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DOE OSTI2024

Dynamic Modeling of Power Conversion Stages for an Exascale Supercomputer

In this paper a power conversion and energy consumption model for an exascale supercomputer is investigated. Power consumption, energy loss and efficiency are derived for the 27.2 MW liquid-cooled, centralized, High Performance Computing (HPC) power system, which is supplied directly from the 480 V three-phase mains. Two energy conversion stages are analyzed, measured and modeled. The model is developed in order to be adapted and implemented in a digital twin platform utilizing a Resource Allocator and Power Simulator (RAPS) module. RAPS enables estimation of potential energy savings in the direct AC power supply architecture via both conventional rectifier load sharing (commonly used in HPC systems), as well as smart rectifier load sharing. Moreover, besides the direct AC supply architecture analysis, the full direct DC supply architecture with with 1 kV DC bus were also studied. Comparison of 10 hour time frame operation of the system, with direct 480 V AC voltage supply with conventional and smart load sharing and medium dc voltage supply were done. For the direct AC supply architecture, with conventional and smart load sharing the predicted power loss was approximately 840 kW and 820 kW, respectively and the predicted total system efficiency was 92.87% and 93.05%, respectively. For the direct DC supply architecture with the 1000 V DC supply bus power loss was approximately 340 kW and the predicted total system efficiency was 97.02%.

Wojda, Rafal↗

DOE OSTI2025

Hidden-strangeness tetraquarks in the dynamical diquark model

The dynamical diquark model describes multiquark exotic hadrons in terms of diquark components nucleated by heavy quarks and successfully explains multiple features of hidden-charm and hidden-bottom exotics. Here, we apply the model to the marginally heavy case of hidden-strange states to probe whether mesons near 2 GeV with peculiar properties, such as 𝜙⁡(2170), 𝑓 2 ⁡(2340), and 𝑋⁡(2370), are possible tetraquark candidates. We calculate spin-multiplet average masses using potentials obtained through lattice simulations and quark models, and we also describe the detailed spectra of the expected multiplets as a diagnostic to discern the nature of future hadrons likely to be discovered in this mass region by experiments at facilities such as BESIII, JLab, and the EIC.

Bound states↗

DOE OSTI2026

Applying Particle Swarm Optimization and Extended Kalman Filtering to Model Kaplan Generation Dynamics for Hydropower Systems

Variable renewable generation is increasing the need for hydropower plants to provide fast and flexible grid support, which places new demands on plant-level dynamic models used for monitoring, control, and operational decision-making. This need is especially important for hydroelectric systems, where turbine and generator dynamics are strongly coupled, nonlinear, and time-varying, making accurate real-time representation difficult. To address this problem, this paper develops a digital twin (DT) framework for a synchronous generator–Kaplan turbine system using an explicit separation of slow turbine dynamics and fast generator dynamics. The turbine subsystem is represented by a six-coefficient model, whose parameters are identified offline using particle swarm optimization, while the generator subsystem is updated online through an extended Kalman filter for real-time state and parameter estimation. These models are integrated within a closed-loop simulation that includes a proportional–integral–derivative–double-derivative governor and excitation system, allowing the DT to track plant behavior under realistic operating conditions. Unlike prior studies that treat turbine and generator modeling separately or rely mainly on simulated inputs, the proposed framework is validated using real operational data from a hydropower plant. Results show that the DT reproduces terminal voltage, active power, and reactive power with a normalized root mean square error of approximately 5%. This hybrid offline–online formulation constitutes the main contribution of the work, providing an adaptive and practically deployable DT for hydropower systems with direct relevance to control improvement, performance monitoring, and grid-support applications under high renewable penetration.

