Comparing approximate methods for mock catalogues and covariance matrices – I. Correlation function
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Martha Lippich↗
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Three CA-Related Covariance Issues and Their Solutions
This presentation addresses three issues that arise in the use of DoD-produced satellite state estimate covariances in the conjunction assessment process: the realism of the provided covariances, how to address correlated error between two satellites' covariance matrices, and how to proceed when a furnished covariance is non-positive-definite. To address the first, DoD has implemented a set of two "consider parameters" with which to alter/expand the covariance to account for atmospheric density forecast error and satellite frontal area uncertainty, the two largest sources of unmodeled position prediction error for LEO orbits; these values are governed by a satellite's orbital parameters and ballistic coefficient histories, as well as the current and predicted space weather situation. This approach substantially improves the realism of the covariance by accounting for expected prediction errors that are not part of the fit process that generates the covariance. The second issue of covariance correlation is a complicated one, but the main shared error source—global atmospheric density error that is common to both satellites can be characterized and, through sensitivity vectors, quantified at the conjunction's time of closest approach and removed from the joint covariance, which is used in the CA probability of collision (Pc) calculation. Finally, while according to the orbit determination theory a non-positive-definite (NPD) covariance is not possible, numerical truncation and covariance interpolation can conspire to produce NPD results, which render the covariance unusable for certain CA risk assessment calculations, such as Monte Carlo Pc determination. Three similar techniques are profiled and the simplest of them recommended as a reasonable remediation technique when NPD covariances are encountered by CA practitioners.
Dynamic Consider Parameter↗
The minimal bound on the estimation error covariance matrix in the presence of correlated driving noise
Increase approximation of estimation errors covariance between discrete measurements due to random forcing function uncertainty
Deckert, J. C.↗
Computation of the factorized error covariance of the difference between correlated estimators
A state estimation problem where some of the measurements may be common to two or more data sets is considered. Two approaches for computing the error covariance of the difference between filtered estimates (for each data set) are discussed. The first algorithm is based on postprocessing of the Kalman gain profiles of two correlated estimators. It uses UD factors of the covariance of the relative error. The second algorithm uses a square root information filter applied to relative error analysis. In the absence of process noise, the square root information filter is computationally more efficient and more flexible than the Kalman gain (covariance update) method. Both the algorithms (covariance and information matrix based) are applied to a Venus orbiter simulation, and their performances are compared.
Wolff, Peter J.↗
Determination of statistics for any rotation of axes of a bivariate normal elliptical distribution
Determination of mean, variance, covariance, and correlation estimates for axes rotation of bivariate normal elliptical distribution
Crutcher, H. L.↗
Probabilistic finite elements for fracture mechanics
The probabilistic finite element method (PFEM) is developed for probabilistic fracture mechanics (PFM). A finite element which has the near crack-tip singular strain embedded in the element is used. Probabilistic distributions, such as expectation, covariance and correlation stress intensity factors, are calculated for random load, random material and random crack length. The method is computationally quite efficient and can be expected to determine the probability of fracture or reliability.
Besterfield, Glen↗
A quick look method of detecting water of hydration in small solar system bodies
The action of aqueous alteration of the near-subsurface material on asteroids and probably some satellites in the Solar System constitutes part of the formation history of the Solar System. The C-class asteroids (and subclasses B, G, and F) were initially believed to have undergone aqueous alteration based on their low albedos and neutral broadband visible and near-infrared colors. These spectra exhibit a sharp drop at wavelengths shorter than 0.55 microns due to a strong ferric oxide intervalence change transfer transition. This IVCT comprises multiple absorptions that are not uniquely indicative of phyllosilicates, but rather are present in the spectrum of any object containing Fe(2+) and Fe(3+) in its surface material. A definitive indication of aqueous alteration came when the broad IR absorption feature having a minimum near 3.0 microns indicative of structural hydroxyl (OH) and interlayer and and absorbed water in phyllosilicates was identified in the IR photometry of many C-class asteroids. Additional mineralogical compositional evidence of aqueously-altered asteroids has come as the result of high SNR narrowband spectrophotometry in the visible and near-infrared spectral regions taken using a CCD/spectrograph combination. An absorption feature centered at 0.7 microns indicative of an Fe(2+) - Fe(3+) charge transfer transition in oxidized iron in phyllosilicates in spectra of some low-albedo asteroids, especially C and G class, and CM2 carbonaceous chondrite meteorites was identified in the CCD spectra. The correlation and covariance of the 0.7-micron and 3.0-micron features were examined by comparing observations of asteroids common to both the CCD reflectance spectra and the 3.0-micron multicolor photometry data sets. Thirty-one pairs of observations were included in this training group. The results of the statistical study indicate that with a 95% confidence level, 84% of the objects observed either having or not having the 0.7-micron feature will correspondingly have the 3.0-micron water of hydration feature.
