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All sourcesNASA NTRSDOE OSTI
NASA NTRS2012

Radiative Transfer Through Clouds and Its Applications in Support of the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2) Mission

The Greenland and Antarctic ice sheets, which contain enough ice to raise sea level by about 7 and 60 m, respectively, are losing mass at an increasing rate. To acquire continuous information of the cryosphere, after the Ice, Cloud, and land Elevation Satellite (ICESat) (2003-2010), NASA is actively planning for the ICESat-2 mission. Both ICESat and ICESat-2 are space-borne lidar altimetry systems. The systems measure the time of flight of the arriving photons that are reflected by the surface to deduce the elevation of the underlying terrain. As one of NASA's top priority missions, ICESat-2 is scheduled to launch in 2016. One of the major science goals of ICESat-2 is to quantify the ice sheet mass balance to determine its contributions to the sea level change and its impacts on ocean circulation (Abdalati et al. 2010). Compared to ICESat, which operates at 40 Hz and records the reflected laser energy as a waveform, the significantly improved ICESat-2 lidar employs a 532 nm micro-pulse photon counting system that operates at a high frequency of 10kHz with single photon detectability (Yang et al. 2012). To achieve its science goals, ICESat-2 requires the ability of detecting the elevation change with an accuracy of 0.2 cm/year over the entire ice sheet. Since every photon emitted by the lidar system will travel through the atmosphere, clouds can certainly affect the flight time of the arriving photons. Forward scattering by cloud particles increases the photon path length, thus resulting in biases in ice sheet elevation measurements known as atmospheric path delay (Duta et al. 2001, Yang et al. 2010, 2011). To ensure the accuracy of ICESat-2 surface elevation measurements, it is critical to understand how clouds would affect the travel time of arriving photons. In this talk, we will first present a framework that simulates the behavior of a space-borne 532 mn micro-pulse photon counting lidar in cloudy and clear atmospheres. To investigate the process of laser propagation through clouds, a 3-D Monte Carlo radiative transfer model is used to simulate the photon path distribution and the Poisson distribution is adopted for the number of photon returns. Since the photon counting system only registers the time of the first arriving photon within the detector "dead time", the retrieved average surface elevation tends to bias towards higher values. This is known as the first photon bias. With the scenarios simulated here, the first photon bias for clear sky is about 6.5 cm. Clouds affect surface altimetry in two ways: (1) cloud attenuation lowers the average number of arriving photons and hence reduces the first photon bias; (2) cloud forward scattering increases the photon path length and makes the surface appear further away from the satellite. Compared to clear sky, the average surface elevation detected by the photon counting system for cloudy sky with optical depth 1.0 is 4.0 to 6.0 cm lower for the simulations conducted. The effect of surface roughness on the accuracy of elevation retrievals will also discussed.

Yang, Yuekui↗
NASA NTRS1991

Modelling the spatial distribution of QSO absorption lines

A formalism has been developed for testing specific models for the three-dimensional spatial distributions of Ly-alpha clouds against the line-of-sight observations. The formalism can be used to compute the redshift density of lines and the distribution of line-interval sizes along a line-of-sight for comparison with the observations. A few simple models are considered: a Poisson distribution of Ly-alpha clouds, clouds arising from density fluctuations with a white-noise power spectrum, clouds distributed only on shells of constant comoving radii, clouds restricted to shells that are expanding in a self-similar fashion, and a Poisson distribution of clouds in the 'walls' between small-scale voids.

Babul, Arif↗
NASA NTRSDate not supplied

Redundancy: How Many Unreliable Spares are Needed for High Reliability and Confidence on a Time Limited Mission?