13 HYDRO ENERGY↗

DOE OSTI2025

Moving beyond post hoc explainable artificial intelligence: a perspective paper on lessons learned from dynamical climate modeling

AI models are criticized as being black boxes, potentially subjecting climate science to greater uncertainty. Explainable artificial intelligence (XAI) has been proposed to probe AI models and increase trust. In this review and perspective paper, we suggest that, in addition to using XAI methods, AI researchers in climate science can learn from past successes in the development of physics-based dynamical climate models. Dynamical models are complex but have gained trust because their successes and failures can sometimes be attributed to specific components or sub-models, such as when model bias is explained by pointing to a particular parameterization. We propose three types of understanding as a basis to evaluate trust in dynamical and AI models alike: (1) instrumental understanding, which is obtained when a model has passed a functional test; (2) statistical understanding, obtained when researchers can make sense of the modeling results using statistical techniques to identify input–output relationships; and (3) component-level understanding, which refers to modelers' ability to point to specific model components or parts in the model architecture as the culprit for erratic model behaviors or as the crucial reason why the model functions well. We demonstrate how component-level understanding has been sought and achieved via climate model intercomparison projects over the past several decades. Such component-level understanding routinely leads to model improvements and may also serve as a template for thinking about AI-driven climate science. Currently, XAI methods can help explain the behaviors of AI models by focusing on the mapping between input and output, thereby increasing the statistical understanding of AI models. Yet, to further increase our understanding of AI models, we will have to build AI models that have interpretable components amenable to component-level understanding. We give recent examples from the AI climate science literature to highlight some recent, albeit limited, successes in achieving component-level understanding and thereby explaining model behavior. The merit of such interpretable AI models is that they serve as a stronger basis for trust in climate modeling and, by extension, downstream uses of climate model data.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

ABSTRACT Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase‐field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second‐order phase‐field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth‐order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high‐order phase‐field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high‐order phase‐field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use ‐conforming virtual elements and the generalized‐ time integration method for the momentum balance equation, and adopt ‐conforming virtual elements for the high‐order phase‐field equation. We verify our virtual element framework using classical quasi‐static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.

42 ENGINEERING↗

DOE OSTI2025

The Water Table Model (WTM) (v2.0.1): coupled groundwater and dynamic lake modelling

Abstract. Ice-free land comprises 26 % of the Earth's surface and holds liquid water that delineates ecosystems, affects global geochemical cycling, and modulates sea levels. However, we currently lack the capacity to simulate and predict these terrestrial water changes across the full range of relevant spatial (watershed to global) and temporal (monthly to millennial) scales. To address this knowledge gap, we present the Water Table Model (WTM), which integrates coupled components to compute dynamic lake and groundwater levels. The groundwater component solves the 2D horizontal groundwater flow equation using non-linear equation solvers from the C++ PETSc (Portable, Extensible Toolkit for Scientific Computation) library. The dynamic lake component makes use of the Fill–Spill–Merge (FSM) algorithm to move surface water into lakes, where it may evaporate or affect groundwater flow. In a proof-of-concept application, we demonstrate the continental-scale capabilities of the WTM by simulating the steady-state climate-driven water table for the present day and the Last Glacial Maximum (LGM; 21 000 calendar years before present) across the North American continent. During the LGM, North America stored an additional 14.98 cm of sea-level equivalent (SLE) in lakes and groundwater compared to the climate-driven present-day scenario. We compare the present-day result to other simulations and real-world data. Open-source code for the WTM is available on GitHub and Zenodo.

Callaghan, Kerry L. (ORCID:0000000226740838)↗

DOE OSTI2024

Potential Adoption and Benefits of Co-Optimized Multimode Engines and Fuels for U.S. Light-Duty Vehicles

