random error are propagated to represent errors at coarser time and space resolutions, based upon applications of the algorithm to TRMM Microwave Imager (TMI) data. Errors in instantaneous rain rate estimates at 0.5 deg resolution range from approximately 50% at 1 mm/h to 20% at 14 mm/h. These errors represent about 70-90% of the mean random deviation between collocated passive microwave and spaceborne radar rain rate estimates. The cumulative algorithm error in TMI estimates at monthly, 2.5 deg resolution is relatively small (less than 6% at 5 mm/day) compared to the random error due to infrequent satellite temporal sampling (8-35% at the same rain rate).

Olson, William S.↗

Precipitation and Latent Heating Distributions from Satellite Passive Microwave Radiometry: Improved Method and Uncertainties - Part 1

A revised Bayesian algorithm for estimating surface rain rate, convective rain proportion, and latent heating profiles from satellite-borne passive microwave radiometer observations over ocean backgrounds is described. The algorithm searches a large database of cloud-radiative model simulations to find cloud profiles that are radiatively consistent with a given set of microwave radiance measurements. The properties of these radiatively consistent profiles are then composited to obtain best estimates of the observed properties. The revised algorithm is supported by an expanded and more physically consistent database of cloud-radiative model simulations. The algorithm also features a better quantification of the convective and nonconvective contributions to total rainfall, a new geographic database, and an improved representation of background radiances in rain-free regions. Bias and random error estimates are derived from applications of the algorithm to synthetic radiance data, based upon a subset of cloud-resolving model simulations, and from the Bayesian formulation itself. Synthetic rain-rate and latent heating estimates exhibit a trend of high (low) bias for low (high) retrieved values. The Bayesian estimates of random error are propagated to represent errors at coarser time and space resolutions, based upon applications of the algorithm to TRMM Microwave Imager (TMI) data. Errors in TMI instantaneous rain-rate estimates at 0.5 -resolution range from approximately 50% at 1 mm/h to 20% at 14 mm/h. Errors in collocated spaceborne radar rain-rate estimates are roughly 50%-80% of the TMI errors at this resolution. The estimated algorithm random error in TMI rain rates at monthly, 2.5deg resolution is relatively small (less than 6% at 5 mm day.1) in comparison with the random error resulting from infrequent satellite temporal sampling (8%-35% at the same rain rate). Percentage errors resulting from sampling decrease with increasing rain rate, and sampling errors in latent heating rates follow the same trend. Averaging over 3 months reduces sampling errors in rain rates to 6%-15% at 5 mm day.1, with proportionate reductions in latent heating sampling errors.

Olson, William S.↗

Noise-aware optimization in nominally identical manufacturing and measuring systems for high-throughput parallel workflows

Device-to-device variability in experimental noise critically impacts reproducibility, especially in automated, high-throughput systems like additive manufacturing farms. While manageable in small labs, such variability can escalate into serious risks at larger scales, such as architectural 3D printing, where noise may cause structural or economic failures. This contribution presents a noise-aware decision-making algorithm that quantifies and models device-specific noise profiles to manage variability adaptively. It uses distributional analysis and pairwise divergence metrics with clustering to choose between single-device and robust multi-device Bayesian optimization strategies. Unlike conventional methods that assume homogeneous devices or enforce generic robustness, the proposed framework explicitly determines whether shared optimization across devices is appropriate based on the degree of inter-device noise heterogeneity. This enables improved performance, reproducibility, and efficiency. An experimental case study involving three nominally identical 3D printers (same brand, model, and close serial numbers) demonstrates reduced redundancy, lower resource usage, and improved reliability, along with improved convergence stability and solution quality through the selection of the appropriate optimization strategy based on the degree of inter-device noise heterogeneity. Overall, this framework establishes a general approach for precision- and resource-aware optimization in scalable, automated experimental platforms, demonstrated here on a representative multi-device 3D printing case study.

Schenk, Christina↗

Writing the Programs of Programmable Catalysis

It has long been known that non-steady state and periodic catalytic reactor operation in terms of temperature, pressure, and composition can lead to higher overall productivity and/or product selectivity than the best steady operation. Recently, the emergence of catalysts whose intrinsic properties can be made to oscillate with time, introduces advanced forcing capabilities that can be “programmed” into the catalysts to broaden the scope and applicability of periodic operation to surface chemistry. In this work, an algorithmic approach is implemented to significantly accelerate the discovery and optimization of periodic steady states of catalytic reactors. Decomposition of complex dynamics into fundamental mechanistic fast–slow steps is seen to improve conceptual understanding of the relationship between binding energy oscillation protocols and overall catalytic rates. Finding structured forcing protocols, optimally tailored to the multiple time scales of a given individual mechanism, requires an efficient search of high-dimensional parameter spaces. Here, this is enabled here through active learning (Bayesian optimization, enhanced by our proposed Bayesian continuation). Implementation of these methods is shown to accelerate the evaluation of catalyst programs by up to several orders of magnitude. Faster screening of programmable catalysts to discover periodic steady states enables the optimization of catalytic operating protocols and thus opens the possibility for catalyst engineering based on optimal forcing programs to control rate and product selectivity, even for complex multistep catalytic mechanisms.

