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All sourcesNASA NTRSDOE OSTI
NASA NTRS2013

An Alternative Flight Software Paradigm: Applying Multivariate Logistic Regression to Sense Trigger Conditions using Inaccurate or Scarce Information

In late 2014, NASA will fly the Orion capsule on a Delta IV-Heavy rocket for the Exploration Flight Test-1 (EFT-1) mission. For EFT-1, the Orion capsule will be flying with a new GPS receiver and new navigation software. Given the experimental nature of the flight, the flight software must be robust to the loss of GPS measurements. Once the high-speed entry is complete, the drogue parachutes must be deployed within the proper conditions to stabilize the vehicle prior to deploying the main parachutes. When GPS is available in nominal operations, the vehicle will deploy the drogue parachutes based on an altitude trigger. However, when GPS is unavailable, the navigated altitude errors become excessively large, driving the need for a backup barometric altimeter to improve altitude knowledge. In order to increase overall robustness, the vehicle also has an alternate method of triggering the parachute deployment sequence based on planet-relative velocity if both the GPS and the barometric altimeter fail. However, this backup trigger results in large altitude errors relative to the targeted altitude. Motivated by this challenge, this paper demonstrates how logistic regression may be employed to semi-automatically generate robust triggers based on statistical analysis. Logistic regression is used as a ground processor pre-flight to develop a statistical classifier. The classifier would then be implemented in flight software and executed in real-time. This technique offers improved performance even in the face of highly inaccurate measurements. Although the logistic regression-based trigger approach will not be implemented within EFT-1 flight software, the methodology can be carried forward for future missions and vehicles

Smith, Kelly↗
NASA NTRS2016

A Distant Mirror: Solar Oscillations Observed on Neptune by the Kepler K2 Mission

Starting in 2014 December, Kepler (K2) observed Neptune continuously for 49 days at a 1-minute cadence. The goals consisted of studying its atmospheric dynamics, detecting its global acoustic oscillations, and those of the Sun, which we report on here. We present the first indirect detection of solar oscillations in intensity measurements. Beyond the remarkable technical performance, it indicates how Kepler would see a star like the Sun. The result from the global asteroseismic approach, which consists of measuring the oscillation frequency at maximum amplitude max velocity and the mean frequency separation between mode overtones delta velocity, is surprising as the max velocity measured from Neptune photometry is larger than the accepted value. Compared to the usual reference max velocity of the sun equal to 3100 microhertz, the asteroseismic scaling relations therefore make the solar mass and radius appear larger by 13.8 plus or minus 5.8 percent and 4.3 plus or minus 1.9 percent, respectively. The higher max velocity is caused by a combination of the value of max velocity of the sun, being larger at the time of observations than the usual reference from SOHO/VIRGO/SPM (Variability of solar IRradiance and Gravity Oscillations / on board SOHO (Solar and Heliospheric Observatory) / Sun PhotoMeters) data (3160 plus or minus 10 microhertz), and the noise level of the K2 time series, being 10 times larger than VIRGO's. The peak-bagging method provides more consistent results: despite a low signal-to-noise ratio (S/N), we model 10 overtones for degrees iota equal 0, 1, 2. We compare the K2 data with simultaneous SOHO/VIRGO/SPM photometry and Bison (Birmingham Solar-Oscillations Network) velocity measurements. The individual frequencies, widths, and amplitudes mostly match those from VIRGO and BiSON within 1 sigma, except for the few peaks with the lowest S/N.

planets and satellites: individual (Neptune) – s↗
NASA NTRS2011

An Integrated Hydrological and Water Management Study of the Entire Nile River System - Lake Victoria to Nile Delta