This paper investigates the number of redundant units needed to achieve high reliability with high confidence. The approach applies to the case where the unit failure rate is too high for a single unit to provide the required reliability over the mission duration. To achieve high reliability, the design then uses N redundant units, one operating unit and N – 1 spares. If the unit failure rate is f, the mission length is L, and f * L is small (not the case assumed here), the unit failure probability over the mission duration is F1 = f * L << 1. In this case, the probability that all N units will fail is FN = F1N, and the needed N = LN(FN)/LN(F1). For the case of large f * L assumed here, F1 = f * L > 1, and F1 is the expected number of failures during the mission. The needed redundancy, N, to achieve the specified N unit reliability, FN, can be computed using the cumulative Poisson distribution with mean equal to F1. The number of spares, N - 1, is increased until the probability - that the total number of failures will be less than N -1 - achieves the required reliability. The confidence that this reliability can be achieved can be computed using the cumulative Poisson distribution or the chi-square distribution. Since the measured unit failure rate, f, has some uncertainty, the confidence that the rate is not lower than the actual failure rate and the required reliability is not overestimated is about 50%. Adding more redundant units increases the confidence that the required reliability, FN, will be achieved. For a fixed number of redundant units, the expected reliability and confidence can be traded off, since lower reliability goals have higher confidence in being achieved. Both the required reliability and confidence can be specified initially and the needed number of redundant units computed using the measured failure rate. The unit failure rate is determined by initial reliability growth testing to remove design errors and to better estimate the final constant failure rate. Reducing the failure rate and reducing its variance both reduce the number of redundant units needed for the required reliability and confidence. Since the total cost is the sum of the costs of the units and of the testing, there is an optimum test time that produces minimum cost.

Harry W. Jones↗
NASA NTRS2024

Redundancy: How Many Unreliable Spares are Needed for High Reliability and Confidence?

This paper investigates the number of redundant units needed to achieve high reliability with high confidence. The approach is developed for the case when the system failure rate is too high for a single unit to provide the required reliability over the mission duration. To achieve high reliability, N redundant units can be used, one operating unit and N – 1 spares. If the unit failure rate is f, the mission length is L, and f * L is small (not the case assumed here), the unit failure probability over the mission duration is F1 = f * L << 1. In this case, the probability that all N units will fail is Ffail = F1 N , and the needed redundancy N = LN(F)/LN(F1). For the case of large f * L assumed here, F1 = f * L > 1, and F1 is the expected number of failures during the mission. (When F1 = f * L << 1, F1 is the probability that a unit will fail during the mission. When F1 = f * L > 1, F1 is the expected number of failures during the mission.) The needed redundancy, N, to achieve the required N redundant unit reliability, FN, can be computed using the cumulative Poisson distribution with mean equal to F1. The number of spares, N - 1, is increased until the probability - that the total number of failures will be less than N -1 - is equal to the required reliability. The confidence that this reliability can be achieved can be computed using the cumulative Poisson distribution or the chi-square distribution. Since the measured unit failure rate, f, has some probabilistic uncertainty, the actual failure rate will be randomly higher or lower. This means that the reliability of the N redundant systems will be overestimated about half the time. Adding more redundant units increases the confidence that the required reliability will be achieved. For a fixed number of redundant units, the expected reliability and confidence can be traded off, since lower reliability goals will be achieved with higher confidence. Both the desired reliability and confidence can be specified as initial requirements and the needed number of redundant units estimated using the measured failure rate.

Redundancy↗
NASA NTRS1995

Multiparameter linear least-squares fitting to Poisson data one count at a time

A standard problem in gamma-ray astronomy data analysis is the decomposition of a set of observed counts, described by Poisson statistics, according to a given multicomponent linear model, with underlying physical count rates or fluxes which are to be estimated from the data. Despite its conceptual simplicity, the linear least-squares (LLSQ) method for solving this problem has generally been limited to situations in which the number n(sub i) of counts in each bin i is not too small, conventionally more than 5-30. It seems to be widely believed that the failure of the LLSQ method for small counts is due to the failure of the Poisson distribution to be even approximately normal for small numbers. The cause is more accurately the strong anticorrelation between the data and the wieghts w(sub i) in the weighted LLSQ method when square root of n(sub i) instead of square root of bar-n(sub i) is used to approximate the uncertainties, sigma(sub i), in the data, where bar-n(sub i) = E(n(sub i)), the expected value of N(sub i). We show in an appendix that, avoiding this approximation, the correct equations for the Poisson LLSQ (PLLSQ) problems are actually identical to those for the maximum likelihood estimate using the exact Poisson distribution. We apply the method to solve a problem in high-resolution gamma-ray spectroscopy for the JPL High-Resolution Gamma-Ray Spectrometer flown on HEAO 3. Systematic error in subtracting the strong, highly variable background encountered in the low-energy gamma-ray region can be significantly reduced by closely pairing source and background data in short segments. Significant results can be built up by weighted averaging of the net fluxes obtained from the subtraction of many individual source/background pairs. Extension of the approach to complex situations, with multiple cosmic sources and realistic background parameterizations, requires a means of efficiently fitting to data from single scans in the narrow (approximately = 1.2 keV, HEAO 3) energy channels of a Ge spectrometer, where the expected number of counts obtained per scan may be very low. Such an analysis system is discussed and compared to the method previously used.

