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All sourcesNASA NTRSDOE OSTI
NASA NTRS1987

Analysis of the Bivariate Parameter Wind Differences Between Jimsphere and Windsonde

An analysis is presented for the bivariate parameter differences between the FPS-16 Radar/Jimsphere and the Meteorological Sounding System (MSS) Windsonde. The Jimsphere is used as the standard to measure the ascent wind during the Space Shuttle launches at Kennedy Space Center, Florida, and the Windsonde is the backup system. In addition, a discussion of the terrestrial environment (below 20 km) and a description of the Jimsphere and Windsonde wind sensors are given. Computation of the wind statistics from 64 paired Jimsphere and Windsonde balloon releases in support of 14 Space Shuttle launches shows a good agreement between the two wind sensors. From the analysis of buildup and back-off data for various scales of distance and the comparison of the cumulative percent frequency (CPF) versus wind speed change, it is shown that the wind speed change for various scales of distances for the Jimsphere and Windsonde compare favorably.

Susko, Michael↗
NASA NTRS1993

Improving the chi-squared approximation for bivariate normal tolerance regions

Let X be a two-dimensional random variable distributed according to N2(mu,Sigma) and let bar-X and S be the respective sample mean and covariance matrix calculated from N observations of X. Given a containment probability beta and a level of confidence gamma, we seek a number c, depending only on N, beta, and gamma such that the ellipsoid R = (x: (x - bar-X)'S(exp -1) (x - bar-X) less than or = c) is a tolerance region of content beta and level gamma; i.e., R has probability gamma of containing at least 100 beta percent of the distribution of X. Various approximations for c exist in the literature, but one of the simplest to compute -- a multiple of the ratio of certain chi-squared percentage points -- is badly biased for small N. For the bivariate normal case, most of the bias can be removed by simple adjustment using a factor A which depends on beta and gamma. This paper provides values of A for various beta and gamma so that the simple approximation for c can be made viable for any reasonable sample size. The methodology provides an illustrative example of how a combination of Monte-Carlo simulation and simple regression modelling can be used to improve an existing approximation.

Feiveson, Alan H.↗
NASA NTRS1995

On the joint spectral density of bivariate random sequences

For univariate random sequences, the power spectral density acts like a probability density function of the frequencies present in the sequence. This dissertation extends that concept to bivariate random sequences. For this purpose, a function called the joint spectral density is defined that represents a joint probability weighing of the frequency content of pairs of random sequences. Given a pair of random sequences, the joint spectral density is not uniquely determined in the absence of any constraints. Two approaches to constraining the sequences are suggested: (1) assume the sequences are the margins of some stationary random field, (2) assume the sequences conform to a particular model that is linked to the joint spectral density. For both approaches, the properties of the resulting sequences are investigated in some detail, and simulation is used to corroborate theoretical results. It is concluded that under either of these two constraints, the joint spectral density can be computed from the non-stationary cross-correlation.

Aalfs, David D.↗
DOE OSTI2021

Discrimination of Seismic Events (2006–2020) in North Korea Using P/Lg Amplitude Ratios from Regional Stations and a Bivariate Discriminant Function

Two events of magnitude (m b ) 3.6–3.8 occurred in southern North Korea (NK) on 27 June 2019 and 11 May 2020. Although these events were located ~330–400 km from the known nuclear test site, the fact that they occurred within the territory of NK, a country with a recent history of underground nuclear tests, made them events of interest for the monitoring community. In this work, we used P/Lg ratios from regional stations to categorize seismic events that occurred in NK from 2006 to May 2020, including these two recent events, the six declared NK nuclear tests, and the cavity collapse and triggered earthquakes that followed the 3 September 2017 nuclear explosion. We were able to separate the cavity collapse from the population of nuclear explosions. However, based on P/Lg ratios, the distinction between the earthquakes and the cavity collapse is ambiguous. The performed discriminant analyses suggest that combining Pg/Lg and Pn/Lg ratios results in improved discriminant power compared with any of the ratio types alone. We used the two ratio types jointly in a quadratic discriminant function and successfully classified the six declared nuclear tests and the triggered earthquakes that followed the September 2017 explosion. Our analyses also confirm that the recent southern events of June 2019 and May 2020 are both tectonic earthquakes that occurred naturally.

