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NASA NTRSDate not supplied

Development of An Improved BRDF Hotspot Model and its Use in VLIDORT to Study the Impact of Atmospheric Scattering on Hotspot Directional Signatures in the Atmosphere

The term “hotspot” refers to the sharp increase of reflectance occurring when incident (solar) and reflected (viewing) directions almost coincide in the backscatter direction. The accurate simulation of hotspot directional signatures is important for many remote sensing applications. The RossThick-LiSparse-Reciprocal (RTLSR) Bidirectional Reflectance Distribution Function (BRDF) model is widely used in radiative transfer simulations, and the hotspot model mostly used is from Maignan- Bréon but it typically requires large values of numerical quadrature and Fourier expansion terms in order to represent the hotspot accurately. To improve its use in atmospheric radiative transfer (RT) model simulations, in this paper we have developed a modified version based on the Maignan-Bréon’s hotspot BRDF model that converge much faster numerically, making it more practical for use in RT models that require Fourier expansion of BRDF to simulate the top-of-atmosphere (TOA) hotspot signatures. Using the vector linearized discrete ordinate radiative transfer model (VLIDORT), we found that reasonable TOA hotspot accuracy can be obtained with just 23 Fourier terms for clear atmospheres, and 63 Fourier terms for atmospheres with aerosol scattering. One advantage of this modified model is that the new hotspot model agrees very well with the original RossThick model away the hotspot region, making it is very convenient to use in the condition with and without hotspot in applications. This model can calculate the amplitude of hot spot accurately, and has been added in the most recent version of VLIDORT. However, there are some difference of this modified model with the original model for scattering angle close the hot spot, and it may not be appropriate for those who need an exact representation of the hot spot angular signature close to hot spot.

Xiaozhen (Shawn) Xiong↗

NASA NTRSDate not supplied

Extending an Economical Third-Order Inviscid Nodal-Gradient Cell-Centered Finite-Volume Method to Mixed-Element Grids

In this paper, we extend an economical third-order nodal-gradient cell-centered finite-volume method, originally developed for tetrahedral grids, to mixed-element grids. It is shown that the efficient quadratic interpolation formula essential to eliminating second derivatives from a third-order accurate discretization can be easily extended to an arbitrary cell type. For the surface flux integration, we consider a split-face approach, where a quadrilateral face is split into two triangles, and the third-order method for tetrahedra is directly applied. Also, we derive a second-derivative-free, high-order, volume quadrature formula for an arbitrary cell. Numerical results are presented for accuracy verification and applications with three-dimensional nontetrahedral grids.

Computational Fluid Dynamics↗

NASA NTRSDate not supplied

A New BRDF Hotspot Model and its Use in Improving Radiative Transfer Model Efficiency and Application to VLIDORT-based PCRTM Model

The term “hotspot” refers to the sharp increase of reflectance occurring when incident (solar) and reflected (viewing) directions almost coincide in the backscatter direction. The accurate simulation of hotspot directional signatures is important for many remote sensing applications. The RossThick-LiSparse-Reciprocal (RTLSR) Bidirectional Reflectance Distribution Function (BRDF) model is widely used in radiative transfer simulations, and the hotspot model mostly used is from Maignan- Bréon but it typically requires large values of numerical quadrature and Fourier expansion terms in order to represent the hotspot accurately. To improve its use in atmospheric radiative transfer (RT) model simulations, we have developed a modified version based on the Maignan-Bréon’s hotspot BRDF model that converge much faster numerically, making it more practical for use in RT models that require Fourier expansion of BRDF to simulate the top-of-atmosphere (TOA) hotspot signatures. On basis of the vector linearized discrete ordinate radiative transfer model (VLIDORT), we have built a line-by-line based simulation system that can simulate the TOA radiance for atmospheres with 26 gases, different aerosols, dust and clouds, and surface with different BRDF. Using simulated results under diverse atmospheric conditions, we have developed VLIDORT-PCRTM model.