catalysis↗

A Robust Estimate of Continental-Scale Terrestrial Carbon Sinks Using GOSAT XCO 2 Retrievals

Satellite XCO 2 retrievals could improve the estimates of surface carbon fluxes, but it remains unknown on what scales these estimates are robust. Here, we use the time-dependent Bayesian synthesis top-down method and prior net ecosystem exchanges (NEEs) from 12 terrestrial biosphere models (TBMs) to infer the monthly carbon fluxes of 51 land regions with constraints by GOSAT XCO 2 retrievals. We find that the uncertainty (standard deviation of 12 TBMs) reduction rates (uncertainty reduction rate (URR)) decrease significantly at decreasing spatial scales. On the continental-scale, the mean URR is about 57%, and the annual and seasonal cycle estimates of NEE are rather robust. The evaluation shows that the posterior CO 2 concentrations are significantly improved at the continental scale. Our study suggests that the GOSAT XCO 2 can only promise a robust continental-scale NEE estimate and improving the XCO 2 accuracy is an effective way to achieve robust estimates on smaller scales under current spatial coverage.

58 GEOSCIENCES↗

BeyondPlanck: XII. Cosmological parameter constraints with end-to-end error propagation

We present cosmological parameter constraints estimated using the Bayesian BEYONDPLANCK analysis framework. This method supports seamless end-to-end error propagation from raw time-ordered data onto final cosmological parameters. As a first demonstration of the method, we analyzed time-ordered Planck LFI observations, combined with selected external data (WMAP 33–61 GHz, Planck HFI DR4 353 and 857 GHz, and Haslam 408 MHz) in the form of pixelized maps that are used to break critical astrophysical degeneracies. Overall, all the results are generally in good agreement with previously reported values from Planck 2018 and WMAP, with the largest relative difference for any parameter amounting about 1σ when considering only temperature multipoles between 30 ≤ ℓ ≤ 600. In cases where there are differences, we note that the BEYONDPLANCK results are generally slightly closer to the high-ℓ HFI-dominated Planck 2018 results than previous analyses, suggesting slightly less tension between low and high multipoles. Using low-ℓ polarization information from LFI and WMAP, we find a best-fit value of τ = 0.066 ± 0.013, which is higher than the low value of τ = 0.052 ± 0.008 derived from Planck 2018 and slightly lower than the value of 0.069 ± 0.011 derived from the joint analysis of official LFI and WMAP products. Most importantly, however, we find that the uncertainty derived in the BEYONDPLANCK processing is about 30 % greater than when analyzing the official products, after taking into account the different sky coverage. We argue that this uncertainty is due to a marginalization over a more complete model of instrumental and astrophysical parameters, which results in more reliable and more rigorously defined uncertainties. We find that about 2000 Monte Carlo samples are required to achieve a robust convergence for a low-resolution cosmic microwave background (CMB) covariance matrix with 225 independent modes, and producing these samples takes about eight weeks on a modest computing cluster with 256 cores.

79 ASTRONOMY AND ASTROPHYSICS↗

Multipoint-BAX: a new approach for efficiently tuning particle accelerator emittance via virtual objectives

Abstract Although beam emittance is critical for the performance of high-brightness accelerators, optimization is often time limited as emittance calculations, commonly done via quadrupole scans, are typically slow. Such calculations are a type of multipoint query , i.e. each query requires multiple secondary measurements. Traditional black-box optimizers such as Bayesian optimization are slow and inefficient when dealing with such objectives as they must acquire the full series of measurements, but return only the emittance, with each query. We propose a new information-theoretic algorithm, Multipoint-BAX , for black-box optimization on multipoint queries, which queries and models individual beam-size measurements using techniques from Bayesian Algorithm Execution (BAX). Our method avoids the slow multipoint query on the accelerator by acquiring points through a virtual objective , i.e. calculating the emittance objective from a fast learned model rather than directly from the accelerator. We use Multipoint-BAX to minimize emittance at the Linac Coherent Light Source (LCLS) and the Facility for Advanced Accelerator Experimental Tests II (FACET-II). In simulation, our method is 20× faster and more robust to noise compared to existing methods. In live tests, it matched the hand-tuned emittance at FACET-II and achieved a 24% lower emittance than hand-tuning at LCLS. Our method represents a conceptual shift for optimizing multipoint queries, and we anticipate that it can be readily adapted to similar problems in particle accelerators and other scientific instruments.