The Nile basin River system spans 3 million km(exp 2) distributed over ten nations. The eight upstream riparian nations, Ethiopia, Eretria, Uganda, Rwanda, Burundi, Congo, Tanzania and Kenya are the source of approximately 86% of the water inputs to the Nile, while the two downstream riparian countries Sudan and Egypt, presently rely on the river's flow for most of the their needs. Both climate and agriculture contribute to the complicated nature of Nile River management: precipitation in the headwaters regions of Ethiopia and Lake Victoria is variable on a seasonal and inter-annual basis, while demand for irrigation water in the arid downstream region is consistently high. The Nile is, perhaps, one of the most difficult trans-boundary water issue in the world, and this study would be the first initiative to combine NASA satellite observations with the hydrologic models study the overall water balance in a to comprehensive manner. The cornerstone application of NASA's Earth Science Research Results under this project are the NASA Land Data Assimilation System (LDAS) and the USDA Atmosphere-land Exchange Inverse (ALEXI) model. These two complementary research results are methodologically independent methods for using NASA observations to support water resource analysis in data poor regions. Where an LDAS uses multiple sources of satellite data to inform prognostic simulations of hydrological process, ALEXI diagnoses evapotranspiration and water stress on the basis of thermal infrared satellite imagery. Specifically, this work integrates NASA Land Data Assimilation systems into the water management decision support systems that member countries of the Nile Basin Initiative (NBI) and Regional Center for Mapping of Resources for Development (RCMRD, located in Nairobi, Kenya) use in water resource analysis, agricultural planning, and acute drought response to support sustainable development of Nile Basin water resources. The project is motivated by the recognition that accurate, frequent, and spatially distributed estimates of the water balance are necessary for effective water management. This creates a challenge for watersheds that are large, include data poor regions, and/or span multiple nations. All of these descriptors apply to the Nile River basin, yet successful management of the Nile is critical for development and political stability in the region. For this reason, improved hydrological data to support cooperative water management in the Nile basin is a priority for USAID, the US State Department, the World Bank and other international organizations. In this project, the U.S. based research team is working with partners at RCMRD, Nile Basin Initiative (NBI), and their member national-level agencies to develop satellite-based land cover maps, satellite-derived evapotranspiration estimates (using the ALEXI algorithm), and NASA's Land Data Assimilation System (LDAS) customized to match identified information needs. The cornerstone applied sciences product of the project is the development of a customized "Nile LDAS" that will produce optimal estimates of hydrological states and fluxes, as vetted against the in situ observations of NBI and RCMRD member organizations and independent satellite-derived hydrological estimates. Nile LDAS will be applied to improve the reliability of emerging Decision Support Systems in applications that include drought monitoring, reservoir management, and irrigation planning. The end-users such as RCMRD, NBI, Ethiopian and Kenya Meteorological and Famine Early Warning System Network (FEWSNet) will be the eventual benefactors of this work. There will be a capacity building process involving the above end-user organizations and transfer the models and the results for these organizations to execute for future use. The team has already initiated this study and the early results of first years' work are shown. The plan is to complete this work by late 2013.

Habib, Shahid↗
DOE OSTI2024

Impedance Scan of Inverter-Based Resources and Diesel Generator for Stability Analysis

Impedance-based methods are widely used for power system stability analysis with inverter-based resources (IBRs), e.g., assessing dynamic interactions between the power grid and an IBR, control interactions between multiple IBRs, and the sub-synchronous oscillation and damping phenomenon. Since it is difficult to get a numerical model 100% matching with the hardware IBR, using the hardware inverter directly to obtain its output impedance has become a prominent approach nowadays. Therefore, this article presents the impedance scan using hardware IBRs, and also a hardware diesel generator as it still stays with the grid before the grid completely goes to renewable. The devices under test (DuTs) for the impedance scan includes two 3-..phi.., 480 V, 60 Hz commercial grid-forming IBRs (one of 250 kVA and another of 125 kVA rating) in series with ..delta..-Y transformers, one 3-..phi.., 480 V, 60 Hz commercial grid-following IBR (of 125 kVA rating), and a 3-..phi.., 480 V, 60 Hz commercial diesel generator (of 187.5 kVA rating). Using voltage signals perturbed with sub-, inter-, and higher harmonic components, and measuring the current response, the positive-sequence impedances are computed via an offline-based post-analysis. Moreover, best-fit transfer functions are estimated that closely resemble the measured data points of the positive-sequence impedances. Based on the observations from various outcomes of the hardware experiments, this article also provides some fundamental insights on the equivalent positive-sequence impedance of a combination of multiple hardware components by comparing the estimated and the empirically computed impedances. A comparative insight on the damping capability of the DuTs using the positive-sequence impedances of the hardware is also discussed.