Xiaozhen Xiong↗

NASA NTRSDate not supplied

Effects of Natural Variability on the Use of Standard Deviation to Represent Measurement Uncertainties in Atmospheric Composition Studies

Measurement uncertainty is defined as a “non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand”. It is most common that the uncertainties of GAW hourly measurements, such as greenhouse gas measurements, are reported as standard deviations derived from individual sampling at a higher time resolution (e.g., 1 min). In contrast, the uncertainties of GAW measurements of reactive gases and aerosol properties are reported in percentiles covering the same probability. A quick look at hourly CO 2 data from Cape Grim, Australia yielded some interesting findings: the hourly standard deviation is, on average, more than a factor of 10 higher for the measurements under non-background conditions (over 50% observations), while the difference in average CO 2 amount fraction was less than 2 ppm. The dramatic contrast cannot be explained by the difference in measurement uncertainties, but can largely be attributed the natural variability, or episodic ambient CO 2 variation reflecting changes in meteorological conditions or emissions. These initial findings motivated a more in-depth analysis of the ground-based measurements of trace gases and aerosol properties. This analysis will be using continuous 1 min ground site observations to construct time averaged statistical indicators to evaluate whether the standard deviation is adequate to represent the dispersion, especially under marked influence by natural variability. The suitability of this representation can be determined by examining the difference between the standard deviation and percentiles encompassing the same probability. We will examine time intervals of 1 hour, 3 hours, and 24 hours, with the latter time intervals chosen to match those commonly used in model assessments. We will also investigate how natural variability can alter the probability distribution of the measurands and how adequate the quadrature propagation of uncertainties is under these conditions. The results will include several trace gases (e.g., CO 2 , CO, O 3 , and NO 2 ) with a range of measurement techniques (e.g., PANDORA, in situ), atmospheric lifetimes, and aerosol properties (e.g. scattering coefficient). The findings from this analysis should provide some useful feedback on the best practices for uncertainty reporting.

Measurement Uncertainty↗

NASA NTRS2024

Quantification of Uncertainty and Risk Sensitivity for Safety of Emerging Operations

The growing need to develop and deploy small unmanned aerial vehicles (sUAVs) for various applications in the airspace necessitates reliable tools to accurately predict the flight trajectories of the sUAVs. The knowledge of the predicted trajectories help decision makers anticipate potential conflict, assess the risk, and take appropriate risk mitigation actions. In addition, uncertainties in vehicle models, weather, and controller action further highlights the need for reliable prediction tools. In this project, the application of mixed sparse grid-based quadrature and generalized polynomial chaos(gPC) expansion method for uncertainty quantification and collision assessment in air traffic consisting of fixed-wing small unmanned aerial vehicles (sUAV) was studied. From the results obtained, it can be concluded that this provides a reliable framework to carry out quantitative conflict assessment in an unmanned air traffic, which when employed, can improve the functionalities of the unmanned traffic management system. It was observed that the results from the gPC expansion framework developed in the project can be utilized to conduct rapid probabilistic collision assessment for near real-time unmanned traffic management in the airspace. From the vehicle models, position updates, and wind-field data, a priori gPC based 3-σcon-fidence ellipses can provide estimates of potential conflict at some future instants. The computational costs scaled linearly when the uncertain inputs were fewer. Further, the largest allowable distribution of para-metric uncertainties that leads to the smallest risk of collision in traffic of small unmanned aerial vehicles could be calculated. The time of closest approach between two sUAVs can be established paving way for development of proactive mitigation strategies. The separation between the sUAVs was found to be most significantly affected by uncertainties in the maximum available thrusts, zero-lift drag coefficients, and wing planform areas of the sUAVs. The study of uncertain wind-fields indicated that a heterogeneous traffic mix resulted in an increased probability of conflict. Increased measurement update rate reduced the uncertain-ties in the trajectories of the vehicles, further reducing the probability of conflict but rapid updates of all vehicles in the airspace poses a stringent communication limitation. The gPC framework also provided the means to analyze vehicle impact (crash region) due to loss of control resulting from actuator failure in sUAS traffic, essentially to predict impact and crash zones for representative vehicles. The predicted regions when compared with non-participant density, provides a means to develop an early mitigation strategy, should the sUAV detect an imminent actuator failure.

Rajnish Bhusal↗

NASA NTRSDate not supplied

Comparison of Two Approaches to Constructing Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Methods for Mixed-Element Grids

In this paper, we introduce a third-order nodal-gradient cell-centered ﬁnite volume scheme applicable to mixed-element grids based on a single numerical ﬂux per face. Previously, we considered a split-quadrilateral-face approach, where we split each quadrilateral face of a non-tetrahedral cell into two triangles and then apply second- and third-order schemes designed for tetrahedral grids. This approach is relatively simple to implement but expensive because it requires two numerical ﬂux evaluations per quadrilateral face. In this study, we consider another approach, where the surface ﬂux quadrature is performed with a single numerical ﬂux per face with ﬂux corrections. It requires a 3×3 correction matrix to be stored at each face but is more economical than the split-quadrilateral-face approach. The two approaches are compared for three-dimensional ﬂows on irregular hexahedral grids, and their relative merits are discussed.

