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NASA NTRS1992

A small-scale turbulence model

A model for the small-scale structure of turbulence is reformulated in such a way that it may be conveniently computed. The model is an ensemble of randomly oriented structured two dimensional vortices stretched by an axially symmetric strain flow. The energy spectrum of the resulting flow may be expressed as a time integral involving only the enstrophy spectrum of the time evolving two-dimensional cross section flow, which may be obtained numerically. Examples are given in which a k(exp -5/3) spectrum is obtained by this method without using large wave number asymptotic analysis. The k(exp -5/3) inertial range spectrum is shown to be related to the existence of a self-similar enstrophy preserving range in the two-dimensional enstrophy spectrum. The results are insensitive to time dependence of the strain-rate, including even intermittent on-or-off strains.

Lundgren, T. S.↗
NASA NTRSDate not supplied

Satellite-Assisted Particulate Matter (SAPM) for the Models, In situ, and Remote Sensing of Aerosols (MIRA) Working Group

Models, In situ, and Remote sensing of Aerosols (MIRA) is an international working group that provides a forum for collaborations amongst these three atmospheric aerosol communities. MIRA consists of a collection of interdisciplinary and independently funded Topic/Project groups with clear goals and generally characterized by requests for additional scientific data. The Satellite-Assisted Particulate Matter (SAPM) Topic group focuses on the study of fine particulate matter (PM2.5), as it is a major contributor to air pollution and negatively impacts human health. Our group aims to provide intercomparisons of various methods and techniques for estimating surface PM2.5 assisted by satellite remote sensors (passive and active), global aerosol models, and in situ aerosol measurements. The overall motivation of our work is to enhance PM2.5 coverage beyond in situ ground stations, which can be limited. In this presentation, we provide an overview of various PM2.5 estimation approaches by current SAPM team members, as each approach has its own strengths and limitations. One approach uses near-surface aerosol extinction retrievals from the spaceborne Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) and an assumed value of mass extinction efficiency to derive PM2.5 concentrations (Toth et al 2019). A study using this method found a promising agreement (R=0.60; slope=0.89) with U.S. Environmental Protection Agency (EPA) in situ PM2.5 measurements over the contiguous United States (CONUS) for a 12-year (2007-2018) period (Toth et al 2022). In a different approach combining spaceborne lidar and a model, the Cloud Aerosol Transport System (CATS) lidar and Goddard Earth Observing System (GEOS) model are blended in an ensemble variational assimilation scheme to retrieve aerosol extinction and convert speciated mass concentrations to total PM2.5. A recent study (Matus et al in prep.) found agreement within 2 μg/m3 over the CONUS on the 2016 annual mean when compared to EPA ground-based measurements. Another study explored trends in city aerosol optical depth (AOD) from a spaceborne passive remote sensor, Moderate Resolution Imaging Spectroradiometer (MODIS), and found that these trends agree well with trends in city surface PM2.5, which provides confidence in the use of city AOD for city PM2.5 trend studies (Vohra et al 2021). In a follow-on study, steep and significant trends (~2.5 to 7.8% a-1) in AOD are found for several South Asian cities (e.g., Bangalore and Hyderabad), suggesting rapid growth in PM2.5 (Vohra et al 2022). Also in this presentation, we report results on our current collective SAPM study of India, a country that exhibits high levels of PM2.5 pollution. We show the temporal (i.e., seasonal) and spatial variability of PM2.5 over India using the CALIOP and CATS+model approaches, comparisons between the estimated PM2.5 from the retrieval schemes, and validation with in situ PM2.5 ground-based observations from the Central Pollution Control Board in India. This region provides us with an excellent case study to test our PM2.5 retrievals in heavily polluted scenes and over a complex and varying topography. The SAPM Topic group actively seeks international participants/collaborators in our group, including those working with in situ aerosol measurements (e.g., ACTRIS). We are interested in aerosol datasets in order to either improve our PM2.5 estimates from various approaches (e.g., using in situ mass scattering/absorption coefficient and aerosol hygroscopic properties for various aerosol species) and/or validate the PM2.5 estimates (e.g., using in situ ground-based PM2.5 concentrations).

