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NASA NTRS1988

Estimability and Regulability of Linear Systems

A linear state-space system will be said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix. It will be said to be regulable if the quadratic cost of state feedback control is strictly smaller than the cost when no feedback is used. Estimability and regulability are shown to be dual properties, equivalent to the nonreducibility of the Kalman filter and of the optimal linear quadratic regulator, respectively.

Baram, Yoram↗
NASA NTRS1999

Combined Satellite and Surface-Based Estimation of the Intracloud/ Cloud-to-Ground Lightning Ratio Over the Continental United States

Four years of observations from the NASA Optical Transient Detector (OTD) and Global Atmospherics National Lightning Detection Network (NLDN) are combined to determine the geographic distribution of the intracloud/cloud to ground lightning ratio over the continental United States. The mean ratio over this region is 2.64-2.94, with a standard deviation of 1.1-1.3 and anomalies as low as 1.0 or less over the Rocky and Appalachian mountains and as high as 8-10 in the central-upper midwest. There is some indication that the ratio covaries with ground elevation, although the relationship is nonunique. Little evidence is found to support a latitudinal covariance, despite significant variation in the climatological mean tropopause pressure over the latitudes considered. The dynamic range of local variability is comparable to the range of values cited by previous studies for latitudinal variation from the deep tropics to midlatitudes. Local high anomalies of this ratio in the midwest are coincident with anomalies in the climatological percentage of positive CG occurrence, as well as in the occurrence of large positive CGs characteristic of MCS convective and trailing stratiform regions. This suggests that storm type, morphology and level of organization may dominate over environmental cofactors in the local determination of this ratio.

Boccippio, D. J.↗
NASA NTRS2001

Precise Orbit Determination for GEOSAT Follow-On Using Satellite Laser Ranging Data and Intermission Altimeter Crossovers

The US Navy's GEOSAT Follow-On Spacecraft was launched on February 10, 1998 with the primary objective of the mission to map the oceans using a radar altimeter. Following an extensive set of calibration campaigns in 1999 and 2000, the US Navy formally accepted delivery of the satellite on November 29, 2000. Satellite laser ranging (SLR) and Doppler (Tranet-style) beacons track the spacecraft. Although limited amounts of GPS data were obtained, the primary mode of tracking remains satellite laser ranging. The GFO altimeter measurements are highly precise, with orbit error the largest component in the error budget. We have tuned the non-conservative force model for GFO and the gravity model using SLR, Doppler and altimeter crossover data sampled over one year. Gravity covariance projections to 70x70 show the radial orbit error on GEOSAT was reduced from 2.6 cm in EGM96 to 1.3 cm with the addition of SLR, GFO/GFO and TOPEX/GFO crossover data. Evaluation of the gravity fields using SLR and crossover data support the covariance projections and also show a dramatic reduction in geographically-correlated error for the tuned fields. In this paper, we report on progress in orbit determination for GFO using GFO/GFO and TOPEX/GFO altimeter crossovers. We will discuss improvements in satellite force modeling and orbit determination strategy, which allows reduction in GFO radial orbit error from 10-15 cm to better than 5 cm.

Lemoine, Frank G.↗
NASA NTRS2001

The impact of Doppler lidar wind observations on a single-level meteorological analysis

Through the use of observation operators, modern data assimilation systems have the capability to ingest observations of quantities that are not themselves model variables, but are mathematically related to those variables. An example of this are the so-called LOS (line of sight) winds that a Doppler wind Lidar can provide. The model - or data assimilation system - needs information about both components of the horizontal wind vectors, whereas the observations in this case only provide the projection of the wind vector onto a given direction. The analyzed value is then calculated essentially based on a comparison between the observation itself and the model-simulated value of the observed quantity. However, in order to assess the expected impact of such an observing system, it is important to examine the extent to which a meteorological analysis can be constrained by the LOS winds. The answer to this question depends on the fundamental character of the atmospheric flow fields that are analyzed, but more importantly it also depends on the real and assumed error covariance characteristics of these fields. A single-level wind analysis system designed to explore these issues has been built at the NASA Data Assimilation Office. In this system, simulated wind observations can be evaluated in terms of their impact on the analysis quality under various assumptions about their spatial distribution and error characteristics and about the error covariance of the background fields. The basic design of the system will be presented along with experimental results obtained with it. In particular, the value of simultaneously measuring LOS winds along two different directions for a given location will be discussed.