Exploring a diverse portfolio of technologies for decarbonization is crucial to understanding the potential impacts of different technological solutions and their associated environmental implications. Using high-octane, high-sensitivity biofuel blends in co-optimized multimode engines can increase engine efficiency and reduce vehicle emissions. Here, the multimode engine research focuses on the benefits of light-duty vehicle engines, which can operate in multiple modes depending on the vehicle's load. Low-temperature combustion can improve efficiency and reduce emissions (such as those from oxides of nitrogen and particulate matter) during low-load operation, while spark ignition performance is maintained in high-load operation. These advanced engines can be optimized to run on blends of biobased fuels. This analysis models scenarios for potential market adoption of co-optimized multimode vehicles fueled by three different bioblendstocks: ethanol, isopropanol, and isobutanol. An integrated modeling approach is used to forecast the energy and environmental impacts of the deployment of co-optimized multimode vehicles and fuels in the light-duty sector over the 2020-to-2050 time horizon. The multidisciplinary approach combines vehicle sales modeling, system dynamics modeling of the biorefining industry, and life cycle assessment to estimate the emissions and energy benefits. The models consider market forces such as consumer preferences for vehicle attributes, biofuel supply and demand dynamics subject to biorefinery capacity build-out and bioresource constraints, and forecasted changes to the U.S. bulk energy system over time. Market adoption of co-optimized vehicles is evaluated across a wide parameter space for incremental vehicle cost and engine efficiency improvement. This analysis reveals that the deployment of co-optimized multimode fuels and vehicles results in up to a 5% reduction in annual sector-wide life cycle greenhouse gas (GHG) emissions by 2050, relative to a business-as-usual scenario, but is also indicates environmental trade-offs, such as higher life cycle water-use. Emission benefits could potentially increase beyond 2050, as the new technologies penetrate the market and gain a foothold. Results also show that, under certain circumstances, vehicles with engines co-optimized for use with high-octane, high-sensitivity biofuel blends can be cost-competitive with conventional gasoline, while reducing GHG emissions. Our modeling results indicate that co-optimized multimode fuels and engines can be strategically leveraged in tandem with electrification to decarbonize the light-duty sector. Co-optimized vehicles could play a role in the early years of the time horizon, while electric vehicles (EVs) could become more competitive in the later years, highlighting the complementary benefits of these technologies for GHG reductions.

Oke, Doris↗

DOE OSTI2025

Particle Swarm Optimization of Dynamic Load Model Parameters in Large Systems

This paper considers two dynamic load models that are widely used in industry to account for induction motor behavior: CMLD and CLOD. These models must be parametrized for the specific utility system in a general way so that they can be used in planning studies and provide a conservative but realistic representation of load behavior. This study considers a measurement-based approach to tuning both models. The load modeling study compares the response of the tuned models to generic candidate models using historical events. This study considers one area-based subsystem to simplify the modeling approach and reduce the number of models required for simulations. Additionally, because dynamic load models often produce similar results for different sets of parameters, a sensitivity study was conducted to assess the parameter impacts on the voltage response. The sensitivity study covers the parameters that are tuned using event measurements. The process to estimate the parameters uses the particle-swarm optimization algorithm. Overall, the performance of the tuned model more accurately captures recovery voltage, delayed recovery, and settling voltage than its predecessor models while not being overly tuned so that it remains general for peak summer conditions.

dynamic load modeling↗

DOE OSTI2026

One Earth Energy Static and Dynamic Reservoir Modeling

This report presents the static and dynamic reservoir modeling conducted for the CarbonSAFE Phase III Illinois Storage Corridor project to assess the feasibility of commercial-scale CO 2 storage in the Mt. Simon Sandstone at the One Earth Energy (OEE) site in McLean County, Illinois. Three-dimensional geocellular models of the Mt. Simon storage complex were developed in Petrel ® by integrating petrophysical log data, core analyses, and seismic surveys from the OEE #1 stratigraphic test well and two nearby wells, with multiple model versions created as new data became available. Dynamic reservoir simulations, performed using Landmark's Nexus software, progressed through three phases (preliminary, sensitivity, and UIC Class VI permit studies) evaluating injection scenarios across varying rates, well orientations, permeability models, and multi-well configurations. Results demonstrate that commercial-scale storage is feasible: three injection wells spaced approximately one mile apart can store a total of 90 million tonnes of CO 2 over 20 years, producing a combined plume with an equivalent radius of 3.2 miles and a maximum pressure-front-defined Area of Review of 178 mi 2 at the end of injection that diminishes to 34 mi 2 after 50 years of post-injection monitoring. Sensitivity analyses indicate that a 20% change in porosity or permeability yields approximately a 7% change in AoR radius, and that perforating the high-permeability arkosic zone minimizes the pressure front compared to injection in the upper Mt. Simon Sandstone.