Travis D Toth↗
NASA NTRSDate not supplied

Multiclass Flight Anomaly Detection Using Sensor Fusion Based on Dempster-Shafer Theory

As aviation systems in commercial operations continue to grow in complexity, the anomalies exhibited by these systems become more elaborate and difficult to detect. To address the challenge of detecting these complex anomalies, deep learning models have been used extensively in aviation anomaly detection studies, at the expense of end-user interpretability. Aiming to maintain the same level of interpretability as traditional threshold-exceedance methods, we continue our development of prediction models using ordinal patterns and their distributions throughout the flight. Specifically, this study extends our work into multiclass anomaly detection using sensor fusion based on Dempster-Shafer theory (DST), a second-order probability theory used to combine information from different sources of evidence. Our approach uses DST toreduce the uncertainty in the class predictions of an ensemble of classifiers. These classifiers rely on the similarity between flight data and class templates to make a prediction of the state of the aircraft. Our approach aims to take advantage of simple models trained on interpretable features (ordinal patterns) to correctly predict an anomaly and identify the flight dynamics linked to the anomaly. Our results show an improvement when using DST-based sensor fusion over simple majority voting. Additionally, our results provide insight into aircraft states linked to rare high-risk anomalies.

Risk detection↗
NASA NTRSDate not supplied

Multiclass Flight Anomaly Detection Using Sensor Fusion Based on Dempster-Shafer Theory

As aviation systems in commercial operations continue to grow in complexity, the anomalies exhibited by these systems become more elaborate and difficult to detect. To address the challenge of detecting these complex anomalies, deep learning models have been used extensively in aviation anomaly detection studies, at the expense of end-user interpretability. Aiming to maintain the same level of interpretability as traditional threshold-exceedance methods, we continue our development of prediction models using ordinal patterns and their distributions throughout the flight. Specifically, this study extends our work into multiclass anomaly detection using sensor fusion based on Dempster-Shafer theory (DST), a second-order probability theory used to combine information from different sources of evidence. Our approach uses DST to reduce the uncertainty in the class predictions of an ensemble of classifiers. These classifiers rely on the similarity between flight data and class templates to make a prediction of the state of the aircraft. Our approach aims to take advantage of simple models trained on interpretable features (ordinal patterns) to correctly predict an anomaly and identify the flight dynamics linked to the anomaly. Our results show an improvement when using DST-based sensor fusion over simple majority voting. Additionally, our results provide insight into aircraft states linked to rare high-risk anomalies.

Risk detection↗
NASA NTRS1983

A procedure for combining acoustically induced and mechanically induced loads (first passage failure design criterion)

The combined load statistics are developed by taking the acoustically induced load to be a random population, assumed to be stationary. Each element of this ensemble of acoustically induced loads is assumed to have the same power spectral density (PSD), obtained previously from a random response analysis employing the given acoustic field in the STS cargo bay as a stationary random excitation. The mechanically induced load is treated as either (1) a known deterministic transient, or (2) a nonstationary random variable of known first and second statistical moments which vary with time. A method is then shown for determining the probability that the combined load would, at any time, have a value equal to or less than a certain level. Having obtained a statistical representation of how the acoustic and mechanical loads are expected to combine, an analytical approximation for defining design levels for these loads is presented using the First Passage failure criterion.

Crowe, D. R.↗
NASA NTRS2006

Comet Mineralogy as Inferred from Infrared Spectra of Comets

For most comets, infrared (IR) spectroscopy (remote sensing) is the method through which we diagnose the mineralogy and size distribution of dust in their comae. The shape and contrast of the IR spectral features depend on the particle size: optically active minerals (absorbing of visible and near-IR solar photons) and submicron solid grains or highly porous (> 90% vacuum) grains primarily contribute to the shapes of the observed resonances. Comet mineralogies typically are determined by fitting thermal emission models of ensembles of discrete mineral grains to observed IR spectral energy distributions. The absorptivities (Q-abs) and scattering efficiencies (Q-scat) of the discrete mineral grains are computed using Mie scattering, Maxwell-Garnet mixing, Discrete Dipole Approximation, and Multi-Layered Sphere codes. These techniques when applied to crystalline minerals, specifically olivine (Mg_x, Fe_1-x)2 Si04, x>0.9, require the use of ellipsoidal shaped particles with elongated axial ratios or hollow spheres to produce the shapes of the resonances observed both from comet comae and laboratory samples. The wavelength positions of the distinct resonances from submicron-radii crystalline silicates, as well as their thermal equilibrium temperatures, constrain the crystalline olivine to have a relatively high Mg-content (x>0.9, or Fo>90). Only resonances computed for submicron Mg-rich crystalline olivine and crystalline orthopyroxene match the observed IR spectral features. However, this has led to the interpretation that micron-radii and larger crystals are absent from comet comae. Furthermore, the mass fraction of silicate crystals is dependent upon whether just the submicron portion of the size distribution is being compared or the submicron crystals compare to the aggregates of porous amorphous silicates that are computationally tractable as porous spheres. We will discuss the Deep Impact results as examples of these challenges to interpreting mid-IR spectra of short-period comets.