Vilas, F.↗
A novel ozone sensor for various environmental applications
A small, lightweight, and fast-response ozone sensor for various environmental applications is described. At a flow rate of 100 l/min(-1) the ozone sensor has a response time of significantly better than 0.1 s with a detection limit lower than 100 pptv. The ozone sensor was successfully tested in various environmental applications, i.e. in measuring directly the vertical ozone flux onto agricultural land utilizing the eddy correlation or covariance technique and in monitoring horizontal and vertical ozone profiles in the troposphere and stratosphere.
Guesten, H.↗
Continuum Covariance Propagation for Understanding Variance Loss in Advective Systems
Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance has been attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because16they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results show that this leads to significant, spurious variance loss in certain regions, and gain in others. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.
covariance propagation↗
A conservative bound on the estimation error covariance matrix in the presence of correlated driving noise and correlated discrete measurement noise.
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Potter, J. E.↗
Sensitivity of Lower Stratospheric Assimilated Ozone on Error Covariance Modeling and Data Selection
Assimilated ozone is produced at the NASA/Goddard Data Assimilation Office by blending ozone retrieved from the Solar Backscatter UltraViolet/2 (SBUV/2) instrument and the Earth Probe Total Ozone Mapping Spectrometer (EP TOMS) measurements into an off-line transport model. The current system tends to overestimate the amount of lower stratospheric ozone. This is a region where ozone plays a key role in the forcing of climate. A biased ozone field in this region will adversely impact calculations of the stratosphere-troposphere exchange and, when used as a first guess in retrievals, the values determined from satellite observations. Since these are all important applications of assimilated ozone products, effort is being directed towards reducing this bias. The SBUV ozone data have a coarse vertical resolution with increased uncertainty below the ozone maximum, and TOMS provides only total ozone columns. Thus, the assimilated ozone in the lower stratosphere, and its vertical distribution in particular, are only weakly constrained by the incoming SBUV and TOMS data. Consequently, the assimilated ozone distribution should be sensitive to changes in inputs to the statistical analysis scheme. Accordingly, the sensitivity of the assimilated lower stratospheric ozone fields to changes in the TOMS error-covariance modeling and the SBUV data selection has been investigated. The use of a spatially correlated TOMS error covariance model led to improvements in the product. However, withholding the SBUV/2 data for the layer between 63 and 126 hPa typically degraded the product, a result which vindicates the use of this layer ozone product, despite its known errors. These efforts to improve the lower stratospheric distribution will be extended to include a more advanced forecast error covariance model, and by assimilating ozone products from new instruments on Envisat and EOS Aura.
Stajner, Ivanka↗
Causal correlation of foliar biochemical concentrations with AVIRIS spectra using forced entry linear regression
A major goal of airborne imaging spectrometry is to estimate the biochemical composition of vegetation canopies from reflectance spectra. Remotely-sensed estimates of foliar biochemical concentrations of forests would provide valuable indicators of ecosystem function at regional and eventually global scales. Empirical research has shown a relationship exists between the amount of radiation reflected from absorption features and the concentration of given biochemicals in leaves and canopies (Matson et al., 1994, Johnson et al., 1994). A technique commonly used to determine which wavelengths have the strongest correlation with the biochemical of interest is unguided (stepwise) multiple regression. Wavelengths are entered into a multivariate regression equation, in their order of importance, each contributing to the reduction of the variance in the measured biochemical concentration. A significant problem with the use of stepwise regression for determining the correlation between biochemical concentration and spectra is that of 'overfitting' as there are significantly more wavebands than biochemical measurements. This could result in the selection of wavebands which may be more accurately attributable to noise or canopy effects. In addition, there is a real problem of collinearity in that the individual biochemical concentrations may covary. A strong correlation between the reflectance at a given wavelength and the concentration of a biochemical of interest, therefore, may be due to the effect of another biochemical which is closely related. Furthermore, it is not always possible to account for potentially suitable waveband omissions in the stepwise selection procedure. This concern about the suitability of stepwise regression has been identified and acknowledged in a number of recent studies (Wessman et al., 1988, Curran, 1989, Curran et al., 1992, Peterson and Hubbard, 1992, Martine and Aber, 1994, Kupiec, 1994). These studies have pointed to the lack of a physical link between wavelengths chosen by stepwise regression and the biochemical of interest, and this in turn has cast doubts on the use of imaging spectrometry for the estimation of foliar biochemical concentrations at sites distant from the training sites. To investigate this problem, an analysis was conducted on the variation in canopy biochemical concentrations and reflectance spectra using forced entry linear regression.
Dawson, Terence P.↗
Electromagnetic wave scattering in a two-layer anisotropic random medium
For electromagnetic wave propagation and scattering in an anisotropic random medium, the Dyson equation for the mean field and the Bethe-Salpeter equation for the correlation or the covariance of the field were derived. With the random permittivity expressed in a general anisotropic form, the bilocal and the nonlinear approximations are employed to solve the Dyson equation, and the ladder approximation to solve the Bethe-Salpeter equation. The mean dyadic Green's function for a two-layer anisotropic random medium with arbitrary three-dimensional correlation functions has been investigated with the zeroth-order solutions to the Dyson equation under the nonlinear approximation. The effective propagation constants are calculated for the four characteristic waves associated with the coherent vector fields, propagating in an anisotropic random-medium layer, which are the ordinary and extraordinary waves with upward- and downward-propagating vectors.
