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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Nek5000 developments in support of industry and the NRC

This year, the Nuclear Energy Advanced Modeling Simulation program (NEAMS) thermal-hydraulics verification and validation (V&V) work has focused in three areas of Nek5000 V&V-driven development. First, in a close collaborative effort with the U. S. Nuclear Regulatory Commission (NRC) staff, we have continued V&V efforts for the HYMERES-2 project using the OECD/NEA sponsored testing in the PSI PANDA facility. This year’s focus of ANL-NRC collaboration involves Nek5000 setups and validation for a range of problems relevant to and including the HYMERES-2 benchmark from PSI. The primary outcome of this year efforts is a more efficient geometry and inlet modeling simplification after a careful sensitivity study of the inlet profiles and pipe geometries. The resulting modeling choice of a short recycling/fully-developed turbulent inlet is within the experimental uncertainty estimate. This finding simplifies the next step of the cross-V&V HYMERES-2 project. In addition, the ANL team continue to provide assistance to the NRC staff in the form of Nek5000 application support in general and on the use of the HPC platforms of ALCF and INL in particular. This supports the NRC’s assessment of Nek5000 for use with the NRC Blue CRAB code suite. Second, we have implemented and tested more robust model of URANS, namely the k – τ model, a variant of the k-ω model, along with other improvements to RANS Nek5000 modeling in general. Because of its demonstrated robustness and stability, the k – τ model is the only RANS model that has been implemented in the new GPU version of the Nek5000 code, nekRS. Lastly, we report the initial implementation of Jacobian-free Newton Krylov approach to the direct Newton method for steady fluid solvers aimed at acceleration of RANS modeling and at IC improvement for LES campaigns. Also leveraging the Exascale Computing Project (ECP) ANL/CEED & SMR team’s software development effort to support NEAMS problems at large scale of the advanced computing architectures, NekRS, a GPU variant of Nek5000, built on top of kernels from libParanumal using OCCA for portability, has been successfully run on the full system of Summit (4608 nodes, 27648 GPUs).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Modeling Variable Renewable Energy and Storage in the Power Sector

The emergence of variable renewable energy and battery storage technologies have fundamentally transformed the electric power sector and generated demand for analysis to understand their roles in future energy systems. Although unique characteristics of these resources are well-recognized and require more sophisticated methodologies to capture effectively, guidance is limited on best practices and research gaps. This paper selectively reviews recent literature and draws upon our collective modeling experience to offer recommendations to analysts and consumers of model outputs on approaches for modeling variable renewable energy and storage in long-term electric sector models. We focus on regional- and national-scale models with technological, temporal, and spatial detail given their prevalence in planning and policy analysis, though insights are applicable in other settings. The review highlights how the research frontier has advanced in representing renewables and energy storage over the past decade; however, given the many considerations involved with appropriately capturing salient economic and operational characteristics of renewables, there is a gap between commonly used models and state-of-the-art methods. Model simplifications can materially impact policy analysis associated with power sector decarbonization and high renewables deployment, and improving model representations of variable renewables can enhance insights for policymakers and other stakeholders. This review can point the way for improved methods for established models and designs for emerging ones.

energy and environmental policy↗
NASA NTRS2001

A Simplified Model of ARIS for Optimal Controller Design

Many space-science experiments require active vibration isolation. Boeing's Active Rack Isolation System (ARIS) isolates experiments at the rack (vs. experiment or sub-experiment) level, with multi e experiments per rack. An ARIS-isolated rack typically employs eight actuators and thirteen umbilicals; the umbilicals provide services such as power, data transmission, and cooling. Hampton, et al., used "Kane's method" to develop an analytical, nonlinear, rigid-body model of ARIS that includes full actuator dynamics (inertias). This model, less the umbilicals, was first implemented for simulation by Beech and Hampton; they developed and tested their model using two commercial-off-the-shelf (COTS) software packages. Rupert, et al., added umbilical-transmitted disturbances to this nonlinear model. Because the nonlinear model, even for the untethered system, is both exceedingly complex and "encapsulated" inside these COTS tools, it is largely inaccessible to ARIS controller designers. This paper shows that ISPR rattle-space constraints and small ARIS actuator masses permit considerable model simplification, without significant loss of fidelity. First, for various loading conditions, comparisons are made between the dynamic responses of the nonlinear model (untethered) and a truth model. Then comparisons are made among nonlinear, linearized, and linearized reduced-mass models. It is concluded that these three models all capture the significant system rigid-body dynamics, with the third being preferred due to its relative simplicity.