Riishojgaard, L. P.↗
NASA NTRS2001

The Computational Complexity, Parallel Scalability, and Performance of Atmospheric Data Assimilation Algorithms

The computational complexity of algorithms for Four Dimensional Data Assimilation (4DDA) at NASA's Data Assimilation Office (DAO) is discussed. In 4DDA, observations are assimilated with the output of a dynamical model to generate best-estimates of the states of the system. It is thus a mapping problem, whereby scattered observations are converted into regular accurate maps of wind, temperature, moisture and other variables. The DAO is developing and using 4DDA algorithms that provide these datasets, or analyses, in support of Earth System Science research. Two large-scale algorithms are discussed. The first approach, the Goddard Earth Observing System Data Assimilation System (GEOS DAS), uses an atmospheric general circulation model (GCM) and an observation-space based analysis system, the Physical-space Statistical Analysis System (PSAS). GEOS DAS is very similar to global meteorological weather forecasting data assimilation systems, but is used at NASA for climate research. Systems of this size typically run at between 1 and 20 gigaflop/s. The second approach, the Kalman filter, uses a more consistent algorithm to determine the forecast error covariance matrix than does GEOS DAS. For atmospheric assimilation, the gridded dynamical fields typically have More than 10(exp 6) variables, therefore the full error covariance matrix may be in excess of a teraword. For the Kalman filter this problem can easily scale to petaflop/s proportions. We discuss the computational complexity of GEOS DAS and our implementation of the Kalman filter. We also discuss and quantify some of the technical issues and limitations in developing efficient, in terms of wall clock time, and scalable parallel implementations of the algorithms.

Lyster, Peter M.↗
NASA NTRS2002

On the Choice of Variable for Atmospheric Moisture Analysis

The implications of using different control variables for the analysis of moisture observations in a global atmospheric data assimilation system are investigated. A moisture analysis based on either mixing ratio or specific humidity is prone to large extrapolation errors, due to the high variability in space and time of these parameters and to the difficulties in modeling their error covariances. Using the logarithm of specific humidity does not alleviate these problems, and has the further disadvantage that very dry background estimates cannot be effectively corrected by observations. Relative humidity is a better choice from a statistical point of view, because this field is spatially and temporally more coherent and error statistics are therefore easier to obtain. If, however, the analysis is designed to preserve relative humidity in the absence of moisture observations, then the analyzed specific humidity field depends entirely on analyzed temperature changes. If the model has a cool bias in the stratosphere this will lead to an unstable accumulation of excess moisture there. A pseudo-relative humidity can be defined by scaling the mixing ratio by the background saturation mixing ratio. A univariate pseudo-relative humidity analysis will preserve the specific humidity field in the absence of moisture observations. A pseudorelative humidity analysis is shown to be equivalent to a mixing ratio analysis with flow-dependent covariances. In the presence of multivariate (temperature-moisture) observations it produces analyzed relative humidity values that are nearly identical to those produced by a relative humidity analysis. Based on a time series analysis of radiosonde observed-minus-background differences it appears to be more justifiable to neglect specific humidity-temperature correlations (in a univariate pseudo-relative humidity analysis) than to neglect relative humidity-temperature correlations (in a univariate relative humidity analysis). A pseudo-relative humidity analysis is easily implemented in an existing moisture analysis system, by simply scaling observed-minus background moisture residuals prior to solving the analysis equation, and rescaling the analyzed increments afterward.