current measurement↗
DOE OSTI2024

Impedance Scan of Inverter-Based Resources and Diesel Generator for Stability Analysis: Preprint

Impedance-based methods are widely used for power system stability analysis with inverter-based resources (IBRs), e.g., assessing dynamic interactions between the power grid and an IBR, control interactions between multiple IBRs, and the sub-synchronous oscillation and damping phenomenon. Since it is difficult to get a numerical model 100% matching with the hardware IBR, using the hardware inverter directly to obtain its output impedance has become a prominent approach nowadays. Therefore, this article presents the impedance scan using hardware IBRs, and also a hardware diesel generator as it still stays with the grid before the grid completely goes to renewable. The devices under test (DuTs) for the impedance scan includes two 3-..phi.., 480 V, 60 Hz commercial grid-forming IBRs (one of 250 kVA and another of 125 kVA rating) in series with ..delta..-Y transformers, one 3-..phi.., 480 V, 60 Hz commercial grid-following IBR (of 125 kVA rating), and a 3-..phi.., 480 V, 60 Hz commercial diesel generator (of 187.5 kVA rating). Using voltage signals perturbed with sub-, inter-, and higher harmonic components, and measuring the current response, the positive-sequence impedances are computed via an offline- based post-analysis. Moreover, best-fit transfer functions are estimated that closely resemble the measured data points of the positive-sequence impedances. Based on the observations from various outcomes of the hardware experiments, this article also provides some fundamental insights on the equivalent positive- sequence impedance of a combination of multiple hardware components by comparing the estimated and the empirically computed impedances. A comparative insight on the damping capability of the DuTs using the positive-sequence impedances of the hardware is also discussed.

grid following inverter↗
NASA NTRS2008

Integrated Vehicle Ground Vibration Testing in Support of Launch Vehicle Loads and Controls Analysis

All structural systems possess a basic set of physical characteristics unique to that system. These unique physical characteristics include items such as mass distribution and damping. When specified, they allow engineers to understand and predict how a structural system behaves under given loading conditions and different methods of control. These physical properties of launch vehicles may be predicted by analysis or measured by certain types of tests. Generally, these properties are predicted by analysis during the design phase of a launch vehicle and then verified by testing before the vehicle becomes operational. A ground vibration test (GVT) is intended to measure by test the fundamental dynamic characteristics of launch vehicles during various phases of flight. During the series of tests, properties such as natural frequencies, mode shapes, and transfer functions are measured directly. These data will then be used to calibrate loads and control systems analysis models for verifying analyses of the launch vehicle. NASA manned launch vehicles have undergone ground vibration testing leading to the development of successful launch vehicles. A GVT was not performed on the inaugural launch of the unmanned Delta III which was lost during launch. Subsequent analyses indicated had a GVT been performed, it would have identified instability issues avoiding loss of the vehicle. This discussion will address GVT planning, set-up, execution and analyses, for the Saturn and Shuttle programs, and will also focus on the current and on-going planning for the Ares I and V Integrated Vehicle Ground Vibration Test (IVGVT).