Beech, Geoffrey S.↗
NASA NTRS1992

Application of engineering codes to the assessment of spacecraft charging induced hazards

One of the major hurdles to developing a better understanding of the environmentally induced interactions in spacecraft systems (spacecraft charging) is the lack of simple tools to analyze the behavior of the spacecraft in a sub-storm environment. The available tool, NASA Charging Analyzer Program (NASCAP), is expensive to use and treats only half of the problem - surface effects only. In response to this need, we have developed, under our IRAD Program, a set of engineering tools that operate on personal computers and addresses both the surface and bulk charging aspects of the interactions. These engineering codes utilize circuit analysis techniques and model simplifications to allow rapid computations. The codes were compared to both NASCAP predictions and to bulk charging data with excellent agreement for both phenomena. The models used in our engineering codes are shown, and in the engineering codes, the models are defined in terms of plates or triangles in an unconfined space. The predictions are based upon current balances to each plate. The code has routines to allow interactions between the plates to simulate 'barrier' effects. The agreement for the predicted differential surface voltages for the model is given. In conducting the comparison of engineering codes to NASCAP predictions, it was found that some inconsistencies existed in the NASCAP code that are not obvious to the user. The use of the grid imposes constraints that voltage gradients exist between the grid points. This allows the existence of voltage gradients in conductors and influences surface resistivity predictions. These points are discussed along with the predictions for treating these inconsistencies properly. The model predictions for a variable intensity substorm is also presented.

Stevens, N. J.↗
NASA NTRS1993

Ice Accretion Prediction for a Typical Commercial Transport Aircraft

Ice accretion calculations were made for a modern commercial transport using the NASA Lewis LEWICE3D ice accretion code. The ice accretion calculations were made for the wing and horizontal tail using both isolated flow models and flow models incorporating the entire airplane. The isolated flow model calculations were made to assess the validity of using these simplified models in lieu of the entire model in the ice accretion analysis of full aircraft. Ice shapes typifying a rime and a mixed ice shape were generated for a 30 minute hold condition. In general, the calculated ice shapes looked reasonable and appeared representative of a rime and a mixed ice conditions. The isolated flow model simplification was good for the main wing except at the root where it overpredicted the amount of accreted ice relative to the full aircraft flow model. For the horizontal tail the size and amount of predicted ice compared well for the two flow models, but the position of the accretions were more towards the upper surface for the aircraft flow model relative to the isolated flow model. This was attributed to downwash from the main wing which resulted in a lower effective angle of attack for the aircraft tail.

Bidwell, C. S.↗
NASA NTRS1993

Ice accretion prediction for a typical commercial transport aircraft

Ice accretion calculations were made for a modern commercial transport using the NASA Lewis LEWICE3D ice accretion code. The ice accretion calculations were made for the wing and horizonal tail using both isolated flow models and flow models incorporating the entire airplane. The isolated flow model calculations were made to assess the validity of using these simplified models in lieu of the entire model in the ice accretion analysis for full aircraft. Ice shapes typifying a rime and a mixed ice shape were generated for a 30 minute hold condition. In general, the calculated ice shapes looked reasonable and appeared representative of a rime and a mixed ice conditions. The isolated flow model simplification was good for the main wing except at the root where it overpredicted the amount of accreted ice relative to the full aircraft flow model. For the horizontal tail the size and amount of predicted ice compared well for the two flow models, but the position of the accretions were more towards the upper surface for the aircraft flow model relative to the isolated flow model. This was attributed to downwash from the main wing which resulted in a lower effective angle of attack for the aircraft tail.