Wooden, Diane H.↗
NASA NTRS1989

Determination of longitudinal aerodynamic derivatives using flight data from an icing research aircraft

A flight test was performed with the NASA Lewis Research Center's DH-6 icing research aircraft. The purpose was to employ a flight test procedure and data analysis method, to determine the accuracy with which the effects of ice on aircraft stability and control could be measured. For simplicity, flight testing was restricted to the short period longitudinal mode. Two flights were flown in a clean (baseline) configuration, and two flights were flown with simulated horizontal tail ice. Forty-five repeat doublet maneuvers were performed in each of four test configurations, at a given trim speed, to determine the ensemble variation of the estimated stability and control derivatives. Additional maneuvers were also performed in each configuration, to determine the variation in the longitudinal derivative estimates over a wide range of trim speeds. Stability and control derivatives were estimated by a Modified Stepwise Regression (MSR) technique. A measure of the confidence in the derivative estimates was obtained by comparing the standard error for the ensemble of repeat maneuvers, to the average of the estimated standard errors predicted by the MSR program. A multiplicative relationship was determined between the ensemble standard error, and the averaged program standard errors. In addition, a 95 percent confidence interval analysis was performed for the elevator effectiveness estimates, C sub m sub delta e. This analysis identified the speed range where changes in C sub m sub delta e could be attributed to icing effects. The magnitude of icing effects on the derivative estimates were strongly dependent on flight speed and aircraft wing flap configuration. With wing flaps up, the estimated derivatives were degraded most at lower speeds corresponding to that configuration. With wing flaps extended to 10 degrees, the estimated derivatives were degraded most at the higher corresponding speeds. The effects of icing on the changes in longitudinal stability and control derivat

Ranaudo, R. J.↗
NASA NTRS2002

Protein Kinase Classification with 2866 Hidden Markov Models and One Support Vector Machine

The main application considered in this paper is predicting true kinases from randomly permuted kinases that share the same length and amino acid distributions as the true kinases. Numerous methods already exist for this classification task, such as HMMs, motif-matchers, and sequence comparison algorithms. We build on some of these efforts by creating a vector from the output of thousands of structurally based HMMs, created offline with Pfam-A seed alignments using SAM-T99, which then must be combined into an overall classification for the protein. Then we use a Support Vector Machine for classifying this large ensemble Pfam-Vector, with a polynomial and chisquared kernel. In particular, the chi-squared kernel SVM performs better than the HMMs and better than the BLAST pairwise comparisons, when predicting true from false kinases in some respects, but no one algorithm is best for all purposes or in all instances so we consider the particular strengths and weaknesses of each.

Weber, Ryan↗
NASA NTRS2011

Lessons Learned from Assimilating Altimeter Data into a Coupled General Circulation Model with the GMAO Augmented Ensemble Kalman Filter