Askins, Bruce R.↗
NASA NTRS1993

Near-Threshold Rovibrational Excitation of HF by Electron Impact

The pronounced threshold peaks found in cross sections for electron-impact excitation of the vibrational levels of HF have been studied using an R-matrix method, in which electron exchange, correlation effects and polarization are included. Vibration is modelled inside the R-matrix sphere using the energy-modified adiabatic phase matrix method (EMAP). Rotation of the target molecule is treated outside the R-matrix sphere, using the frame transformation method. The external rovibronic close-coupling equations are solved by analytic methods. Analysis of the multichannel threshold structures indicates that the peaks found for rovibrational excitation of nu = 1 can be interpreted as virtual states while those for nu = 2 correspond to nuclear-excited Feshbach resonances. These two different results correspond simply to a change of sign of a small parameter that is determined by the threshold analysis. Whereas the overall shapes of the excitation peaks agree with earlier theoretical results, we have found detailed rotational fine structures for scattering energies on the order of the rotational constant above and below the opening of the vibrational channels. The partial and total cross sections for rovibrational state-to-state transitions in the energy region of the peaks are analysed. Characteristic cusp structures and excitation peaks are found at the rovibrational thresholds for transitions with small angular momentum transfer (Delta)j =0, 1 and 2. Differential cross sections are similar to recent measurements. These experimental data do not show the relatively sharp rise at the vibrational thresholds indicated by theory. Theory and experiment are in good agreement for the shape and magnitude of the angular dependence of cross sections for rovibrational state-to-state transitions, especially at larger energies.

Thummel, Helmar T.↗
NASA NTRS2007

Real-time Signal-to-noise Ratio (SNR) Estimation for BPSK and QPSK Modulation Using the Active Communications Channel

Method and apparatus for estimating signal-to-noise ratio (SNR) gamma of a composite input signal e(t) on a phase modulated (e.g., BPSK) communications link. A first demodulator receives the composite input signal and a stable carrier signal and outputs an in-phase output signal; a second demodulator receives the composite input signal and a phase-shifted version of the carrier signal and outputs a quadrature-phase output signal; and phase error theta(sub E)(t) contained within the composite input signal e(t) is calculated from the outputs of the first and second demodulators. A time series of statistically independent phase error measurements theta(sub E)(t(sub 1)), theta (sub E)(t(sub 2)),..., theta (sub E)(t(sub k)) is obtained from the composite input signal subtending a time interval delta t = t(sub k) - t(sub 1) whose value is small enough such that gamma(t) and sigma(t) can be taken to be constant in delta t. A biased estimate gamma(sup *) for the signal-to-noise ratio (SNR) gamma if the composite input signal is calculated using maximum likelihood (ML) estimation techniques, and an unbiased estimate gamma(sup ^) for the signal-to-noise ratio (SNR) gamma of the composite input signal is determined from the biased estimate gamma(sup *), such as by use of a look-up table.

Manning, Robert M.↗
NASA NTRS2000

A Method for Assessing the Quality of Model-Based Estimates of Ground Temperature and Atmospheric Moisture Using Satellite Data

A method is developed for validating model-based estimates of atmospheric moisture and ground temperature using satellite data. The approach relates errors in estimates of clear-sky longwave fluxes at the top of the Earth-atmosphere system to errors in geophysical parameters. The fluxes include clear-sky outgoing longwave radiation (CLR) and radiative flux in the window region between 8 and 12 microns (RadWn). The approach capitalizes on the availability of satellite estimates of CLR and RadWn and other auxiliary satellite data, and multiple global four-dimensional data assimilation (4-DDA) products. The basic methodology employs off-line forward radiative transfer calculations to generate synthetic clear-sky longwave fluxes from two different 4-DDA data sets. Simple linear regression is used to relate the clear-sky longwave flux discrepancies to discrepancies in ground temperature ((delta)T(sub g)) and broad-layer integrated atmospheric precipitable water ((delta)pw). The slopes of the regression lines define sensitivity parameters which can be exploited to help interpret mismatches between satellite observations and model-based estimates of clear-sky longwave fluxes. For illustration we analyze the discrepancies in the clear-sky longwave fluxes between an early implementation of the Goddard Earth Observing System Data Assimilation System (GEOS2) and a recent operational version of the European Centre for Medium-Range Weather Forecasts data assimilation system. The analysis of the synthetic clear-sky flux data shows that simple linear regression employing (delta)T(sub g)) and broad layer (delta)pw provides a good approximation to the full radiative transfer calculations, typically explaining more thin 90% of the 6 hourly variance in the flux differences. These simple regression relations can be inverted to "retrieve" the errors in the geophysical parameters, Uncertainties (normalized by standard deviation) in the monthly mean retrieved parameters range from 7% for (delta)T(sub g) to approx. 20% for the lower tropospheric moisture between 500 hPa and surface. The regression relationships developed from the synthetic flux data, together with CLR and RadWn observed with the Clouds and Earth Radiant Energy System instrument, ire used to assess the quality of the GEOS2 T(sub g) and pw. Results showed that the GEOS2 T(sub g) is too cold over land, and pw in upper layers is too high over the tropical oceans and too low in the lower atmosphere.