58 GEOSCIENCES↗
NASA NTRS1976

Determination of statistics for any rotation of axes of a bivariate normal elliptical distribution

Transformation of statistics from a dimensional set to another dimensional set involves linear functions of the original set of statistics. Similarly, linear functions will transform statistics within a dimensional set such that the new statistics are relevant to a new set of coordinate axes. A restricted case of the latter is the rotation of axes in a coordinate system involving any two correlated random variables. A special case is the transformation for horizontal wind distributions. Wind statistics are usually provided in terms of wind speed and direction (measured clockwise from north) or in east-west and north-south components. A direct application of this technique allows the determination of appropriate wind statistics parallel and normal to any preselected flight path of a space vehicle. Among the constraints for launching space vehicles are critical values selected from the distribution of the expected winds parallel to and normal to the flight path. These procedures are applied to space vehicle launches at Cape Kennedy, Florida.

Falls, L. W.↗
NASA NTRS1982

A bivariate gamma probability distribution with application to gust modeling

A five-parameter gamma distribution (BGD) having two shape parameters, two location parameters, and a correlation parameter is investigated. This general BGD is expressed as a double series and as a single series of the modified Bessel function. It reduces to the known special case for equal shape parameters. Practical functions for computer evaluations for the general BGD and for special cases are presented. Applications to wind gust modeling for the ascent flight of the space shuttle are illustrated.

Smith, O. E.↗
NASA NTRS1989

Bivariate conditional sampling of buoyancy flux during an intense cold-air outbreak

The joint frequency distribution technique was used to analyze buoyancy fluxes in the marine atmospheric boundary layer (MABL) for the cloud street regime noted during the Genesis of Atlantic Lows Experiment. It is found that for the lower half of the MABL, the buoyancy flux is mainly generated by the rising thermals and the sinking compensating ambient air, and is mainly consumed by the entrainment and detrainment of thermals, penetrative convection, and the entrainment from the MABL top. If the buoyancy flux is primarily driven by the temperature flux, these buoyancy-flux generating processes should be the same for the lower boundary layers over land and ocean. The results of the scale analysis of the buoyancy flux agree well with those obtained for mesoscale cellular convection during the Air-Mass Transformation Experiment.

Chou, Shu-Hsien↗
NASA NTRS1992

A method of smooth bivariate interpolation for data given on a generalized curvilinear grid

A method of locally bicubic interpolation is presented for data given at the nodes of a two-dimensional generalized curvilinear grid. The physical domain is transformed to a computational domain in which the grid is uniform and rectangular by a generalized curvilinear coordinate transformation. The metrics of the transformation are obtained by finite differences in the computational domain. Metric derivatives are determined by repeated application of the chain rule for partial differentiation. Given the metrics and the metric derivatives, the partial derivatives required to determine a locally bicubic interpolant can be estimated at each data point using finite differences in the computational domain. A bilinear transformation is used to analytically transform the individual quadrilateral cells in the physical domain into unit squares, thus allowing the use of simple formulas for bicubic interpolation.

Zingg, David W.↗
NASA NTRS1972

Optimum runway orientation relative to crosswinds

Specific magnitudes of crosswinds may exist that could be constraints to the success of an aircraft mission such as the landing of the proposed space shuttle. A method is required to determine the orientation or azimuth of the proposed runway which will minimize the probability of certain critical crosswinds. Two procedures for obtaining the optimum runway orientation relative to minimizing a specified crosswind speed are described and illustrated with examples. The empirical procedure requires only hand calculations on an ordinary wind rose. The theoretical method utilizes wind statistics computed after the bivariate normal elliptical distribution is applied to a data sample of component winds. This method requires only the assumption that the wind components are bivariate normally distributed. This assumption seems to be reasonable. Studies are currently in progress for testing wind components for bivariate normality for various stations. The close agreement between the theoretical and empirical results for the example chosen substantiates the bivariate normal assumption.