Third-order scheme↗

NASA NTRS2025

Assessment of Radiation Scattering by Dust in Mars Entry

The impact of gas phase radiation scattering by dust particle presence in Mars entry scenarios is assessed using a suite of numerical tools. A hypersonic gas and dilute particle multiphase flow is simulated using a mixed continuum model and an efficient size distribution quadrature scheme. Radiative transport is simulated with a hybrid ray-tracing and P1 spherical harmonics model. The NASA HARA radiation code and a Lorentz-Mie scattering model are used to predict the gas-particle mixture radiative properties. Application of the numerical suite to a representative case based on the Mars 2020 entry shows that, even at conditions in excess of the strongest recorded dust storm, the augmentation in surface radiative heat flux due to scattering alone is lower than 1%, and therefore negligible. Predictions with even higher dust loading conditions confirm that even highly uncertain dust mass profiles are unlikely to be a concern for the scattering augmentation mechanism.

Andrew Hinkle↗

NASA NTRSDate not supplied

The Design of a Flexible, Interoperable Navigation Signal for Future Lunar Missions

The LunaNet Interoperability Specification (LNIS) is a set of standards currently under development by NASA, ESA, and JAXA, which define a common, interoperable set of services and interfaces for lunar communication and navigation. The LNIS includes specifications for the GNSS-like Augmented Forward Signal (AFS). The LANS (Lunar Augmented Navigation Service) will be comprised of Multiple LunaNet Service Provider (LNSP) nodes broadcasting the AFS, such as NASA’s LCRNS (Lunar Communications Relay and Navigation Systems), ESA’s Moonlight LCNS (Lunar Communication and Navigation Services) and the Japan LNSS (Lunar Navigation Satellite System). The LANS will provide a GNSS-like capability enabling orbiting and surface users in lunar space (such as Artemis) to estimate their position, velocity and time as described in Giordano et al., (2023). Initial capabilities will focus on providing service to the lunar South pole region. The specification of AFS defines two orthogonal signal components on a single carrier, with the in-phase component (AFS-I) being a lower-chip-rate data channel tailored for applications where low SWaP is critical (e.g., IoT devices or search and rescue), and the quadrature component (AFS-Q) being a high-chip-rate data-less pilot signal for high-precision, robust lunar navigation and positioning applications. An initial description of AFS was provided in the LNIS, (2023), and initial analysis results were shown in Dafesh, et al., (2024). In this work, we provide rationale for updates to the LNIS that define key aspects of the signal including the primary spreading code designs for the data and pilot channels, and a three-tiered overlay code approach for the pilot channel that provides flexible signal acquisition alternatives, rapid time dissemination and robust frame Sync. The paper also describes a robust data sync word that is designed to enable frame Sync. for low-SWaP receivers that only use the I channel, as well as a low-density parity check code (LDPC) data message encoding design and interleaving definition. The work further describes the impact of the updated AFS design in terms of improved acquisition performance, interference resistance, navigation message capabilities and rapid absolute time dissemination for users able to access clock and ephemeris data over an external network. The cross-correlation and synchronization performance of the AFS design is also compared to potential alternatives, further providing rationale for the final signal design configuration.

LANS↗

NASA NTRS2025

The Design of a Flexible, Interoperable Navigation Signal for Future Lunar Missions