Bidwell, C. S.↗
DOE OSTI2023

Urban Flood Modeling: Uncertainty Quantification and Physics‐Informed Gaussian Processes Regression Forecasting

Abstract Estimating uncertainty in flood model predictions is important for many applications, including risk assessment and flood forecasting. We focus on uncertainty in physics‐based urban flooding models. We consider the effects of the model's complexity and uncertainty in key input parameters. The effect of rainfall intensity on the uncertainty in water depth predictions is also studied. As a test study, we choose the Interconnected Channel and Pond Routing (ICPR) model of a part of the city of Minneapolis. The uncertainty in the ICPR model's predictions of the floodwater depth is quantified in terms of the ensemble variance using the multilevel Monte Carlo (MC) simulation method. Our results show that uncertainties in the studied domain are highly localized. Model simplifications, such as disregarding the groundwater flow, lead to overly confident predictions, that is, predictions that are both less accurate and uncertain than those of the more complex model. We find that for the same number of uncertain parameters, increasing the model resolution reduces uncertainty in the model predictions (and increases the MC method's computational cost). We employ the multilevel MC method to reduce the cost of estimating uncertainty in a high‐resolution ICPR model. Finally, we use the ensemble estimates of the mean and covariance of the flood depth for real‐time flood depth forecasting using the physics‐informed Gaussian process regression method. We show that even with few measurements, the proposed framework results in a more accurate forecast than that provided by the mean prediction of the ICPR model.

Kohanpur, Amir H.↗
DOE OSTI2024

DECOVALEX-2023: Task F1 Final Report

DECOVALEX-2023 Task F is a comparison of models and methods for post-closure performance assessment (PA) of a deep geologic repository for radioactive waste. The general aims of Task F are to build confidence in the models, methods, and software used for PA and to stimulate additional research and development in PA methodologies. The task objectives are to motivate development of PA modelling skills and capabilities, to examine the influence of model choices on calculated repository performance, and to compare the uncertainties introduced by model choices to other sources of uncertainty. Task F involves no actual experiment or site. It is a PA modelling exercise that requires the conceptual development of hypothetical repository designs and geologic settings. Because three of the teams were interested in salt and the rest of the teams were interested in crystalline rock, Task F was split into two branches: Task F1 for crystalline rock and Task F2 for salt. This report is for Task F1, crystalline rock. Teams from seven countries (Canada, Czech Republic, Germany, Korea, Sweden, Taiwan, and United States) participated in Task F1. The teams worked together to define the features, events, and processes of the reference case repository and established a set of performance measures. In addition, they defined a set of benchmark problems designed to test and compare modelling capabilities for fracture flow and transport at different scales. The repository design and benchmark problems are documented in a Task Specification that evolved over time as the group honed the specifications. The benchmark problems verified that each team can aptly model flow and transport in fractured media in 1-, 2-, and 3-dimensions. Two general approaches were used for the 3-dimensional benchmarks: discrete fracture network (DFN) and equivalent continuous porous medium (ECPM). DFN modelling involves explicit meshing of each fracture while ECPM modelling aims to capture the effective porosity and directional permeability of each cell in a space-filling mesh as affected by intersecting fractures. In some models, a combination of the two is used, i.e., DFN for large known fractures and ECPM for the rest of the domain. Transport is solved by using either the advection-dispersion equation or particle tracking. Although some variation is observed among model breakthrough curves in the benchmark problems, there is strong agreement in breakthrough behaviour up to at least the 75 th percentile for all benchmarks. At the 90 th percentile, breakthrough results show larger differences, suggesting several models retain substantially higher fractions of tracer in regions of slower moving water. In addition to the flow and transport benchmarks, several teams completed the source term benchmark, verifying capabilities for modelling radionuclide decay and ingrowth, waste package breach, instant release fractions, fuel matrix degradation rates, and radionuclide solubility limitations. The reference case is conceptualized as a generic spent fuel repository at a depth of 450 m in fractured crystalline rock. The repository has 50 parallel backfilled drifts, each with 50 deposition holes 6 m apart. Each deposition hole contains a 4-PWR waste package and bentonite buffer. The rock domain is 5 km in length, 2 km in width, and 1 km in depth. It has 6 deterministic fractured deformation zones and a multitude of stochastic fractures. Teams generally used the ECPM approach for the entire rock or a hybrid approach in which the deterministic fracture zones are modelled with a DFN and the rest of the rock is modelled by ECPM. Of the reference case problems specified, only the results of the initial reference case problem are compared in this report. The initial problem focuses on transport from the deposition holes to the surface, i.e., it neglects waste package performance. Tracers are released at all waste package locations at time zero and tracked for their releases to the near field and ground surface. The water fluxes calculated at the ground surface entry and exit regions of the domain are similar for all models except for two that have considerably lower fluxes. For tracer transport, large differences are observed among models in the magnitude of tracer transported. Much of the difference appears to be due to how the repository is implemented and hence the different degrees of repository simplification. Models that exclude the drifts, buffer, and backfill from the domain tend to show greater release of tracers and radionuclides from the repository. The initial study presented here indicates that major differences in modelling important processes within the repository (e.g., diffusion through buffer and backfill) can produce broadly different release and transport results, especially when those processes are excluded. Even for the models that included all specified features, events, and processes, the results show significant differences and demonstrate the importance of examining multiple modelling approaches in performance assessment. The differences in results observed in this study are expected to motivate teams to either increase complexity in future versions of the reference case models or to improve methods to account for the effects of simplified features and processes. Either way, future improvements in these models are expected to produce results that more closely agree.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2023