Wu, Man Li C.↗
NASA NTRS1998

Time-resolved diffusion tomographic imaging in highly scattering turbid media

A method for imaging objects in highly scattering turbid media. According to one embodiment of the invention, the method involves using a plurality of intersecting source/detectors sets and time-resolving equipment to generate a plurality of time-resolved intensity curves for the diffusive component of light emergent from the medium. For each of the curves, the intensities at a plurality of times are then inputted into the following inverse reconstruction algorithm to form an image of the medium: X.sup.(k+1).spsp.T =?Y.sup.T W+X.sup.(k).spsp.T .LAMBDA.!?W.sup.T W+.LAMBDA.!.sup.-1 wherein W is a matrix relating output at detector position r.sub.d, at time t, to source at position r.sub.s, .LAMBDA. is a regularization matrix, chosen for convenience to be diagonal, but selected in a way related to the ratio of the noise, to fluctuations in the absorption (or diffusion) X.sub.j that we are trying to determine: .LAMBDA..sub.ij =.lambda..sub.j .delta..sub.ij with .lambda..sub.j = /<.DELTA.Xj.DELTA.Xj> Here Y is the data collected at the detectors, and X.sup.k is the kth iterate toward the desired absoption information.

Alfano, Robert R.↗
DOE OSTI2023

Identifying COVID-19 cases and extracting patient reported symptoms from Reddit using natural language processing

We used social media data from “covid19positive” subreddit, from 03/2020 to 03/2022 to identify COVID-19 cases and extract their reported symptoms automatically using natural language processing (NLP). We trained a Bidirectional Encoder Representations from Transformers classification model with chunking to identify COVID-19 cases; also, we developed a novel QuadArm model, which incorporates Question-answering, dual-corpus expansion, Adaptive rotation clustering, and mapping, to extract symptoms. Our classification model achieved a 91.2% accuracy for the early period (03/2020-05/2020) and was applied to the Delta (07/2021–09/2021) and Omicron (12/2021–03/2022) periods for case identification. We identified 310, 8794, and 12,094 COVID-positive authors in the three periods, respectively. The top five common symptoms extracted in the early period were coughing (57%), fever (55%), loss of sense of smell (41%), headache (40%), and sore throat (40%). During the Delta period, these symptoms remained as the top five symptoms with percent authors reporting symptoms reduced to half or fewer than the early period. During the Omicron period, loss of sense of smell was reported less while sore throat was reported more. Our study demonstrated that NLP can be used to identify COVID-19 cases accurately and extracted symptoms efficiently.

60 APPLIED LIFE SCIENCES↗
NASA NTRS1988

A full-potential theory analysis of the supersonic aerodynamics of a 60-deg delta wing-body configuration

Experimental spanwise pressure distributions for a 60-deg delta wing/body of approximate fineness ratio 7.6 have been obtained and compared to predictions using full-potential theory. Analysis was performed at Mach 1.6 for angles of attack in the range 0.8 to 10 deg, and for Mach numbers ranging from 1.4 to 1.8 at lift coefficients 0.3 and 0.4. The intent of the study was to examine an attached flow approach for maneuver wing design in the presence of a fuselage. For the Mach number, angle-of-attack conditions considered, the full-potential theory accurately modeled the pressure distributions provided the flow remained attached. By combining the full-potential theory results with an empirical shock-induced separation criterion, it was found that the onset of shock-induced separation can be predicted. The investigation showed that, if an attached-flow approach is used with an empirical method of indicating shock-induced separation, the full-potential method is capable of being used as an effective tool for designing maneuver wings.