LunaNet↗

NASA NTRSDate not supplied

Performance Assessment of LunaNet’s Augmented Forward Signal

LunaNet provides a common set of interoperable specifications for communication and position, navigation and time (PNT) services and interfaces soon to be implemented in lunar vicinity. The LunaNet Interoperability Specification (LNIS) provides the design for the GNSS-like Augmented Forward Signal (AFS), which enables orbiting and surface users in lunar space, such as Artemis, to estimate their position, velocity, and time. The specification of AFS defines two orthogonal signal components on a single carrier: the in-phase component (AFS-I), a lower-chip-rate data channel tailored for applications where low SWaP (Size, Weight, and Power) is critical (e.g., IoT devices or search and rescue), and the quadrature component (AFS-Q), a high-chip-rate data-less pilot signal for high-precision, robust lunar navigation and positioning applications. An initial description of AFS was provided in [1], with initial analysis results shown in [2] and [3] and the current signal in space description provided in [4]. As part of NASA's Lunar Communication Relay and Navigation Systems (LCRNS) project, this work expands upon the initial analysis results and proposes a new expanded set of AFS-Q spreading codes that exceed the cross-correlation and autocorrelation sidelobe performance of L1C and other GNSS signals, while providing additional expansion capabilities for future provider satellites. A set of 420 codes was selected from a Weil-based code derived from the prime number 10247, which is larger than the 10243 prime number used to derive Beidou’s B1C Weil sequences. Both the initial set of 210 codes and the expanded set of 420 codes are shown to provide the best cross-correlation of any 10230-chip satellite navigation codes. The performance is demonstrated for hierarchical sets of spreading codes optimized and organized in sets of 30 codes. The new codes were developed using an optimization approach and correlation methodology described in [5]. The work also compares LunaNet’s AFS to terrestrial GNSS signals in terms of acquisition, tracking, and data demodulation performance. Performance is evaluated for receivers that only track the 1.023 MCPS data channel spreading code for low SWaP IoT use cases, as well as for receivers that track both the 1.023 MCPS data channel and the 5.115 MCPS pilot channel spreading code for high-performance use cases. Performance is assessed in the presence of interference and thermal noise. The analysis is performed in terms of expected operating conditions on the lunar surface. Several unique flexibility aspects of the augmented forward signal are described, including the use of the Q channel’s secondary and tertiary codes to enable variable coherent integrations during acquisition. This is compared to GNSS signals such as L5/E5 and MBOC in terms of achievable processing gain for interference mitigation versus acquisition complexity. The work details acquisition and tracking techniques used to optimally acquire and track the primary, secondary, and tertiary codes on the Q channel, as well as acquisition of the I channel spreading code. Acquisition of the 8 ms Q channel spreading code is also compared to joint acquisition of the I and Q channel primary codes in noise and interference environments.

LCRNS↗

NASA NTRSDate not supplied

Performance Assessment of LunaNet’s Augmented Forward Signal

LunaNet provides a common set of interoperable specifications for communication and position, navigation and time (PNT) services and interfaces soon to be implemented in lunar vicinity. The LunaNet Interoperability Specification (LNIS) provides the design for the GNSS-like Augmented Forward Signal (AFS), which enables orbiting and surface users in lunar space, such as Artemis, to estimate their position, velocity, and time. The specification of AFS defines two orthogonal signal components on a single carrier: the in-phase component (AFS-I), a lower-chip-rate data channel tailored for applications where low SWaP (Size, Weight, and Power) is critical (e.g., IoT devices or search and rescue), and the quadrature component (AFS-Q), a high-chip-rate data-less pilot signal for high-precision, robust lunar navigation and positioning applications. An initial description of AFS was provided in LNIS 2023, with initial analysis results shown in Dafesh 2024 and Dafesh 2025, and the current signal in space description provided in LNIS 2025. As part of NASA's Lunar Communication Relay and Navigation Systems (LCRNS) project, this work expands upon the initial analysis results and proposes a new expanded set of AFS-Q spreading codes that exceed the cross-correlation and autocorrelation sidelobe performance of L1C and other GNSS signals, while providing additional expansion capabilities for future service satellites. A set of 420 codes was selected from a Weil-based code derived from the prime number 10247, which is larger than the 10243 prime number used to derive BeiDou’s B1C Weil sequences. Both the initial set of 210 codes and the expanded set of 420 codes are shown to provide the best cross-correlation of any 10230-chip satellite navigation codes. The performance is demonstrated for hierarchical sets of spreading codes optimized and organized in sets of 30 codes. The work also compares LunaNet’s AFS to terrestrial GNSS signals in terms of acquisition, tracking, and data demodulation performance. Performance is evaluated for receivers that only track the 1.023 MCPS data channel spreading code for low SWaP IoT use cases, as well as for receivers that track both the 1.023 MCPS data channel and the 5.115 MCPS pilot channel spreading code for high-performance use cases. Performance is assessed in the presence of interference and thermal noise. The analysis is performed in terms of expected operating conditions on the lunar surface. Several unique flexibility aspects of the augmented forward signal are described, including the use of the Q channel’s secondary and tertiary codes to enable variable coherent integrations during acquisition. This is compared to GNSS signals such as L5/E5 and MBOC in terms of achievable processing gain for interference mitigation versus acquisition complexity. The work details acquisition and tracking techniques used to optimally acquire and track the primary, secondary, and tertiary codes on the Q channel, as well as acquisition of the I channel spreading code. Acquisition of the 8 ms, Q channel spreading code is also compared to joint acquisition of the I and Q channel primary codes in noise and interference environments