Wheaton, Wm. A.↗
NASA NTRS1990

Compound estimation procedures in reliability

At NASA, components and subsystems of components in the Space Shuttle and Space Station generally go through a number of redesign stages. While data on failures for various design stages are sometimes available, the classical procedures for evaluating reliability only utilize the failure data on the present design stage of the component or subsystem. Often, few or no failures have been recorded on the present design stage. Previously, Bayesian estimators for the reliability of a single component, conditioned on the failure data for the present design, were developed. These new estimators permit NASA to evaluate the reliability, even when few or no failures have been recorded. Point estimates for the latter evaluation were not possible with the classical procedures. Since different design stages of a component (or subsystem) generally have a good deal in common, the development of new statistical procedures for evaluating the reliability, which consider the entire failure record for all design stages, has great intuitive appeal. A typical subsystem consists of a number of different components and each component has evolved through a number of redesign stages. The present investigations considered compound estimation procedures and related models. Such models permit the statistical consideration of all design stages of each component and thus incorporate all the available failure data to obtain estimates for the reliability of the present version of the component (or subsystem). A number of models were considered to estimate the reliability of a component conditioned on its total failure history from two design stages. It was determined that reliability estimators for the present design stage, conditioned on the complete failure history for two design stages have lower risk than the corresponding estimators conditioned only on the most recent design failure data. Several models were explored and preliminary models involving bivariate Poisson distribution and the Consael Process (a bivariate Poisson process) were developed. Possible short comings of the models are noted. An example is given to illustrate the procedures. These investigations are ongoing with the aim of developing estimators that extend to components (and subsystems) with three or more design stages.

Barnes, Ron↗
NASA NTRS1991

Avalanche photodiode photon counting receivers for space-borne lidars

Avalanche photodiodes (APD) are studied for uses as photon counting detectors in spaceborne lidars. Non-breakdown APD photon counters, in which the APD's are biased below the breakdown point, are shown to outperform: (1) conventional APD photon counters biased above the breakdown point; (2) conventional APD photon counters biased above the breakdown point; and (3) APD's in analog mode when the received optical signal is extremely weak. Non-breakdown APD photon counters were shown experimentally to achieve an effective photon counting quantum efficiency of 5.0 percent at lambda = 820 nm with a dead time of 15 ns and a dark count rate of 7000/s which agreed with the theoretically predicted values. The interarrival times of the counts followed an exponential distribution and the counting statistics appeared to follow a Poisson distribution with no after pulsing. It is predicted that the effective photon counting quantum efficiency can be improved to 18.7 percent at lambda = 820 nm and 1.46 percent at lambda = 1060 nm with a dead time of a few nanoseconds by using more advanced commercially available electronic components.

Sun, Xiaoli↗
NASA NTRS1973

The asymptotic distribution of maxima in bivariate samples

The joint distribution (as n tends to infinity) of the maxima of a sample of n independent observations of a bivariate random variable (X,Y) is studied. A method is developed for deriving the asymptotic distribution of the maxima, assuming that X and Y possess asymptotic extreme-value distributions and that the probability element dF(x,y) can be expanded in a canonical series. Applied both to the bivariate normal distribution and to the bivariate gamma and compound correlated bivariate Poisson distributions, the method shows that maxima from all these distributions are asymptotically uncorrelated.

Campbell, J. W.↗
NASA NTRS1984

The distribution of catchment coverage by stationary rainstorms

The occurrence of wetted rainstorm area within a catchment is modeled as a Poisson arrival process in which each storm is composed of stationary, nonoverlapping, independent random cell clusters whose centers are Poisson-distributed in space and whose areas are fractals. The two Poisson parameters and hence the first two moments of the wetted fraction are derived in terms of catchment average characteristics of the (observable) station precipitation. The model is used to estimate spatial properties of tropical air mass thunderstorms on six tropical catchments in the Sudan.

Eagleson, P. S.↗
NASA NTRS1990

Newton/Poisson-Distribution Program

NEWTPOIS, one of two computer programs making calculations involving cumulative Poisson distributions. NEWTPOIS (NPO-17715) and CUMPOIS (NPO-17714) used independently of one another. NEWTPOIS determines Poisson parameter for given cumulative probability, from which one obtains percentiles for gamma distributions with integer shape parameters and percentiles for X(sup2) distributions with even degrees of freedom. Used by statisticians and others concerned with probabilities of independent events occurring over specific units of time, area, or volume. Program written in C.