Rose, O. C.↗
NASA NTRS1973

Hypersonic aerodynamic characteristics of an all-body research aircraft configuration

An experimental investigation was conducted at Mach 6 to determine the hypersonic aerodynamic characteristics of an all-body, delta-planform, hypersonic research aircraft (HYFAC configuration). The aerodynamic characteristics were obtained at Reynolds numbers based on model length of 2.84 million and 10.5 million and over an angle-of-attack range from minus 4 deg to 20 deg. The experimental results show that the HYFAC configuration is longitudinally stable and can be trimmed over the range of test conditions. The configuration had a small degree of directional stability over the angle-of-attack range and positive effective dihedral at angles of attack greater than 2 deg. Addition of canards caused a decrease in longitudinal stability and an increase in directional stability. Oil-flow studies revealed extensive areas of separated and vortex flow on the fuselage lee surface. A limited comparison of wind-tunnel data with several hypersonic approximations indicated that, except for the directional stability, the tangent-cone method gave adequate agreement at control settings between 5 deg and minus 5 deg and positive lift coefficient. A limited comparison indicated that the HYFAC configuration had greater longitudinal stability than an elliptical-cross-section configuration, but a lower maximum lift-drag ratio.

Clark, L. E.↗
NASA NTRS1984

An analytical design procedure for the determination of effective leading edge extensions on thick delta wings

An analytical design procedure for leading edge extensions (LEE) was developed for thick delta wings. This LEE device is designed to be mounted to a wing along the pseudo-stagnation stream surface associated with the attached flow design lift coefficient of greater than zero. The intended purpose of this device is to improve the aerodynamic performance of high subsonic and low supersonic aircraft at incidences above that of attached flow design lift coefficient, by using a vortex system emanating along the leading edges of the device. The low pressure associated with these vortices would act on the LEE upper surface and the forward facing area at the wing leading edges, providing an additional lift and effective leading edge thrust recovery. The first application of this technique was to a thick, round edged, twisted and cambered wing of approximately triangular planform having a sweep of 58 deg and aspect ratio of 2.30. The panel aerodynamics and vortex lattice method with suction analogy computer codes were employed to determine the pseudo-stagnation stream surface and an optimized LEE planform shape.

Farhad Ghaffari↗
NASA NTRS1994

Near tip stress and strain fields for short elastic cracks

Recent experimental fatigue crack growth studies have concluded an apparent anomalous behavior of short cracks. To investigate the reasons for this unexpected behavior, the present paper focuses on identifying the crack length circumstances under which the requirements for a single parameter (K(sub I) or delta K(sub I) if cyclic loading is considered) characterization are violated. Furthermore, an additional quantity, the T stress, as introduced by Rice, and the related biaxiality ratio, B, are calculated for several crack lengths and two configurations, the single-edge-cracked and the centrally-cracked specimen. It is postulated that a two-parameter characterization by K and T (or B) is needed for the adequate description of the stress and strain field around a short crack. To further verify the validity of this postulate, the influence of the third term of the Williams series on the stress, strain and displacement fields around the crack tip and in particular on the B parameter is also examined. It is found that the biaxiality ratio would be more negative if the third term effects are included in both geometries. The study is conducted using the finite element method with linearly elastic material and isoparametric elements and axial (mode I) loading. Moreover, it is clearly shown that it is not proper to postulate the crack size limits for 'short crack' behavior as a normalized ratio with the specimen width, a/w; it should instead be stated as an absolute, or normalized with respect to a small characteristic dimension such as the grain size. Finally, implications regarding the prediction of cyclic (fatigue) growth of short cracks are discussed.