Satellite altimetry measurements have provided global, evenly distributed observations of the ocean surface since 1993. However, the difficulties introduced by the presence of model biases and the requirement that data assimilation systems extrapolate the sea surface height (SSH) information to the subsurface in order to estimate the temperature, salinity and currents make it difficult to optimally exploit these measurements. This talk investigates the potential of the altimetry data assimilation once the biases are accounted for with an ad hoc bias estimation scheme. Either steady-state or state-dependent multivariate background-error covariances from an ensemble of model integrations are used to address the problem of extrapolating the information to the sub-surface. The GMAO ocean data assimilation system applied to an ensemble of coupled model instances using the GEOS-5 AGCM coupled to MOM4 is used in the investigation. To model the background error covariances, the system relies on a hybrid ensemble approach in which a small number of dynamically evolved model trajectories is augmented on the one hand with past instances of the state vector along each trajectory and, on the other, with a steady state ensemble of error estimates from a time series of short-term model forecasts. A state-dependent adaptive error-covariance localization and inflation algorithm controls how the SSH information is extrapolated to the sub-surface. A two-step predictor corrector approach is used to assimilate future information. Independent (not-assimilated) temperature and salinity observations from Argo floats are used to validate the assimilation. A two-step projection method in which the system first calculates a SSH increment and then projects this increment vertically onto the temperature, salt and current fields is found to be most effective in reconstructing the sub-surface information. The performance of the system in reconstructing the sub-surface fields is particularly impressive for temperature, but not as satisfactory for salt.

Keppenne, Christian↗
NASA NTRS2016

Prediction of Weather Impacts on Airport Arrival Meter Fix Capacity

This paper introduces a data driven model for predicting airport arrival capacity with a look-ahead time 2-8 hour forecast. The model is suitable for air traffic flow management by explicitly investigating the impact of convective weather on airport arrival meter fix throughput. Estimation of the arrival airport capacity under arrival meter fix flow constraints due to severe weather is an important part of Air Traffic Management (ATM). Airport arrival capacity can be reduced if one or more airport arrival meter fixes are partially or completely blocked by convective weather. When the predicted airport arrival demands exceed the predicted available airport's arrival capacity for a sustained period, Ground Delay Program (GDP) operations will be triggered by ATM system. Serious imbalances between demand and capacity occur most frequently when the airport capacity is severely degraded due to either bad airport terminal surface weather or inclement convective weather around airport arrival fixes. A model that predicts the weather-impacted airport arrival meter fix throughput may help ATM personnel to plan GDP operations more efficiently. This paper identifies the characteristics of air traffic flow across arrival meter fixes at Newark Liberty International Airport (EWR). The proposed approach, based on machine-learning methods, is developed to predict the weather impacted EWR arrival Meter Fix (MF) throughput. Sector forecast coverage is used to envision the weather impact on airport arrival MF flow, and the validation is accomplished by using Convective Weather Avoidance Model (CWAM) 0.5 to 2-hour and Collaborative Convective Forecast Product (CCFP) 4 to 8-hour look-ahead forecast data for the period of April-September in 2014. Furthermore, the regression tree ensemble learning of random forests approach for translating a sector forecast coverage model to an EWR arrival meter fix throughput model is examined. The results suggest that ATM decision makers in charge of MF flow control and GDP planning may benefit from adopting the airport arrival meter capacity prediction models to estimate the inclement weather impacts.

Wang, Yao X.↗
NASA NTRS2020

Subseasonal-to-Seasonal Hindcast Skill Assessment of Ridging Events Related to Drought Over the Western United States

Persistent atmospheric ridging events centered near the western United States are associated with widespread precipitation deficits and meteorological drought. Due to the relatively low skill of- dynamical models in forecasting precipitation on subseasonal‐to‐seasonal (S2S) time scales across this region, forecasts of ridging are explored in this study as a potential bridge for early warning drought prediction. To assess skill, we evaluate deterministic and probabilistic S2S hindcasts (out to 6 week lead time) of ridging events and geopotential height anomalies over the western United States in five ensemble hindcast systems. Prediction skill for ridging events is shown to be highly variable across models. For some models, longer‐time‐averaged patterns of geopotential height anomalies across the first 6 weeks can be skillfully simulated, when evaluated probabilistically against climatology. The most skillful models show modest skill in forecasting above normal ridging occurrences at lead times of Weeks 3–4 and Weeks 5–6,with some sensitivity to the specific ridge location and method for determining skill. Using the European Centre for Medium‐Range Weather Forecasts (ECMWF) model as a case study, longer lead time forecast busts are shown to often occur under extended periods of highly active La Nina‐like tropical convection. Model errors at simulating these tropical convection features may have a disproportionately large impact and degrade downstream forecasts over the western United States. Despite the documented model shortcomings, our results highlight an opportunity to target skillful ridging forecasts from dynamical models for improving early warning drought forecasting at S2S lead times.