Bowerman, Paul N.↗
NASA NTRS1992

Detecting change as it occurs

Traditionally climate changes have been detected from long series of observations and long after they have happened. Our 'inverse sequential' procedure, for detecting change as soon as it occurs, describes the existing or most recent data by their frequency distribution. Its parameter(s) are estimated both from the existing set of observations and from the same set augmented by 1,2,....j new observations. Individual-value probability products ('likelihoods') are used to form ratios which yield two probabilities for erroneously accepting the existing parameter(s) as valid for the augmented data set, and vice versa. A genuine parameter change is signaled when these probabilities (or a more stable compound probability) show a progressive decrease. New parameter values can then be estimated from the new observations alone using standard statistical techniques. The inverse sequential procedure will be illustrated for global annual mean temperatures (assumed normally distributed), and for annual numbers of North Atlantic hurricanes (assumed to represent Poisson distributions). The procedure was developed, but not yet tested, for linear or exponential trends, and for chi-squared means or degrees of freedom, a special measure of autocorrelation.

Radok, Uwe↗
NASA NTRS1975

Cost performance satellite design using queueing theory

A modified Poisson arrival, infinite server queuing model is used to determine the effects of limiting the number of broadcast channels (C) of a direct broadcast satellite used for public service purposes (remote health care, education, etc.). The model is based on the reproductive property of the Poisson distribution. A difference equation has been developed to describe the change in the Poisson parameter. When all initially delayed arrivals reenter the system a (C plus 1) order polynomial must be solved to determine the effective value of the Poisson parameter. When less than 100% of the arrivals reenter the system the effective value must be determined by solving a transcendental equation. The model was used to determine the minimum number of channels required for a disaster warning satellite without degradation in performance. Results predicted by the queuing model were compared with the results of digital simulation.

Hein, G. F.↗
DOE OSTI2024

Regularized Differentiation for Bioburden Density Estimation in Planetary Protection

In this paper, we propose and investigate the performance of two novel shrinkage estimators for bioburden density estimation in planetary protection. The estimators are based on the regularized differentiation of a cumulative count of colony forming units collected throughout the data collecting session or the life cycle of the entire mission. The regularized differentiation recasts the problem of bioburden density estimation as a linear least squares problem. The least squares problem is then solved through regularization techniques, such as truncated singular value decomposition and penalized least squares. The regularization is necessary to avoid noise amplification during the differentiation of noisy data. The two regularization estimators are compared with four other commonly used estimators to simultaneously evaluate the means of multivariable independent Poisson distributions: the maximum likelihood, noninformative Bayes estimator with Jeffreys prior, Empirical Bayes using conjugate gamma-Poisson model with gamma parameters selected by method of moments, and the Clevenson-Zidek estimator. It is shown through computer-simulated data that the regularized differentiation based on ridge regression has the smallest mean-squared error among all estimators. The analysis of shrinkage mechanism implemented by regularized differentiation is performed, and it is shown that the regularized differentiation amounts to performing a weighted averaging of all the samples. The weights are determined by the regularization parameter automatically selected by the L-curve technique. Since the method of least squares makes no distributional assumptions about the data, it presents an attractive technique for bioburden density estimation when there are concerns about the misspecification of the distributional model. The paper concludes with the analysis of the bioburden data collected during InSight mission and directions for future work.

97 - MATHEMATICS AND COMPUTING↗
NASA NTRS2005

A Methodology for Quantifying Certain Design Requirements During the Design Phase

A methodology for developing and balancing quantitative design requirements for safety, reliability, and maintainability has been proposed. Conceived as the basis of a more rational approach to the design of spacecraft, the methodology would also be applicable to the design of automobiles, washing machines, television receivers, or almost any other commercial product. Heretofore, it has been common practice to start by determining the requirements for reliability of elements of a spacecraft or other system to ensure a given design life for the system. Next, safety requirements are determined by assessing the total reliability of the system and adding redundant components and subsystems necessary to attain safety goals. As thus described, common practice leaves the maintainability burden to fall to chance; therefore, there is no control of recurring costs or of the responsiveness of the system. The means that have been used in assessing maintainability have been oriented toward determining the logistical sparing of components so that the components are available when needed. The process established for developing and balancing quantitative requirements for safety (S), reliability (R), and maintainability (M) derives and integrates NASA s top-level safety requirements and the controls needed to obtain program key objectives for safety and recurring cost (see figure). Being quantitative, the process conveniently uses common mathematical models. Even though the process is shown as being worked from the top down, it can also be worked from the bottom up. This process uses three math models: (1) the binomial distribution (greaterthan- or-equal-to case), (2) reliability for a series system, and (3) the Poisson distribution (less-than-or-equal-to case). The zero-fail case for the binomial distribution approximates the commonly known exponential distribution or "constant failure rate" distribution. Either model can be used. The binomial distribution was selected for modeling flexibility because it conveniently addresses both the zero-fail and failure cases. The failure case is typically used for unmanned spacecraft as with missiles.