Lee, J. K.↗
Remote sensing of earth terrain
Progress on the investigation of the anisotropy of the terrain media, such as vegetation canopy and sea ice, and the study of the fluctuation-dissipation theorem in conjunction with the application of strong fluctuation theory for passive remote sensing of snowpacks is reported. The Feynman diagrammatic technique is used to derive the Dyson equation for the mean field and the Bethe-Salpeter equation for the correlation or the covariance of the field for electromagnetic wave propagation and scattering in an anisotropic random medium. With the random permittivity expressed in a general form, the bilocal and the nonlinear approximations are employed to solve the Dyson equation and the ladder approximation to the Bethe-Salpeter equation. The mean dyadic Green's function for a two layer anisotropic random medium with arbitrary three dimensional correlation function was investigated with the zeroth-order solutions to the Dyson equation under the four characteristic waves associated with the coherent vector fields propagating in an anisotropic random medium layer, which are the ordinary and extraordinary waves with upward and downward propagating vectors.
Kong, J. A.↗
Macroscopic Violation of Three Cauchy-Schwarz Inequalities Using Correlated Light Beams From an Infra-Red Emitting Semiconductor Diode Array
We briefly review quantum mechanical and semi-classical descriptions of experiments which demonstrate the macroscopic violation of the three Cauchy-Schwarz inequalities: g(sup 2)(sub 11)(0) greater than or equal to 1; g(sup 2)(sub 11)(0) greater than or equal to g(sup 2)(sub 11)(t), (t approaches infinity); (the absolute value of g(sup 2)(sub 11)(0))(exp 2) less than or equal to g(sup 2)(sub 11)(0) g(sup 2)(sub 11)(0). Our measurements demonstrate the violation, at macroscopic intensities, of each of these inequalities. We show that their violation, although weak, can be demonstrated through photodetector current covariance measurements on correlated sub-Poissonian Poissonian, and super Poissonian light beams. Such beams are readily generated by a tandem array of infrared-emitting semiconductor junction diodes. Our measurements utilize an electrically coupled array of one or more infrared-emitting diodes, optically coupled to a detector array. The emitting array is operated in such a way as to generate highly correlated beams of variable photon Fano Factor. Because the measurements are made on time scales long compared with the first order coherence time and with detector areas large compared with the corresponding coherence areas, first order interference effects are negligible. The first and second inequalities are violated, as expected, when a sub-Poissonian light beam is split and the intensity fluctuations of the two split beams are measured by two photodetectors and subsequently cross-correlated. The third inequality is violated by bunched (as well as anti-bunched) beams of equal intensity provided the measured cross correlation coefficient exceeds (F - 1)/F, where F is the measured Fano Factor of each beam. We also investigate the violation for the case of unequal beams.
Edwards, P. J.↗
A dual frequency radar for ocean roughness sampling
A microwave technique for determining the roughness (rms wave height) of a randomly varying air-water interface has been developed theoretically, verified with laboratory wavetank studies and is currently being implemented for ocean surface measurements in a series of applications flight experiments. These aircraft observations will be near the Chesapeake Light Tower and will include a range of altitudes and sea conditions. The measurement concept involves cross-correlating the envelope fluctuations on two received carriers that are monochromatic when transmitted in a normal direction to the interface and are observed in a backscatter direction after reflection by the large number of randomly distributed specular points on the surface. The measured correlation coefficient (normalized covariance) as a function of carrier frequency separation will depend on the surface roughness, specifically the probability density function of the specular point height. Details of the flight system are discussed, and preliminary flight results are presented.
Weissman, D. E.↗
A study of sound generation in subsonic rotors, volume 2
Computer programs were developed for use in the analysis of sound generation by subsonic rotors. Program AIRFOIL computes the spectrum of radiated sound from a single airfoil immersed in a laminar flow field. Program ROTOR extends this to a rotating frame, and provides a model for sound generation in subsonic rotors. The program also computes tone sound generation due to steady state forces on the blades. Program TONE uses a moving source analysis to generate a time series for an array of forces moving in a circular path. The resultant time series are than Fourier transformed to render the results in spectral form. Program SDATA is a standard time series analysis package. It reads in two discrete time series and forms auto and cross covariances and normalizes these to form correlations. The program then transforms the covariances to yield auto and cross power spectra by means of a Fourier transformation.
Chalupnik, J. D.↗
Continuum Covariance Propagation for Understanding Variance Loss in Advective Systems
We demonstrate for state dynamics governed by the continuity equation and related hyperbolic partial differential equations that significant, spurious variance loss occurs during covariance propagation by traditional methods used in data assimilation, even at full rank. This inaccurate variance evolution is caused not by numerical dissipation, but rather by a discontinuous change in the continuum covariance dynamics as correlation lengths tend to zero.
Shay Gilpin↗