S2S↗
NASA NTRS2000

Low-dimensional Representation of Error Covariance

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time-dependent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady-state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting nonmodal transient growth. Failure to observe growing modes leads to increased steady-state analysis error variances. Leading eigenvectors of the steady-state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady-state analysis error covariance matrix.

Tippett, Michael K.↗
NASA NTRS1982

Random element method for numerical modeling of diffusional processes

The random element method is a generalization of the random vortex method that was developed for the numerical modeling of momentum transport processes as expressed in terms of the Navier-Stokes equations. The method is based on the concept that random walk, as exemplified by Brownian motion, is the stochastic manifestation of diffusional processes. The algorithm based on this method is grid-free and does not require the diffusion equation to be discritized over a mesh, it is thus devoid of numerical diffusion associated with finite difference methods. Moreover, the algorithm is self-adaptive in space and explicit in time, resulting in an improved numerical resolution of gradients as well as a simple and efficient computational procedure. The method is applied here to an assortment of problems of diffusion of momentum and energy in one-dimension as well as heat conduction in two-dimensions in order to assess its validity and accuracy. The numerical solutions obtained are found to be in good agreement with exact solution except for a statistical error introduced by using a finite number of elements, the error can be reduced by increasing the number of elements or by using ensemble averaging over a number of solutions.

Ghoniem, A. F.↗
NASA NTRS2011

Estimation of Smoothing Error in SBUV Profile and Total Ozone Retrieval

Data from the Nimbus-4, Nimbus-7 Solar Backscatter Ultra Violet (SBUV) and seven of the NOAA series of SBUV/2 instruments spanning 41 years are being reprocessed using V8.6 algorithm. The data are scheduled to be released by the end of August 2011. An important focus of the new algorithm is to estimate various sources of errors in the SBUV profiles and total ozone retrievals. We discuss here the smoothing errors that describe the components of the profile variability that the SBUV observing system can not measure. The SBUV(/2) instruments have a vertical resolution of 5 km in the middle stratosphere, decreasing to 8 to 10 km below the ozone peak and above 0.5 hPa. To estimate the smoothing effect of the SBUV algorithm, the actual statistics of the fine vertical structure of ozone profiles must be known. The covariance matrix of the ensemble of measured ozone profiles with the high vertical resolution would be a formal representation of the actual ozone variability. We merged the MLS (version 3) and sonde ozone profiles to calculate the covariance matrix, which in general case, for single profile retrieval, might be a function of the latitude and month. Using the averaging kernels of the SBUV(/2) measurements and calculated total covariance matrix one can estimate the smoothing errors for the SBUV ozone profiles. A method to estimate the smoothing effect of the SBUV algorithm is described and the covariance matrixes and averaging kernels are provided along with the SBUV(/2) ozone profiles. The magnitude of the smoothing error varies with altitude, latitude, season and solar zenith angle. The analysis of the smoothing errors, based on the SBUV(/2) monthly zonal mean time series, shows that the largest smoothing errors were detected in the troposphere and might be as large as 15-20% and rapidly decrease with the altitude. In the stratosphere above 40 hPa the smoothing errors are less than 5% and between 10 and 1 hPa the smoothing errors are on the order of 1%. We validate our estimated smoothing errors by comparing the SBUV ozone profiles with other ozone profiling sensors.

Kramarova, N. A.↗
NASA NTRS2010

Stretch Reflex as a Simple Measure to Evaluate the Efficacy of Potential Flight Countermeasures Using the Bed Rest Environment

INTRODUCTION: Spaceflight is acknowledged to have significant effects on the major postural muscles. However, it has been difficult to separate the effects of ascending somatosensory changes caused by the unloading of these muscles during flight from changes in sensorimotor function caused by a descending vestibulo-cerebellar response to microgravity. It is hypothesized that bed rest is an adequate model to investigate postural muscle unloading given that spaceflight and bed rest may produce similar results in both nerve axon and muscle tissue. METHODS: To investigate this hypothesis, stretch reflexes were measured on 18 subjects who spent 60 to 90 days in continuous 6 head-down bed rest. Using a motorized system capable of rotating the foot around the ankle joint (dorsiflexion) through an angle of 10 deg at a peak velocity of approximately 250 deg/sec, a stretch reflex was recorded from the subject's left triceps surae muscle group. Using surface electromyography, about 300 reflex responses were obtained and ensemble-averaged on 3 separate days before bed rest, 3 to 4 times in bed, and 3 times after bed rest. The averaged responses for each test day were examined for reflex latency and conduction velocity (CV) across gender and compared with spaceflight data. RESULTS: Although no gender differences were found, bed rest induced changes in reflex latency and CV similar to the ones observed during spaceflight. Also, a relationship between CV and loss of muscle strength in the lower leg was observed for most bed rest subjects. CONCLUSION: Even though bed rest (limb unloading) alone may not mimic all of the synaptic and muscle tissue loss that is observed as a result of spaceflight, it can serve as a working analog of flight for the evaluation of potential countermeasures that may be beneficial in mitigating unwanted changes in the major postural muscles that are observed post flight.