Development of Building Design Optimization Methodology: Residential Building Applications

Building design optimization is a highly complex problem, requiring long computational running processes because of the many options that exist when a building is being designed. This paper introduces an integrated approach through which to perform this optimization within an acceptable time frame. The approach includes the methods of variable selection, model simplification, and a sequential optimization process. Using singular value decomposition, a large number of design variables is reduced to a smaller subset that can be solved more quickly through the optimization algorithm. To expedite the variable selection process, a modeling approach that quickly simulates annual energy consumption was developed to replace full annual energy simulations. The developed methodology was applied to two residential buildings in the US, and the results are discussed herein. To assess the accuracy of the integrated optimization methodology, the optimized life cycle costs are compaa variables demonstrating the strongest contributions in the optimization study were identified. The proposed methodology significantly shortened the time requirements for the optimization processes of the two case studies by 74% and 84%; the optimized life cycle costs were within 0.05% and 0.06%, respectively, of the optimum point.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2024

The MUSIC Critical Benchmark and Nuclear Data

The Measurement of Uranium Subcritical and Critical (MUSIC) experiment was a series of measurements of critical and subcritical configurations of bare highly enriched uranium. The goal was to compare measurement methods, analysis techniques, and simulation methods across regimes of criticality and to provide high-quality validation of 235 U nuclear data. A benchmark evaluation of the two critical configurations of the MUSIC experiments will soon be published in the release of the International Criticality Safety Benchmark Evaluation Project Handbook. The recent execution of the experiment aids in proper quantification of model simplifications and all uncertainties associated with the experiment. Historical benchmark evaluations are heavily relied on for uranium nuclear data validation despite the fact that the same level of documentation and comparable uncertainty analysis may not be present. The MUSIC evaluation is less likely to include “unknown unknowns” that could impede accurately modeling the system. Presented are both highly detailed and very simplified models, which represent the experimental configurations accurately, aiding the users of the benchmark for nuclear data or transport code validation. The sensitivities of k eff to nuclear data and nuclear data–related uncertainties are very similar between this experiment and previous bare uranium sphere experiments. In addition, the nuclear data uncertainties to any nuclides other than 235 U are small. For all these reasons, the recently evaluated MUSIC benchmark critical configurations could prove very useful for 235 U nuclear data validation. Currently, major libraries have good agreement with the experimental results, within 200 pcm for all nuclear data libraries, and within one standard deviation of the experimental result for most. Suggested nuclear data adjustments based on MUSIC and Lady Godiva are also presented, with posterior improvements to both the agreement in k eff and the uncertainty associated with the nuclear data.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