09 BIOMASS FUELS↗

DOE OSTI2024

Uncertainty-informed selection of CMIP6 Earth System Model subsets for use in multisectoral and impact models

Earth system models (ESMs) and general circulation models (GCMs) are heavily used to provide inputs to sectoral impact and multisector dynamic models, which include representations of energy, water, land, economics, and their interactions. Therefore, representing the full range of model uncertainty, scenario uncertainty, and interannual variability that ensembles of these models capture is critical to the exploration of the future co-evolution of the integrated human–Earth system. The pre-eminent source of these ensembles has been the Coupled Model Intercomparison Project (CMIP). With more modeling centers participating in each new CMIP phase, the size of the model archive is rapidly increasing, which can be intractable for impact modelers to effectively utilize due to computational constraints and the challenges of analyzing large datasets. In this work, we present a method to select a subset of the latest phase, CMIP6, featuring models for use as inputs to a sectoral impact or multisector dynamics models, while prioritizing preservation of the range of model uncertainty, scenario uncertainty, and interannual variability in the full CMIP6 ensemble results. This method is intended to help impact modelers select climate information from the CMIP archive efficiently for use in downstream models that require global coverage of climate information. This is particularly critical for large-ensemble experiments of multisector dynamic models that may be varying additional features beyond climate inputs in a factorial design, thus putting constraints on the number of climate simulations that can be used. We focus on temperature and precipitation outputs of CMIP6 models, as these are two of the most used variables among impact models, and many other key input variables for impacts are at least correlated with one or both of temperature and precipitation (e.g., relative humidity). Besides preserving the multi-model ensemble variance characteristics, we prioritize selecting CMIP6 models in the subset that preserve the very likely distribution of equilibrium climate sensitivity values as assessed by the latest Intergovernmental Panel on Climate Change (IPCC) report. This approach could be applied to other output variables of climate models and, possibly when combined with emulators, offers a flexible framework for designing more efficient experiments on human-relevant climate impacts. It can also provide greater insight into the properties of existing CMIP6 models.

Snyder, Abigail C.↗

DOE OSTI2024

Towards dynamical low-rank approximation for neutrino kinetic equations. Part I: Analysis of an idealized relaxation model

Dynamical low-rank approximation (DLRA) is an emerging tool for reducing computational costs and provides memory savings when solving high-dimensional problems. Here, in this work, we propose and analyze a semi-implicit dynamical low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous kinetic equation with a relaxation operator, modeling the emission and absorption of particles by a background medium. Both DLRA and the discontinuous Galerkin (DG) scheme can be formulated as Galerkin equations. To ensure their consistency, a weighted DLRA is introduced so that the resulting DLR-DG solution is a solution to the fully discrete DG scheme in a subspace of the standard DG solution space. Similar to the standard DG method, we show that the proposed DLR-DG method is well-posed. We also identify conditions such that the DLR-DG solution converges to the equilibrium. Numerical results are presented to demonstrate the theoretical findings.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

First principles free energy model with dynamic magnetism for δ -plutonium

We present an ab initio free energy model derived from a fully relativistic density functional theory (DFT) electronic structure with dynamic magnetism for δ -plutonium (face-centered cubic, fcc). The DFT model is extended with orbital-orbital interaction in a parameter free orbital polarization (OP) mechanism consistent with previous modeling of plutonium. Gibbs free energy is built from components associated with the temperature dependence of the electronic structure and the corresponding electronic entropy, lattice vibrations within an anharmonic lattice dynamics model, and dynamical fluctuations of the magnetization density, i.e. magnetic fluctuations. The fluctuation model consists of transverse and longitudinal modes driven by temperature induced excitations of the DFT + OP electronic structure. The ab initio model thus incorporates fluctuating states beyond the electronic ground state. Thanks to the dynamic magnetism, the theory predicts excellent thermodynamic properties and a Gibbs free energy in accord with CALPHAD and semi-empirical modeling developed from the thermodynamic observables. The magnetic fluctuations further explain anomalous behaviors of the thermal expansion in plutonium. Specifically, a thermal expansion for the δ -plutonium system turning from positive to negative at temperatures above room temperature, a tendency for gallium to reduce and remove the negative thermal expansion depending on composition, and a positive thermal expansion for the high temperature ϵ phase.