Dee, Dick P.↗
NASA NTRS2002

Climatology, Natural Cycles, and Modes of Interannual Variability of the Great Plains Low-Level Jet as Assimilated by the GEOS-1 Data Analysis System

Despite the fact that the low-level jet of the southern Great Plains (the GPLLJ) of the U.S. is primarily a nocturnal phenomenon that virtually vanishes during the daylight hours, it is one of the most persistent and stable features of the low-level continental flow during the warm-season months, May through August. We have first used significant-level data to validate the skill of the GEOS-1 Data Assimilation System (DAS) in realistically detecting this jet and inferring its structure and evolution. We have then carried out a 15-year reanalysis with the GEOS-1 DAS to determine and validate its climatology and mean diurnal cycle and to study its interannual variability. Interannual variability of the GPLLJ is much smaller than mean diurnal and random intraseasonal variability and comparable in magnitude, but not location, to mean seasonal variability. There are three maxima of interannual low-level meridional flow variability of the GPLLJ over the upper Great Plains, southeastern Texas, and the western Gulf of Mexico. Cross-sectional profiles of mean southerly wind through the Texas maximum remain relatively stable and recognizable from year to year with only its eastward flank showing significant variability. This variability, however, exhibits a distinct, biennial oscillation during the first six years of the reanalysis period and only then. Each of the three variability maxima corresponds to a spatially coherent, jet-like pattern of low-level flow interannual variability. There are three prominent modes of interannual. variability. These include the intermittent biennial oscillation (IBO), local to the Texas maximum. Its signal is evident in surface pressure, surface temperature, ground wetness and upper air flow, as well. A larger-scale continental convergence pattern (CCP) of covariance, exhibiting strong anti-correlation between the flow near the Texas and the upper Great Plains variability maxima, is revealed only when the IBO is removed from the interannual time series. A third, subtropical mode of covariance is associated with the Gulf of Mexico variability maximum. Significant interannual anti-correlations of the southeasterly flow over the Arizona/New Mexico region with the CCP and the subtropical mode are enhanced when restricted to the month of July. These anti-correlations may relate to an observed out-of-phase precipitation relationship between the Great Plains and the southwestern U.S.. The typical duration of interannual low-level meridional wind anomalies within a given season increases over the continent with decreasing latitude from two to three weeks over the upper Great Plains to six to seven weeks over eastern Texas.

Helfand, H. M.↗
NASA NTRS2003

Orbit Determination Support for the Microwave Anisotropy Probe (MAP)

NASA's Microwave Anisotropy Probe (MAP) was launched from the Cape Canaveral Air Force Station Complex 17 aboard a Delta II 7425-10 expendable launch vehicle on June 30, 2001. The spacecraft received a nominal direct insertion by the Delta expendable launch vehicle into a 185-km circular orbit with a 28.7deg inclination. MAP was then maneuvered into a sequence of phasing loops designed to set up a lunar swingby (gravity-assisted acceleration) of the spacecraft onto a transfer trajectory to a lissajous orbit about the Earth-Sun L2 Lagrange point, about 1.5 million km from Earth. Because of its complex orbital characteristics, the mission provided a unique challenge for orbit determination (OD) support in many orbital regimes. This paper summarizes the premission trajectory covariance error analysis, as well as actual OD results. The use and impact of the various tracking stations, systems, and measurements will be also discussed. Important lessons learned from the MAP OD support team will be presented. There will be a discussion of the challenges presented to OD support including the effects of delta-Vs at apogee as well as perigee, and the impact of the spacecraft attitude mode on the OD accuracy and covariance analysis.

Bauer, Frank↗
NASA NTRS2003

First Year Wilkinson Microwave Anisotropy Probe(WMAP) Observations: Data Processing Methods and Systematic Errors Limits

We describe the calibration and data processing methods used to generate full-sky maps of the cosmic microwave background (CMB) from the first year of Wilkinson Microwave Anisotropy Probe (WMAP) observations. Detailed limits on residual systematic errors are assigned based largely on analyses of the flight data supplemented, where necessary, with results from ground tests. The data are calibrated in flight using the dipole modulation of the CMB due to the observatory's motion around the Sun. This constitutes a full-beam calibration source. An iterative algorithm simultaneously fits the time-ordered data to obtain calibration parameters and pixelized sky map temperatures. The noise properties are determined by analyzing the time-ordered data with this sky signal estimate subtracted. Based on this, we apply a pre-whitening filter to the time-ordered data to remove a low level of l/f noise. We infer and correct for a small (approx. 1 %) transmission imbalance between the two sky inputs to each differential radiometer, and we subtract a small sidelobe correction from the 23 GHz (K band) map prior to further analysis. No other systematic error corrections are applied to the data. Calibration and baseline artifacts, including the response to environmental perturbations, are negligible. Systematic uncertainties are comparable to statistical uncertainties in the characterization of the beam response. Both are accounted for in the covariance matrix of the window function and are propagated to uncertainties in the final power spectrum. We characterize the combined upper limits to residual systematic uncertainties through the pixel covariance matrix.