Soediono, A. H.↗
NASA NTRS1976

Theoretical and experimental study of twisted and cambered delta wings designed for a Mach number of 3.5

Data are provided for the evaluation of the aerodynamic performance of a series of twisted and cambered delta wings designed for a Mach number of 3.5. Systematic force and pressure data are also presented for comparison with theory. Force tests were made at Mach numbers of 2.3, 3.0, 3.5, 4.0, and 4.6. Design lift coefficients of 0.0 and 0.1 were employed on the 55 deg and 68 deg sweep wings, and design lift coefficients of 0.0, 0.05, and 0.1 were employed on the 76 deg sweep wings. Pressure tests were conducted on the 55 deg and 76 deg sweep flat wings and on the 0.1 design lift coefficient 76 deg sweep wing. The results indicate that for the sweep angles tested, an increase in the zero-lift pitching-moment coefficient is the primary benefit of twist and camber at a Mach number of 3.5. Comparison of the experimental results with results obtained from several lift theories indicates that the Carlson-Middleton linear theory method gave the best overall agreement. The pressure data indicate, however, that there is a cancellation of error at high angle of attack where the lower surface pressures are significantly underpredicted over the inboard region of the wing and where the upper and lower surface pressures are overpredicted over the outboard region of the wing.

Sorrells, R. B., III↗
DOE OSTI2023

TDCOSMO. XII. Improved Hubble constant measurement from lensing time delays using spatially resolved stellar kinematics of the lens galaxy

Strong-lensing time delays enable measurement of the Hubble constant ($H_{0}$) independently of other traditional methods. The main limitation to the precision of time-delay cosmography is mass-sheet degeneracy (MSD). Some of the previous TDCOSMO analyses broke the MSD by making standard assumptions about the mass density profile of the lens galaxy, reaching 2% precision from seven lenses. However, this approach could potentially bias the $H_0$ measurement or underestimate the errors. For this work, we broke the MSD for the first time using spatially resolved kinematics of the lens galaxy in RXJ1131$-$1231 obtained from the Keck Cosmic Web Imager spectroscopy, in combination with previously published time delay and lens models derived from Hubble Space Telescope imaging. This approach allowed us to robustly estimate $H_0$, effectively implementing a maximally flexible mass model. Following a blind analysis, we estimated the angular diameter distance to the lens galaxy $D_{\rm d} = 865_{-81}^{+85}$ Mpc and the time-delay distance $D_{\Delta t} = 2180_{-271}^{+472}$ Mpc, giving $H_0 = 77.1_{-7.1}^{+7.3}$ km s$^{-1}$ Mpc$^{-1}$ - for a flat $\Lambda$ cold dark matter cosmology. The error budget accounts for all uncertainties, including the MSD inherent to the lens mass profile and the line-of-sight effects, and those related to the mass-anisotropy degeneracy and projection effects. Our new measurement is in excellent agreement with those obtained in the past using standard simply parametrized mass profiles for this single system ($H_0 = 78.3^{+3.4}_{-3.3}$ km s$^{-1}$ Mpc$^{-1}$) and for seven lenses ($H_0 = 74.2_{-1.6}^{+1.6}$ km s$^{-1}$ Mpc$^{-1}$), or for seven lenses using single-aperture kinematics and the same maximally flexible models used by us ($H_0 = 73.3^{+5.8}_{-5.8}$ km s$^{-1}$ Mpc$^{-1}$). This agreement corroborates the methodology of time-delay cosmography.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1983

The fundamental structure function of oscillator noise models

Continuous-time models of oscillator phase noise x(t) usually have stationary nth differences, for some n. The covariance structure of such a model can be characterized in the time domain by the structure function: D sub n (t;gamma sub 1, gamma sub 2) = E delta (n) sub gamma sub 1 x(s+t) delta(n) sub gamma sub 2 x (s). Although formulas for the special case D sub 2 (0;gamma,gamma) (the Allan variance times 2 gamma(2)) exist for power-law spectral models, certain estimation problems require a more complete knowledge of (0). Exhibited is a much simpler function of one time variable, D(t), from which (0) can easily be obtained from the spectral density by uncomplicated integrations. Believing that D(t) is the simplest function of time that holds the same information as (0), D(t) is called the fundamental structure function. D(t) is computed for several power-law spectral models. Two examples are D(t) = K/t/(3) for random walk FM, D(t) = Kt(2) 1n/t/ for flicker FM. Then, to demonstrate its use, a BASIC program is given that computes means and variances of two Allan variance estimators, one of which incorporates a method of frequency drift estimation and removal.

Greenhall, C. A.↗