dynamic↗

DOE OSTI2024

Inverter Model Validation and Calibration Using Phasor Measurement Unit Data

As the penetration of inverter-based renewable energy resources increases in the power grid, especially at the distribution and microgrid levels, the need to accurately represent them in planning studies increases as well. However, due to the lack of well-established standard procedures, and vendor reluctance towards the detailed sharing of proprietary models, automated dynamic model validation and parameter calibration tools for inverter based resources (IBRs) remain scarce. This work presents a model validation and parameter calibration platform for representing IBRs with generic phasor-domain models. Phasor measurements of power system events are used for continuous validation using the data playback method, and model parameters are re-calibrated if a significant mismatch between measurements and model response is observed. Unique features of the proposed platform include- (a) an iterative Bayesian optimization approach towards parameter calibration to address a possible mismatch between the structures of generic models implemented in simulation softwares and actual commercial inverters, (b) error metrics designed to account for a possible mismatch between the time resolution of simulation and measurements, and (c) analysis of the measurement-simulation mismatch to provide guidance to engineering personnel regarding model shortcomings. The performance of the platform has been illustrated using both simulated data and field measurements to validate/calibrate inverter models in GridLAB-D.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2024

Experimentation in Exploring Photovoltaic Inverter Dynamics Under Different Irradiance Levels Through a Data-Driven Approach

As conventional direct connections of synchronous generators are being phased out, inverter-based resources (IBRs) with grid support functions are increasingly being integrated into power systems. This transition requires the development of accurate dynamic models for IBRs to predict how power systems will adapt to varying levels of IBRs penetration, establish grid code requirements, and ensure compliance. Here, this study introduces an active probing signal-based data-driven modeling technique to accurately derive the dynamics model of a smart photovoltaic inverter operating in Volt-Watt and Freq-Watt modes, in compliance with the IEEE 1547–2018 standard. The paper focuses on investigating how the dynamics of the PV inverter model respond to fluctuations in solar irradiance, utilizing real-time digital simulator experimentation. The experimental analysis demonstrates that the amplitude of dynamics fluctuates with changes in irradiance across both operational modes and confirms the active power’s dependence on irradiance levels. Furthermore, the nature of inverter dynamics varies distinctly between the different modes of activation. Critically, our findings indicate that dynamic models require DC-gain adjustments to accommodate contrasting irradiance levels, highlighting a negative gradient linear relationship between the DC-gain of each model and the irradiance.

14 SOLAR ENERGY↗

DOE OSTI2025

Development of Predictive Models for Advanced Reactor Autonomous Control

Advanced reactor designs including microreactors and small modular reactors will contribute to the clean production of cheap energy, and autonomous control for advanced reactors is an appealing option for reducing cost. However, there is a lack of industry experience applying autonomous control for advanced nuclear reactors. To accelerate the development and industry acceptance of autonomous control software for nuclear reactors, we aim to demonstrate autonomous control of the Purdue University research reactor (PUR-1) using INL-developed model predictive control (MPC) methods. To prepare for this demonstration, data-driven predictive models based on process data collected from PUR-1 have been developed and integrated with MPC and used to control a physics-based model of PUR-1. A data-driven dynamics model and a gated recurrent unit (GRU) network were both trained on process data from PUR-1. The dynamics model was shown to effectively control the reactor model with MPC when provided reactivity as a control variable but failed to control the model through the control rod positions. The GRU network produced more accurate predictions than the dynamics model when evaluated on operational data, and future work will include the evaluation of the GRU network in the controller.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2026

$\overline{\Sigma }^{\pm }$ production in $\text {pp}$ and $\text {p}{-}\text{Pb}$ collisions at $\sqrt{s_{\textrm{NN}}} = 5.02$ TeV with ALICE