Hinshaw, G.↗
NASA NTRS2003

Impact of Flow-Dependent Error Correlations and Tropospheric Chemistry on Assimilated Ozone

The presentation compares different versions of a global three-dimensional ozone data assimilation system developed at NASA's Data Assimilation Office. The Solar Backscatter Ultraviolet/2 (SBUV/2) total and partial ozone column retrievals are the sole data assimilated in all of the experiments presented. We study the impact of changing the forecast error covariance model from a version assuming static correlations with a one that captures a short-term Lagrangian evolution of those correlations. This is further combined with a study of the impact of neglecting the tropospheric ozone production, loss and dry deposition rates, which are obtained from the Harvard GEOS-CHEM model. We compare statistical characteristics of the assimilated data and the results of validation against independent observations, obtained from WMO balloon-borne sondes and the Polar Ozone and Aerosol Measurement (POAM) III instrument. Experiments show that allowing forecast error correlations to evolve with the flow results in positive impact on assimilated ozone within the regions where data were not assimilated, particularly at high latitudes in both hemispheres. On the other hand, the main sensitivity to tropospheric chemistry is in the Tropics and sub-Tropics. The best agreement between the assimilated ozone and the in-situ sonde data is in the experiment using both flow-dependent error covariances and tropospheric chemistry.

Wargan, K.↗
NASA NTRS2003

Estimation of Model Error Variances During Data Assimilation

Data assimilation is all about understanding the error characteristics of the data and models that are used in the assimilation process. Reliable error estimates are needed to implement observational quality control, bias correction of observations and model fields, and intelligent data selection. Meaningful covariance specifications are obviously required for the analysis as well, since the impact of any single observation strongly depends on the assumed structure of the background errors. Operational atmospheric data assimilation systems still rely primarily on climatological background error covariances. To obtain error estimates that reflect both the character of the flow and the current state of the observing system, it is necessary to solve three problems: (1) how to account for the short-term evolution of errors in the initial conditions; (2) how to estimate the additional component of error caused by model defects; and (3) how to compute the error reduction in the analysis due to observational information. Various approaches are now available that provide approximate solutions to the first and third of these problems. However, the useful accuracy of these solutions very much depends on the size and character of the model errors and the ability to account for them. Model errors represent the real-world forcing of the error evolution in a data assimilation system. Clearly, meaningful model error estimates and/or statistics must be based on information external to the model itself. The most obvious information source is observational, and since the volume of available geophysical data is growing rapidly, there is some hope that a purely statistical approach to model error estimation can be viable. This requires that the observation errors themselves are well understood and quantifiable. We will discuss some of these challenges and present a new sequential scheme for estimating model error variances from observations in the context of an atmospheric data assimilation system.

Dee, Dick↗
NASA NTRS2004

Recent Climate Variability in Antarctica from Satellite-derived Temperature Data

Recent Antarctic climate variability on month-to-month to interannual time scales is assessed through joint analysis of surface temperatures from satellite thermal infrared observations (T(sub IR)) and passive microwave brightness temperatures (T(sub B)). Although Tw data are limited to clear-sky conditions and T(sub B) data are a product of the temperature and emissivity of the upper approx. 1m of snow, the two data sets share significant covariance. This covariance is largely explained by three empirical modes, which illustrate the spatial and temporal variability of Antarctic surface temperatures. T(sub B) variations are damped compared to TIR variations, as determined by the period of the temperature forcing and the microwave emission depth; however, microwave emissivity does not vary significantly in time. Comparison of the temperature modes with Southern Hemisphere (SH) 500-hPa geopotential height anomalies demonstrates that Antarctic temperature anomalies are predominantly controlled by the principal patterns of SH atmospheric circulation. The leading surface temperature mode strongly correlates with the Southern Annular Mode (SAM) in geopotential height. The second temperature mode reflects the combined influences of the zonal wavenumber-3 and Pacific South American (PSA) patterns in 500-hPa height on month-to-month timescales. ENSO variability projects onto this mode on interannual timescales, but is not by itself a good predictor of Antarctic temperature anomalies. The third temperature mode explains winter warming trends, which may be caused by blocking events, over a large region of the East Antarctic plateau. These results help to place recent climate changes in the context of Antarctica's background climate variability and will aid in the interpretation of ice core paleoclimate records.