43 PARTICLE ACCELERATORS↗

PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments

This article describes PyOED, a highly extensible scientific package that enables developing and testing model-constrained optimal experimental design (OED) for inverse problems. Specifically, PyOED aims to be a comprehensive Python toolkit for model-constrained OED. The package targets scientists and researchers interested in understanding the details of OED formulations and approaches. It is also meant to enable researchers to experiment with standard and innovative OED technologies with a wide range of test problems (e.g., simulation models). OED, inverse problems (e.g., Bayesian inversion), and data assimilation (DA) are closely related research fields, and their formulations overlap significantly. Thus, PyOED is continuously being expanded with a plethora of Bayesian inversion, DA, and OED methods as well as new scientific simulation models, observation error models, and observation operators. These pieces are added such that they can be permuted to enable testing OED methods in various settings of varying complexities. The PyOED core is completely written in Python and utilizes the inherent object-oriented capabilities; however, the current version of PyOED is meant to be extensible rather than scalable. Specifically, PyOED is developed to “enable rapid development and benchmarking of OED methods with minimal coding effort and to maximize code reutilization.” This article provides a brief description of the PyOED layout and philosophy and provides a set of exemplary test cases and tutorials to demonstrate the potential of the package.

97 MATHEMATICS AND COMPUTING↗

EGS Stimulation Design with Uncertainty Quantification at the EGS Collab Site

Engineering a robust hydraulic connection between wells is one of the most difficult aspects of enhanced geothermal systems (EGS). Designing and constructing such hydraulic connections requires and understanding of the in situ state of stress and the heterogeneities and discontinuities that naturally exist and may control the stimulation. Even with comprehensive stress and formation characterization programs substantial uncertainty remains in these key parameters. This is especially the case in high-temperature EGS environments where drilling conditions are often difficult and a far fewer logging and testing options are available. This paper presents a new approach for explicitly quantifying the uncertainties in the state of stress using a Bayesian Markov Chain Monte Carlo method. This approach produces a probability distribution for the stress tensor, including a general 3D orientation, that reflects the uncertainties in all the observations or indicators used to constrain the stress state. This method is demonstrated on the characterization data for the EGS Collab Experiment 2 site. The output of the analysis is used to guide the design of the planned stimulations. In the case of research projects like EGS Collab, explicitly quantifying the uncertainties in the stress state allow for more rigorous hypothesis testing by allowing conclusions drawn from the experiments to be interpreted in the context of the uncertain knowledge about conditions in the test bed.

Burghardt, Jeffrey A.↗

Reliability-based econometrics of aerospace structural systems: Design criteria and test options

The design criteria and test options for aerospace structural reliability were investigated. A decision methodology was developed for selecting a combination of structural tests and structural design factors. The decision method involves the use of Bayesian statistics and statistical decision theory. Procedures are discussed for obtaining and updating data-based probabilistic strength distributions for aerospace structures when test information is available and for obtaining subjective distributions when data are not available. The techniques used in developing the distributions are explained.

Thomas, J. M.↗

Real-time sensor data validation

This report describes the status of an on-going effort to develop software capable of detecting sensor failures on rocket engines in real time. This software could be used in a rocket engine controller to prevent the erroneous shutdown of an engine due to sensor failures which would otherwise be interpreted as engine failures by the control software. The approach taken combines analytical redundancy with Bayesian belief networks to provide a solution which has well defined real-time characteristics and well-defined error rates. Analytical redundancy is a technique in which a sensor's value is predicted by using values from other sensors and known or empirically derived mathematical relations. A set of sensors and a set of relations among them form a network of cross-checks which can be used to periodically validate all of the sensors in the network. Bayesian belief networks provide a method of determining if each of the sensors in the network is valid, given the results of the cross-checks. This approach has been successfully demonstrated on the Technology Test Bed Engine at the NASA Marshall Space Flight Center. Current efforts are focused on extending the system to provide a validation capability for 100 sensors on the Space Shuttle Main Engine.

Bickmore, Timothy W.↗

Utilization of Hydrologic Remote Sensing Data in Land Surface Modeling and Data Assimilation: Current Status and Challenges

Recent advances in remote sensing technologies have enabled the monitoring and measurement of the Earth's land surface at an unprecedented scale and frequency. The myriad of these land surface observations must be integrated with the state-of-the-art land surface model forecasts using data assimilation to generate spatially and temporally coherent estimates of environmental conditions. These analyses are of critical importance to real-world applications such as agricultural production, water resources management and flood, drought, weather and climate prediction. This need motivated the development of NASA Land Information System (LIS), which is an expert system encapsulating a suite of modeling, computational and data assimilation tools required to address challenging hydrological problems. LIS integrates the use of several community land surface models, use of ground and satellite based observations, data assimilation and uncertainty estimation techniques and high performance computing and data management tools to enable the assessment and prediction of hydrologic conditions at various spatial and temporal scales of interest. This presentation will focus on describing the results, challenges and lessons learned from the use of remote sensing data for improving land surface modeling, within LIS. More specifically, studies related to the improved estimation of soil moisture, snow and land surface temperature conditions through data assimilation will be discussed. The presentation will also address the characterization of uncertainty in the modeling process through Bayesian remote sensing and computational methods.

Kumar, Sujay V.↗