The transverse momentum spectra and integrated yields of anti-$Σ$ hyperons ($\overline{\Sigma}^{\pm}$) have been measured in and collisions at $\sqrt{s_{\textrm{NN}}} = 5.02$ TeV with the ALICE experiment. Measurements are performed via the newly accessed decay channel $\overline{\Sigma}^{\pm}$ → $\bar{\textrm{n}}π^±$. A new method of antineutron reconstruction with the PHOS electromagnetic spectrometer is developed and applied to this analysis. The p T spectra of $\overline{\Sigma}^{\pm}$ are measured in the range 0.5 < p T < 3 GeV/c and compared to predictions of the PYTHIA 8, DPMJET, PHOJET, EPOS LHC and EPOS4 models. The EPOS LHC and EPOS4 models provide the best descriptions of the measured spectra both in pp and p-Pb collisions, while models which do not account for multiparton interactions provide a considerably worse description at high p T . The total yields of $\overline{\Sigma }^{\pm }$ in both pp and p-Pb collisions are compared to predictions of the Thermal-FIST model and dynamical models PYTHIA 8, DPMJET, PHOJET, EPOS LHC and EPOS4. All models reproduce the total yields in both colliding systems within uncertainties. The nuclear modification factors R pPb for both $\overline{\Sigma}^{+}$ and $\overline{\Sigma}^{-}$ are evaluated and compared to those of protons, $Λ$ and $Ξ$ hyperons, and predictions of EPOS LHC and EPOS4 models. No deviations of R pPb for $\overline{\Sigma}^{\pm}$ from the model predictions or measurements for other hadrons are found within uncertainties.

Abualrob, I. J. [University of Houston] (ORCID:000↗

DOE OSTI2026

Best practices in software development for robust and reproducible geoscientific models based on insights from the Global Carbon Budget's dynamic vegetation models

Computational models play an increasingly vital role in scientific research by enabling the numerical simulation of complex processes. Such models are also fundamental in geosciences. For instance, they offer critical insights into the impacts of global change on the Earth system today and in the future. Beyond their value as research tools, models are also software products and should therefore adhere to certain established software engineering standards. However, scientists are rarely trained as software developers, which can lead to potential deficiencies in software quality like unreadable, inefficient, or erroneous code. The complexity of models, coupled with their integration into broader workflows, also often makes it challenging to reproduce results, evaluate processes, and build upon them. In this paper, we review the state and current practices of the development processes of the state-of-the-art land surface models used by the Global Carbon Budget. We combine the experience of modelers from the respective research groups with the expertise of software engineers from tech companies to outline key principles and tools for improving software quality in research. We explore four main areas: (1) model testing and validation, (2) scientific, technical, and user documentation, (3) version control, continuous integration, and code review, and (4) the portability and reproducibility of workflows. Our review reveals that while modeling communities are incorporating many best practices, significant room for improvement remains in areas such as automated testing, automated documentation, and reproducibility. Therefore, we here identify and promote essential software engineering practices, including numerous examples of practices from within the community that can serve as guidelines for other models and could help streamline processes across the entire community. We conclude with an open-source example implementation of these principles, demonstrating portable and reproducible data flows, a continuous integration setup, and web-based visualizations. This example may serve as a practical resource for model developers, users, and all scientists engaged in scientific programming.

Gregor, Konstantin [Technical Univ. of Munich (Ger↗

DOE OSTI2025

Dynamical coupled-channel models for hadron dynamics

Dynamical coupled-channel (DCC) approaches parametrize the interactions and dynamics of two and more hadrons and their response to different electroweak probes. The inclusion of unitarity, three-body channels, and other properties from scattering theory allows for a reliable extraction of resonance spectra and their properties from data. Here, we review the formalism and application of the ANL-Osaka, the Juelich-Bonn-Washington, and other DCC approaches in the context of light baryon resonances from meson, (virtual) photon, and neutrino-induced reactions, as well as production reactions, strange baryons, light mesons, heavy meson systems, exotics, and baryon-baryon interactions. Finally, we also provide a connection of the formalism to study finite-volume spectra obtained in Lattice QCD, and review applications involving modern statistical and machine learning tools.

Amplitude analysis↗