Schneider, David P.↗
NASA NTRS2003

An Evaluation of Recent Gravity Models wrt. Altimeter Satellite Missions

With the launch of CHAMP and GRACE, we have entered a new phase in the history of satellite geodesy. For the first time, geopotential models are now available based almost exclusively on satellite-satellite tracking either with GPS in the case of the CHAMP-based geopotential models, or co-orbital intersatellite ultra-precise ranging in the case of GRACE. Different groups have analyzed these data, and produced a series of geopotential models (e.g., EIGENlS, EIGEN2, GGM0lS, GGMOlC) that incorporate the new data. We will compare the performance of these "newer" geopotential models with the standard models now used for computations, (e.g., JGM-3, BGM-96, PGS7727, and GRIMS-C1) for TOPEX, JASON, Geosat-Follow-On (GFO), and Envisat using standard metrics such as SLR RMS of fit, altimeter crossovers, and orbit overlaps. Where covariances are available we can evaluate the predicted geographically correlated orbit error. These predicted results can be compared with the Earth-fixed differences between dynamic and reduced-dynamic orbits to test the predictive accuracy of the covariances, as well as to calibrate the error of the solutions.

Lemoine, Frank G.↗
NASA NTRS1990

Compound-specific isotopic analyses: a novel tool for reconstruction of ancient biogeochemical processes

Patterns of isotopic fractionation in biogeochemical processes are reviewed and it is suggested that isotopic fractionations will be small when substrates are large. If so, isotopic compositions of biomarkers will reflect those of their biosynthetic precursors. This prediction is tested by consideration of results of analyses of geoporphyrins and geolipids from the Greenhorn Formation (Cretaceous, Western Interior Seaway of North America) and the Messel Shale (Eocene, lacustrine, southern Germany). It is shown (i) that isotopic compositions of porphyrins that are related to a common source, but which have been altered structurally, cluster tightly and (ii) that isotopic differences between geolipids and porphyrins related to a common source are equal to those observed in modern biosynthetic products. Both of these observations are consistent with preservation of biologically controlled isotopic compositions during diagenesis. Isotopic compositions of individual compounds can thus be interpreted in terms of biogeochemical processes in ancient depositional environments. In the Cretaceous samples, isotopic compositions of n-alkanes are covariant with those of total organic carbon, while delta values for pristane and phytane are covariant with those of porphyrins. In this unit representing an open marine environment, the preserved acyclic polyisoprenoids apparently derive mainly from primary material, while the extractable, n-alkanes derive mainly from lower levels of the food chain. In the Messel Shale, isotopic compositions of individual biomarkers range from -20.9 to -73.4% vs PDB. Isotopic compositions of specific compounds can be interpreted in terms of origin from methylotrophic, chemautotrophic, and chemolithotrophic microorganisms as well as from primary producers that lived in the water column and sediments of this ancient lake.

Non-NASA Center↗
NASA NTRS1997

Relative Navigation In Elliptical Orbits Using An Iterative Nonlinear Filter

The two step filter is applied to process intersatellite radar measurements to determine the motion of one satellite relative to another in close elliptical orbits. This filter breaks a nonlinear estimation problem into two state vectors. The 'first step' state is chosen so as to have a linear measurement equation. This is nonlinearly related to the 'second step' state which describes the dynamics. Two different forms are used. In one, the first step state is the second step state vector augmented by the measurement equation. In the other, the first step and second step state vectors are of equal dimension. The two step filter is compared against an iterated extended Kalman filter and a Kalman filter using a change of variables. Analytical differences between the two step estimator and these conventional filters are highlighted. Special concerns for initializing the first step state covariance matrix and handling the possibility of numerically rank deficient covariance matrices are addressed. Numerical simulations are performed which show that the Two Step estimator produces a lower estimation bias under two circumstances; large apriori initial error; and small dimension observation vectors which require a longer arc of measurements to generate observability of the state.