LCRNS↗

DOE OSTI2020

Quantum random number generator

A system and method according to one embodiment are provided for random number generation based on measuring quadrature fluctuations of a single mode thermal state using an optical homodyne detector.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

A 1.2-V 6-GHz Dual-Path Charge-Pump PLL Frequency Synthesizer for Quantum Control and Readout in CMOS 65-nm Process

This paper presents a low jitter dual-path chargepump phase locked loop (PLL) synthesizer in a CMOS 65-nm process for quantum readout applications. The PLL incorporates a programmable dual chargepump and a loop filter with both proportional and integral paths that can be driven independently providing flexible control of the loop bandwidth to achieve low jitter performance. The design is implemented at 300 K and critical blocks like voltage-controlled oscillator (VCO) and chargepump (CP) are analyzed at 77 K based on the characterized results. The LC-VCO is realized with the class-C NMOS only architecture with 5-bit coarse control and quadrature signals are generated with poly phase filter. The VCO is designed with the tuning range of 1 GHz around the center frequency of 6 GHz with Phase Noise of -123 dBc/Hz and -132 dBc/Hz at 1MHz offset at 300 K and 77 K temperature. The simulated PLL rms jitter is 125 fs at 6 GHz with a power consumption of 8 mW at the 1.2 V power supply.

47 OTHER INSTRUMENTATION↗

DOE OSTI2021

Enabling Long-range Exploration in Minimization of Multimodal Functions

We consider the problem of minimizing multi-modal loss functions with a large number of local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss’s landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea of DGS is to conducts 1D long-range exploration with a large smoothing radius along d orthogonal directions in Rd, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. The d directional derivatives are then assembled to form the nonlocal gradient. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. The superior performance of our method is demonstrated in three sets of examples, including benchmark functions for global optimization, and two real-world scientific problems.

Zhang, Jiaxin↗

DOE OSTI2021

Truncated nonlinear interferometer-based atomic force microscopes

A truncated non-linear interferometer-based atomic force microscope (AFM) includes an input port and a non-linear amplifier that renders a probe beam and a conjugate beam. The AFM includes local oscillators having a relationship with the probe beam and the conjugate beam. The displacement of the AFM's cantilever is transduced by the probe beam, and/or the conjugate beam or their respective local oscillators. The AFM's phase-sensitive detectors detect a phase modulation between the respective local oscillators and the probe beam and the conjugate beam. The detected phase modulation corresponds to the change in phase. The AFM's circuitry measures phase signals that are indicative of the cantilever displacement. The resulting measurement signals exhibit a quantum noise reduction in either the intensity difference or phase sum quadrature.

Pooser, Raphael C.↗

DOE OSTI2022

Continuous variable quantum secret sharing

Continuous variable quantum secret sharing (CV-QSS) technologies are described that use laser sources and homodyne detectors. Here, a Gaussian-modulated coherent state (GMCS) prepared by one device passes through secure stations of other devices sequentially on its way to a trusted device, and each of the other devices coherently adds a locally prepared, independent GMCS to the group of propagating GMCSs. Finally, the trusted device measures both the amplitude and the phase quadratures of the received group of coherent GMCSs using double homodyne detectors. The trusted device suitably uses the measurement results to establish a secure key for encoding secret messages to be broadcast to the other devices. The devices cooperatively estimate, based on signals corresponding to their respective Gaussian modulations, the trusted device's secure key, so that the cooperative devices can decode the broadcast secret messages with the secure key.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Truncated non-linear interferometer-based sensor system

A truncated non-linear interferometer-based sensor system includes an input that receives an optical beam and a non-linear amplifier that generates a probe beam and a conjugate beam from the optical beam. The system's local oscillators are related to the probe beam and the conjugate beam. The system includes a sensor that transduces an input with the probe beam and the conjugate beam. The transduction detects changes in the phase of each of the probe beam and the conjugate beam. The system's phase sensitive detectors detect phase modulations between the respective local oscillators, the probe beam, and the conjugate beam and outputs phase signals based on detected phase modulations. The system measures phase signals indicative of the sensor's input resulting from a sum or difference of the phase signals. The measurement exhibits a quantum noise reduction in an intensity difference, a phase sum, or an amplitude difference quadrature.