Falls, L. W.↗
NASA NTRS2014

Comparison of the Variations of Sunspot Number, Number of Sunspot Groups, and Sunspot Area, 1875 -2013

Examined are the yearly variations and ratios of sunspot number, the number of sunspot groups, and the total corrected sunspot area for the interval 1875-2013. While yearly sunspot number independently correlates strongly (r = 0.98) with the yearly number of sunspot groups (y = -2 + 11.99x) and the total corrected sunspot area (y = 5 + 0.059x), the strongest correlation (Ry12 = 0.99) is the one based on the bivariate fit of sunspot number against the combined variations of the number of sunspot groups and sunspot area (y = 1 + 5.88x1 + 0.031x2, where y refers to sunspot number, x1 refers to the number of sunspot groups, and x2 refers to the sunspot area). While all cycle minima based on the bivariate fit are concurrent with the observed minimum in sunspot number, cycle maxima are sometimes found to differ. For sunspot cycles 12, 19, 20, and 23, cycle maximum is inferred to have occurred in 1884, 1958, 1970, and 2002, respectively, rather than in 1883, 1957, 1968, and 2000, based on the observed sunspot number. Also, cycle 19's maximum amplitude based on observed sunspot number seems too high in comparison to that found using the bivariate fit. During the 139-year interval 1875-2013, the difference between the observed and predicted sunspot number based on the bivariate fit is <1 standard error of estimate (se) (<6.4) for 111 years, between 1 and <2 se (6.4 to <12.8) for 28 years, and ≥2 se (≥12.8) for only 4 years, these years being 1957 (16.6), 1978 (-15.8), 1980 (23), and 1982 (-16.3). For sunspot cycle 24, the difference between observed and predicted values has been only -0.7 and 3.2 (≤0.5 se).

Wilson, Robert M.↗
DOE OSTI2020

Characterization of Extreme Wave Conditions for Wave Energy Converter Design and Project Risk Assessment

Best practices and international standards for determining n-year return period extreme wave (sea states) conditions allow wave energy converter designers and project developers the option to apply simple univariate or more complex bivariate extreme value analysis methods. The present study compares extreme sea state estimates derived from univariate and bivariate methods and investigates the performance of spectral wave models for predicting extreme sea states at buoy locations within several regional wave climates along the US East and West Coasts. Two common third-generation spectral wave models are evaluated, a WAVEWATCH III® model with a grid resolution of 4 arc-minutes (6–7 km), and a Simulating WAves Nearshore model, with a coastal resolution of 200–300 m. Both models are used to generate multi-year hindcasts, from which extreme sea state statistics used for wave conditions characterization can be derived and compared to those based on in-situ observations at National Data Buoy Center stations. Comparison of results using different univariate and bivariate methods from the same data source indicates reasonable agreement on average. Discrepancies are predominantly random. Large discrepancies are common and increase with return period. There is a systematic underbias for extreme significant wave heights derived from model hindcasts compared to those derived from buoy measurements. This underbias is dependent on model spatial resolution. However, simple linear corrections can effectively compensate for this bias. A similar approach is not possible for correcting model-derived environmental contours, but other methods, e.g., machine learning, should be explored.