The Aerosol Model Benchmarking Repository: A toolkit for model intercomparison

The Aerosol Model Benchmarking Repository and Standards (AMBRS) project was initiated to provide tools and to establish community standards for benchmarking aerosol models. This report describes a set of open-source tools for building, running, and analyzing aerosol box model simulations in a standardized framework. The framework consists of three core components: AMBuilder, a CMake-based build system that compiles supported models consistently; AMBRS, a Python module that defines unified numerical experiments and executes them with aligned inputs; and PyParticle, an aerosol analysis package that standardizes output, computes diagnostics, and visualizes simulation results. Together, these tools enable reproducible intercomparison of aerosol schemes and support process-level evaluation of how model simplifications affect predictions of size distributions, cloud condensation nuclei activity, and other relevant properties relevant for the Earth-Energy system. Beyond its role in benchmarking, AMBRS provides a platform for studying aerosol processes across scales and can be used to generate training data for AI/ML applications in support of a broader hierarchical aerosol modeling strategy.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1976

Vibroacoustic test plan evaluation: Parameter variation study

Statistical decision models are shown to provide a viable method of evaluating the cost effectiveness of alternate vibroacoustic test plans and the associated test levels. The methodology developed provides a major step toward the development of a realistic tool to quantitatively tailor test programs to specific payloads. Testing is considered at the no test, component, subassembly, or system level of assembly. Component redundancy and partial loss of flight data are considered. Most and probabilistic costs are considered, and incipient failures resulting from ground tests are treated. Optimums defining both component and assembly test levels are indicated for the modified test plans considered. modeling simplifications must be considered in interpreting the results relative to a particular payload. New parameters introduced were a no test option, flight by flight failure probabilities, and a cost to design components for higher vibration requirements. Parameters varied were the shuttle payload bay internal acoustic environment, the STS launch cost, the component retest/repair cost, and the amount of redundancy in the housekeeping section of the payload reliability model.

Stahle, C. V.↗
DOE OSTI2021

Recovering the effects of subgrid heterogeneity in simulations of radionuclide transport through fractured media

Groundwater flow and contaminant transport through fractured media can be simulated using Discrete Fracture Network (DFN) models which provide a natural description of structural heterogeneity. However, this approach is computationally expensive, with the large number of intersecting fractures necessitated by many real-world applications requiring modeling simplifications to be made for calculations to be tractable. Upscaling methods commonly used for this purpose can result in some loss of local-scale variability in the groundwater flow velocity field, resulting in underestimation of particle travel times, transport resistance and retention in transport calculations. In this paper, a transport downscaling algorithm to recover the transport effects of heterogeneity is tested on a synthetic Brittle Fault Zone model, motivated by the problem of large safety assessment calculations for geological repositories of spent nuclear fuel. We show that the variability in the local-scale velocity field which is lost by upscaling can be recovered by sampling from a library of DFN transport paths, accurately reproducing DFN transport statistic distributions and radionuclide breakthrough curves in an upscaled model.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Dynamic modelling of slip in a wind turbine spherical roller main bearing

This paper considers the problem of the dynamic modelling of macro slip in spherical roller bearings. By revisiting the fundamental physics which drive these systems, potential issues in existing models have been identified. Furthermore, in pure rolling conditions it was found that governing differential equations become "stiff", requiring the use of implicit methods of time integration. The problem of individual roller macro slip in a wind turbine main bearing is then investigated using a simplified representation of system dynamics. Model results indicate clear links between slip/friction and the operational strategy of the wind turbine, as well as significantly higher frictional effects in the downwind main bearing row. Due to modelling simplifications, these results should not yet be considered conclusive, with further work required.

17 WIND ENERGY↗
DOE OSTI2021

Modeling of Am-241 as an alternative fuel source in a radioisotope thermoelectric generator

Radioisotope thermoelectric generators (RTGs) have been utilized in the USA to power satellites and space exploration equipment for over half a century. In our preceding paper, Dustin and Borrelli (2021), we assessed the performance of 241 Am as an alternative to 238 Pu by modeling the specific decay-heat and gamma spectra of several potential radioisotopes in SCALE-ORIGEN, and comparing these with 238 Pu An analysis of heat output, heat stability, safety, availability and required mass showed that of the isotopes reviewed, 241 Am had the greatest potential for replacing 238 Pu, with additional design considerations. In our current work, we have assessed radiation transport for two configurations of an 241 Am powered Multi-Mission Radioisotope Thermoelectric Generator (MMRTG), due to the potential health effects for operators, fabricators, and others with the potential to be exposed to neutrons and lower energy gamma rays. We have modeled the use of americium as a fuel source replacement for plutonium in current MMRTG designs. The assessment has been performed using MCNP6, in terms of neutron and gamma emissions that might interfere with sensitive equipment and power system production. The radiation transport behavior in terms of dose rate of the 241 Am fueled models is compared with that of a 238 Pu fueled model. Finally, additional work was performed to validate modeling simplifications with regards to thermocouple material configurations.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