47 OTHER INSTRUMENTATION↗

DOE OSTI2023

Beam dynamics corrections in the measurement of the anomalous precession frequency at the Muon $g-2$ experiment at Fermilab

The Muon $g-2$ experiment at Fermilab (E989) aims to measure the anomalous magnetic moment of the muon with an accuracy of 140 ppb (parts per billions). This accuracy, obtained by adding in quadrature a statistical and a systematic contribution of comparable value (100 ppb), will allow to reduce the experimental uncertainty (from the previous E821 experiment at BNL \cite{bnl}) of a factor of 4, and represents one of the most precise tests of the Standard Model (SM) theory of elementary particles. The first result on the Run-1 dataset \cite{prl} was released on April 7, 2021, showing a very good agreement with the previous result from BNL experiment, with a slightly better uncertainty. The corresponding experimental average increases the significance of the discrepancy between the measured and Standard Model prediction of 4.2$\sigma$ \cite{white_paper}.\\ \noindent The measured quantity is the muon magnetic anomaly a$_\mu$=$\frac{g_\mu-2}{2}$ where g$_\mu$ is the gyromagnetic factor of the muon. Dirac's equation predicts g$_\mu$ = 2, while radiative corrections, dominated by the QED contribution due to an exchange of a virtual photon, causes a per-mille correction on this quantity. By including all the SM contributions, a$_\mu$ is known at 370 ppb. The E989 experiment measures $a_\mu$ injecting positive muons with momentum of 3.1 GeV/$c$ polarized longitudinally in bunches (called $fills$) with an average rate of 12 Hz, in a storage ring of 14 meters diameter. Due to the parity violation in the weak muon decay, high energy positrons produced are emitted preferably in the muon's spin direction. By counting the number of positrons with energy greater then 1.7 GeV in function of the time, the frequency precession of the muon spin is measured, that together with the measurement of the magnetic field, allows to extract $a_\mu$. The positrons are detected with 24 electromagnetic calorimeters, that measure the energy and the arrival time of the positrons, each made of 54 crystals of lead fluoride (PbF$_2$) read by silicon photomultipliers (SiPM). Together with calorimeters, two tracking detectors are used to make non-destructive measurements of the muon beam characteristic by reconstructing the muon decay position extrapolating backward the decay positrons. The knowledge of the beam motion inside the ring plays a fundamental role in the analysis of $a_{\mu}$, where the measured anomalous precession frequency $\omega_a$ must be corrected for four main beam dynamics effects. A first correction is associated with the presence of an electric field responsible of the vertical focusing on the storage ring, where vertical direction is orthogonal to the orbit and horizontal direction is along the storage ring radius. Due to the oscillations in the vertical direction, the so called vertical betatron oscillations, a second correction is necessary to account for an average angle associated with the muon motion off the ideal orbit. A third correction is caused by lost muons in the ring which have a different spin phase at the injection respect to the decay ones. Finally, due to the correlation between the vertical and horizontal beam motion of the muons and the acceptance of the calorimeter, a correction (called ``phase-acceptance") arises. In Run1, due to the presence of two damaged resistors in one electrostatic quadrupole, this correction had a prominent role. The replacement of the damaged resistors before Run2 decreased this effect by one order of magnitude, and a further reduction in Run-3 was provided by an improved orbit.\\ \noindent The work of this Thesis focuses on the beam dynamics corrections on $\omega_a$. Due to the prominent role in Run-1 a special attention has been put to the phase-acceptance correction\footnote{The effect of this correction on the E821 BNL measurement of the $g-2$ has been evaluated to be within the quoted systematic error.}. Chapter \ref{ch:anomalous_magnetic_moment} introduces the anomalous magnetic moment of the muon. Chapter \ref{ch:early_experiment} describes the history of the Muon $g-2$ experiments. Chapter \ref{ch:theory} discusses the Standard Model prediction and possible new physics scenario. Chapter \ref{ch:muon_g2_experiment} describes the E989 experiment, whose experimental technique and the apparatus are discussed focusing on the improvements needed to reach the final goal on $a_\mu$ measurement. The original work of the Thesis is discussed in the last 5 chapters. Chapter \ref{ch:experiment_technique} presents the analysis technique to extract $\omega_a$, Chapter \ref{ch:beam_dynamics} describes the beam dynamics in the E989 experiment, Chapter \ref{ch:beam_dynamics_correction} discusses the beam dynamics correction to $\omega_a$, and Chapter \ref{sec:pa_corr} focuses on the phase acceptance correction. Finally Chapter \ref{ch:run23_analysis_improvements} contains the Run-2/3 improvements which are expected to allow for an increased precision ons the beam dynamics co...

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