Adams, Timothy↗
NASA NTRS1975

Bayesian estimation - A sensitivity analysis

The robustness of the assigned prior distribution in a Bayesian estimation problem is examined. A Bayesian analysis for a stochastic intensity parameter of a Poisson distribution is summarized in which the natural conjugate is assigned as the prior distribution of the random parameter. The sensitivity analysis is carried out by assuming the existence of a true prior which is different in form from that of the assigned prior distribution. By using mean-squared error as a measure of performance, the ensuing Bayes decision function is compared to the corresponding minimum variance unbiased estimator. Results indicate that the Bayes estimator is largely robust to deviations from the assigned prior and remains squared-error superior to the MVU type within a broad region.

Canavos, G. C.↗
NASA NTRS1993

The spatial distribution of coronae and related features on Venus

Coronae are large quasi-circular geologica features that are common on Venus. They appear to be the surface tectonic and volcanic expressions of mantle diapirs that have impinged on the underside of the venusian lithosphere. We have investigated the spatial distribution of 335 coronae and related features identified in Magellan radar data. It is more clustered than a Poisson distribution, with a statistical certainty of more than 99%. It is dominated by a single large cluster centered near the equator at about 245 deg longitude. The features are preferentially found at elevation and geoid values close to the planetary mean, with a paucity at both the highest and lowest levels of topgraphy and geoid. Some coronae appear aligned in quasi-linear chains. We attribute the clustering of coronae and related features to preferential formation of these features above regions of broad-scale mantle upwelling, and suggest that a major mantle upwelling underlies the one large cluster. We suggest that coronae are rare at the lowest elevations because these may be regions of mantle downwelling. The shortage of coronae at the highest elevations may result both from obscuration by other intense tectonism there and from suppression there of their formation by an unusually thick crust. Corona chains may be produced by enhanced passive mantle uplift below failed or incipient rifts.

Squyres, Steven W.↗
NASA NTRS2007

Second Cancers After Fractionated Radiotherapy: Stochastic Population Dynamics Effects

When ionizing radiation is used in cancer therapy it can induce second cancers in nearby organs. Mainly due to longer patient survival times, these second cancers have become of increasing concern. Estimating the risk of solid second cancers involves modeling: because of long latency times, available data is usually for older, obsolescent treatment regimens. Moreover, modeling second cancers gives unique insights into human carcinogenesis, since the therapy involves administering well characterized doses of a well studied carcinogen, followed by long-term monitoring. In addition to putative radiation initiation that produces pre-malignant cells, inactivation (i.e. cell killing), and subsequent cell repopulation by proliferation can be important at the doses relevant to second cancer situations. A recent initiation/inactivation/proliferation (IIP) model characterized quantitatively the observed occurrence of second breast and lung cancers, using a deterministic cell population dynamics approach. To analyze ifradiation-initiated pre-malignant clones become extinct before full repopulation can occur, we here give a stochastic version of this I I model. Combining Monte Carlo simulations with standard solutions for time-inhomogeneous birth-death equations, we show that repeated cycles of inactivation and repopulation, as occur during fractionated radiation therapy, can lead to distributions of pre-malignant cells per patient with variance >> mean, even when pre-malignant clones are Poisson-distributed. Thus fewer patients would be affected, but with a higher probability, than a deterministic model, tracking average pre-malignant cell numbers, would predict. Our results are applied to data on breast cancers after radiotherapy for Hodgkin disease. The stochastic IIP analysis, unlike the deterministic one, indicates: a) initiated, pre-malignant cells can have a growth advantage during repopulation, not just during the longer tumor latency period that follows; b) weekend treatment gaps during radiotherapy, apart from decreasing the probability of eradicating the primary cancer, substantially increase the risk of later second cancers.

Sachs, Rainer K.↗