Voltage positioning using co-optimization of controllable grid assets in radial networks

With increasing penetration of solar PV, some distribution feeders are experiencing highly variable net-load flows and even reverse flows. To optimize distribution systems under such conditions, the scheduling of mechanical devices, such as OLTCs and capacitor banks, needs to take into account forecasted solar PV and actual grid conditions. However, these legacy switching assets are operated on a daily or hourly timescale, due to the wear and tear associated with mechanical switching, which makes them unsuitable for real-time control. Therefore, there is a natural timescale- separation between these slower mechanical assets and the responsive nature of inverter-based resources. In this paper, we present a network admissible convex formulation for holistically scheduling controllable grid assets to position voltage optimally against solar PV. An optimal hourly schedule is presented that utilizes mechanical resources to position the predicted voltages close to nominal values, while minimizing the use of inverter-based resources (i.e., DERs), making them available for control at a faster time-scale (after the uncertainty reveals itself). A convex, inner approximation of the OPF problem is adapted to a mixed-integer linear program that minimizes voltage deviations from nominal (i.e., maximizes voltage margins). Here, the resulting OPF solution respects all the network constraints and is, hence, robust against modeling simplifications. Simulation based analysis on IEEE distribution feeders validates the approach.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Projecting the Thermal Response in a HTGR-Type System during Conduction Cooldown Using Graph-Laplacian Based Machine Learning

Accurate prediction of an off-normal event in a nuclear reactor is dependent upon the availability of sensory data, reactor core physical condition, and understanding of the underlying phenomenon. This work presents a method to project the data from some discrete sensory locations to the overall reactor domain during conduction cooldown scenarios similar to High Temperature Gas-cooled Reactors (HTGRs). The existing models for conductive cooldown in a heterogeneous multi-body system, such as an assembly of prismatic blocks or pebble beds relies on knowledge of the thermal contact conductance, requiring significant knowledge of local thermal contacts and heat transport possibilities across those contacts. With a priori knowledge of bulk geometry features and some discrete sensors, a machine learning approach was devised. The presented work uses an experimental facility to mimic conduction cooldown with an assembly of 68 cylindrical rods initially heated to 1200 K. High-fidelity temperature data were collected using an infrared (IR) camera to provide training data to the model and validate the predicted temperature data. The machine learning approach used here first converts the macroscopic bulk geometry information into Graph-Laplacian, and then uses the eigenvectors of the Graph-Laplacian to develop Kernel functions. Support vector regression (SVR) was implemented on the obtained Kernels and used to predict the thermal response in a packed rod assembly during a conduction cooldown experiment. The usage of SVR modeling differs from most models today because of its representation of thermal coupling between rods in the core. When trained with thermographic data, the average normalized error is less than 2% over 400 s, during which temperatures of the assembly have dropped by more than 500 K. The rod temperature prediction performance was significantly better for rods in the interior of the assembly compared to those near the exterior, likely due to the model simplification of the surroundings.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
NASA NTRS1977

Evaluation of algorithms for geological thermal-inertia mapping

The errors incurred in producing a thermal inertia map are of three general types: measurement, analysis, and model simplification. To emphasize the geophysical relevance of these errors, they were expressed in terms of uncertainty in thermal inertia and compared with the thermal inertia values of geologic materials. Thus the applications and practical limitations of the technique were illustrated. All errors were calculated using the parameter values appropriate to a site at the Raft River, Id. Although these error values serve to illustrate the magnitudes that can be expected from the three general types of errors, extrapolation to other sites should be done using parameter values particular to the area. Three surface temperature algorithms were evaluated: linear Fourier series, finite difference, and Laplace transform. In terms of resulting errors in thermal inertia, the Laplace transform method is the most accurate (260 TIU), the forward finite difference method is intermediate (300 TIU), and the linear Fourier series method the least accurate (460 TIU).

Miller, S. H.↗