16 TIDAL AND WAVE POWER↗
NASA NTRS1976

Vector wind and vector wind shear models 0 to 27 km altitude for Cape Kennedy, Florida, and Vandenberg AFB, California

The techniques are presented to derive several statistical wind models. The techniques are from the properties of the multivariate normal probability function. Assuming that the winds can be considered as bivariate normally distributed, then (1) the wind components and conditional wind components are univariate normally distributed, (2) the wind speed is Rayleigh distributed, (3) the conditional distribution of wind speed given a wind direction is Rayleigh distributed, and (4) the frequency of wind direction can be derived. All of these distributions are derived from the 5-sample parameter of wind for the bivariate normal distribution. By further assuming that the winds at two altitudes are quadravariate normally distributed, then the vector wind shear is bivariate normally distributed and the modulus of the vector wind shear is Rayleigh distributed. The conditional probability of wind component shears given a wind component is normally distributed. Examples of these and other properties of the multivariate normal probability distribution function as applied to Cape Kennedy, Florida, and Vandenberg AFB, California, wind data samples are given. A technique to develop a synthetic vector wind profile model of interest to aerospace vehicle applications is presented.

Smith, O. E.↗
NASA NTRS1981

Vector wind profile gust model

To enable development of a vector wind gust model suitable for orbital flight test operations and trade studies, hypotheses concerning the distributions of gust component variables were verified. Methods for verification of hypotheses that observed gust variables, including gust component magnitude, gust length, u range, and L range, are gamma distributed and presented. Observed gust modulus has been drawn from a bivariate gamma distribution that can be approximated with a Weibull distribution. Zonal and meridional gust components are bivariate gamma distributed. An analytical method for testing for bivariate gamma distributed variables is presented. Two distributions for gust modulus are described and the results of extensive hypothesis testing of one of the distributions are presented. The validity of the gamma distribution for representation of gust component variables is established.

Adelfang, S. I.↗
NASA NTRS1983

Statistical modeling of space shuttle environmental data

Statistical models which use a class of bivariate gamma distribution are examined. Topics discussed include: (1) the ratio of positively correlated gamma varieties; (2) a method to determine if unequal shape parameters are necessary in bivariate gamma distribution; (3) differential equations for modal location of a family of bivariate gamma distribution; and (4) analysis of some wind gust data using the analytical results developed for modeling application.

Tubbs, J. D.↗
NASA NTRS1983

Range reference atmosphere models

A description is given of the methods used to establish the statistical parameters and models for wind and various thermodynamic quantities at an altitude of 0-70 km for nine geographical locations. It is noted that wind is modeled as a vector quantity using the bivariate normal probability function. With the five parameters of the bivariate normal distribution, the distribution for wind speed is derived as a generalized Rayleigh distribution. In addition, the frequency of wind direction is derived, and the conditional distribution of wind speed given the wind direction is derived. It is pointed out that these and other wind models are consistent with the rigorous mathematical properties of the bivariate normal probability theory. The thermodynamic quantities are consistent with the hydrostatic equation and the equation of state for the mean values. With these methods, many statistical relationships can be derived.

Smith, O. E.↗
DOE OSTI2020

Stochastic Simulation of Daily Suspended Sediment Concentration Using Multivariate Copulas

Estimation of daily suspended sediment concentration (SSC) is required for water resources and environment management. In this paper, a copula-based stochastic method was proposed for daily SSC simulation. Here, the multivariate copula function, constructed based on a bivariate copula and two bivariate conditional probability distributions, was used to model the temporal and cross dependence structures in daily SSCs. Then, the daily SSCs were generated by sampling from the multivariate conditional distribution. As a result, synthetic long-term SSCs data beyond the limited observation period can be provided for water resources managers, which plays a critical role in accurately estimating frequency and magnitude of extreme SSCs events. The proposed method was under rigorous examination by applying to a case study at Pingshan station in the Jinsha River Basin, China. Results showed that the generated daily SSC sequences not only had a high degree of accuracy in preserving the statistical characteristics of the daily SSC observations, but also captured both the temporal correlation and the cross-correlation between the daily streamflow and daily SSC. Specifically, the average daily relative error values corresponding to mean, standard deviation, skewness, lag-1 temporal correlation, and cross correlation were 0.87%, 4.24%, 7.52%, 0.51% and 2.02%, respectively. The multivariate copula framework proposed here can accurately and efficiently generate long-term daily SSC data for water resources management such as frequency analysis and risk assessment of extreme SSC events.

54 ENVIRONMENTAL SCIENCES↗