Garrison, James L.↗
NASA NTRS1992

Correlation, Cost Risk, and Geometry

The geometric viewpoint identifies the choice of a correlation matrix for the simulation of cost risk with the pairwise choice of data vectors corresponding to the parameters used to obtain cost risk. The correlation coefficient is the cosine of the angle between the data vectors after translation to an origin at the mean and normalization for magnitude. Thus correlation is equivalent to expressing the data in terms of a non orthogonal basis. To understand the many resulting phenomena requires the use of the tensor concept of raising the index to transform the measured and observed covariant components into contravariant components before vector addition can be applied. The geometric viewpoint also demonstrates that correlation and covariance are geometric properties, as opposed to purely statistical properties, of the variates. Thus, variates from different distributions may be correlated, as desired, after selection from independent distributions. By determining the principal components of the correlation matrix, variates with the desired mean, magnitude, and correlation can be generated through linear transforms which include the eigenvalues and the eigenvectors of the correlation matrix. The conversion of the data to a non orthogonal basis uses a compound linear transformation which distorts or stretches the data space. Hence, the correlated data does not have the same properties as the uncorrelated data used to generate it. This phenomena is responsible for seemingly strange observations such as the fact that the marginal distributions of the correlated data can be quite different from the distributions used to generate the data. The joint effect of statistical distributions and correlation remains a fertile area for further research. In terms of application to cost estimating, the geometric approach demonstrates that the estimator must have data and must understand that data in order to properly choose the correlation matrix appropriate for a given estimate. There is a general feeling by employers and managers that the field of cost requires little technical or mathematical background. Contrary to that opinion, this paper demonstrates that a background in mathematics equivalent to that needed for typical engineering and scientific disciplines at the masters or doctorate level is appropriate within the field of cost risk.

Dean, Edwin B.↗
NASA NTRS1998

Effect of altered sensory conditions on multivariate descriptors of human postural sway

Multivariate descriptors of sway were used to test whether altered sensory conditions result not only in changes in amount of sway but also in postural coordination. Eigenvalues and directions of eigenvectors of the covariance of shnk and hip angles were used as a set of multivariate descriptors. These quantities were measured in 14 healthy adult subjects performing the Sensory Organization test, which disrupts visual and somatosensory information used for spatial orientation. Multivariate analysis of variance and discriminant analysis showed that resulting sway changes were at least bivariate in character, with visual and somatosensory conditions producing distinct changes in postural coordination. The most significant changes were found when somatosensory information was disrupted by sway-referencing of the support surface (P = 3.2 x 10(-10)). The resulting covariance measurements showed that subjects not only swayed more but also used increased hip motion analogous to the hip strategy. Disruption of vision, by either closing the eyes or sway-referencing the visual surround, also resulted in altered sway (P = 1.7 x 10(-10)), with proportionately more motion of the center of mass than with platform sway-referencing. As shown by discriminant analysis, an optimal univariate measure could explain at most 90% of the behavior due to altered sensory conditions. The remaining 10%, while smaller, are highly significant changes in posture control that depend on sensory conditions. The results imply that normal postural coordination of the trunk and legs requires both somatosensory and visual information and that each sensory modality makes a unique contribution to posture control. Descending postural commands are multivariate in nature, and the motion at each joint is affected uniquely by input from multiple sensors.

NASA Discipline Neuroscience↗
NASA NTRS2004

Kalman Filter Estimation of Spinning Spacecraft Attitude using Markley Variables

There are several different ways to represent spacecraft attitude and its time rate of change. For spinning or momentum-biased spacecraft, one particular representation has been put forward as a superior parameterization for numerical integration. Markley has demonstrated that these new variables have fewer rapidly varying elements for spinning spacecraft than other commonly used representations and provide advantages when integrating the equations of motion. The current work demonstrates how a Kalman filter can be devised to estimate the attitude using these new variables. The seven Markley variables are subject to one constraint condition, making the error covariance matrix singular. The filter design presented here explicitly accounts for this constraint by using a six-component error state in the filter update step. The reduced dimension error state is unconstrained and its covariance matrix is nonsingular.

Sedlak, Joseph E.↗