Cerisano, J. M.↗
NASA NTRS1978

Zodiacal light as an indicator of interplanetary dust

The most striking feature of the night sky in the tropics is the zodiacal light, which appears as a cone in the west after sunset and in the east before sunrise. It is caused by sunlight scattered or absorbed by particles in the interplanetary medium. The zodiacal light is the only source of information about the integrated properties of the whole ensemble of interplanetary dust. The brightness and polarization in different directions and at different colors can provide information on the optical properties and spatial distribution of the scattering particles. The zodiacal light arises from two independent physical processes related to the scattering of solar continuum radiation by interplanetary dust and to thermal emission which arises from solar radiation that is absorbed by interplanetary dust and reemitted mainly at infrared wavelengths. Attention is given to observational parameters of zodiacal light, the methods of observation, errors and absolute calibration, and the observed characteristics of zodiacal light.

Weinberg, J. L.↗
NASA NTRS2010

Forecasting Lightning Threat Using WRF Proxy Fields

Objectives: Given that high-resolution WRF forecasts can capture the character of convective outbreaks, we seek to: 1. Create WRF forecasts of LTG threat (1-24 h), based on 2 proxy fields from explicitly simulated convection: - graupel flux near -15 C (captures LTG time variability) - vertically integrated ice (captures LTG threat area). 2. Calibrate each threat to yield accurate quantitative peak flash rate densities. 3. Also evaluate threats for areal coverage, time variability. 4. Blend threats to optimize results. 5. Examine sensitivity to model mesh, microphysics. Methods: 1. Use high-resolution 2-km WRF simulations to prognose convection for a diverse series of selected case studies. 2. Evaluate graupel fluxes; vertically integrated ice (VII). 3. Calibrate WRF LTG proxies using peak total LTG flash rate densities from NALMA; relationships look linear, with regression line passing through origin. 4. Truncate low threat values to make threat areal coverage match NALMA flash extent density obs. 5. Blend proxies to achieve optimal performance 6. Study CAPS 4-km ensembles to evaluate sensitivities.

McCaul, E. W., Jr.↗
NASA NTRS1992

Least Reliable Bits Coding (LRBC) for high data rate satellite communications

An analysis and discussion of a bandwidth efficient multi-level/multi-stage block coded modulation technique called Least Reliable Bits Coding (LRBC) is presented. LRBC uses simple multi-level component codes that provide increased error protection on increasingly unreliable modulated bits in order to maintain an overall high code rate that increases spectral efficiency. Further, soft-decision multi-stage decoding is used to make decisions on unprotected bits through corrections made on more protected bits. Using analytical expressions and tight performance bounds it is shown that LRBC can achieve increased spectral efficiency and maintain equivalent or better power efficiency compared to that of Binary Phase Shift Keying (BPSK). Bit error rates (BER) vs. channel bit energy with Additive White Gaussian Noise (AWGN) are given for a set of LRB Reed-Solomon (RS) encoded 8PSK modulation formats with an ensemble rate of 8/9. All formats exhibit a spectral efficiency of 2.67 = (log2(8))(8/9) information bps/Hz. Bit by bit coded and uncoded error probabilities with soft-decision information are determined. These are traded with with code rate to determine parameters that achieve good performance. The relative simplicity of Galois field algebra vs. the Viterbi algorithm and the availability of high speed commercial Very Large Scale Integration (VLSI) for block codes indicates that LRBC using block codes is a desirable method for high data rate implementations.

Vanderaar, Mark↗