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NASA NTRS2002

Definition of Contravariant Velocity Components

This is an old issue in computational fluid dynamics (CFD). What is the so-called contravariant velocity or contravariant velocity component? In the article, we review the basics of tensor analysis and give the contravariant velocity component a rigorous explanation. For a given coordinate system, there exist two uniquely determined sets of base vector systems - one is the covariant and another is the contravariant base vector system. The two base vector systems are reciprocal. The so-called contravariant velocity component is really the contravariant component of a velocity vector for a time-independent coordinate system, or the contravariant component of a relative velocity between fluid and coordinates, for a time-dependent coordinate system. The contravariant velocity components are not physical quantities of the velocity vector. Their magnitudes, dimensions, and associated directions are controlled by their corresponding covariant base vectors. Several 2-D (two-dimensional) linear examples and 2-D mass-conservation equation are used to illustrate the details of expressing a vector with respect to the covariant and contravariant base vector systems, respectively.

Hung, Ching-Mao↗
NASA NTRS2002

On the Singularity in the Estimation of the Quaternion-of-Rotation

It has been claimed in the archival literature that the covariance matrix of a Kalman filter, which is designed to estimate the quaternion-of-rotation, is necessarily rank, deficient because the normality constraint of the quaternion produces dependence between the quaternion elements. In reality, though, this phenomenon does not occur. The covariance matrix is not singular, and the filter is well behaved. Several simple examples are presented th at demonstrate the regularity of the covariance matrix. First, a Kalman filter is designed to estimate variables subject to a functional relationship. Then the particular problem of quaternion estimation is analyzed. It is shown that the discrepancy stems from the fact that the functional relationship exists between the elements of the quaternion but not between its estimate elements.

Bar-Itzhack, Itzhack Y.↗
NASA NTRS2003

Comparison of Flow-Dependent and Static Error Correlation Models in the DAO Ozone Data Assimilation System

In a data assimilation system the forecast error covariance matrix governs the way in which the data information is spread throughout the model grid. Implementation of a correct method of assigning covariances is expected to have an impact on the analysis results. The simplest models assume that correlations are constant in time and isotropic or nearly isotropic. In such models the analysis depends on the dynamics only through assumed error standard deviations. In applications to atmospheric tracer data assimilation this may lead to inaccuracies, especially in regions with strong wind shears or high gradient of potential vorticity, as well as in areas where no data are available. In order to overcome this problem we have developed a flow-dependent covariance model that is based on short term evolution of error correlations. The presentation compares performance of a static and a flow-dependent model applied to a global three- dimensional ozone data assimilation system developed at NASA s Data Assimilation Office. We will present some results of validation against 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 and in the troposphere. We will also discuss statistical characteristics of both models; in particular we will argue that including evolution of error correlations leads to stronger internal consistency of a data assimilation ,

Wargan, K.↗
NASA NTRS1994

Lessons from an evolving rRNA: 16S and 23S rRNA structures from a comparative perspective

The 16S and 23S rRNA higher-order structures inferred from comparative analysis are now quite refined. The models presented here differ from their immediate predecessors only in minor detail. Thus, it is safe to assert that all of the standard secondary-structure elements in (prokaryotic) rRNAs have been identified, with approximately 90% of the individual base pairs in each molecule having independent comparative support, and that at least some of the tertiary interactions have been revealed. It is interesting to compare the rRNAs in this respect with tRNA, whose higher-order structure is known in detail from its crystal structure (36) (Table 2). It can be seen that rRNAs have as great a fraction of their sequence in established secondary-structure elements as does tRNA. However, the fact that the former show a much lower fraction of identified tertiary interactions and a greater fraction of unpaired nucleotides than the latter implies that many of the rRNA tertiary interactions remain to be located. (Alternatively, the ribosome might involve protein-rRNA rather than intramolecular rRNA interactions to stabilize three-dimensional structure.) Experimental studies on rRNA are consistent to a first approximation with the structures proposed here, confirming the basic assumption of comparative analysis, i.e., that bases whose compositions strictly covary are physically interacting. In the exhaustive study of Moazed et al. (45) on protection of the bases in the small-subunit rRNA against chemical modification, the vast majority of bases inferred to pair by covariation are found to be protected from chemical modification, both in isolated small-subunit rRNA and in the 30S subunit. The majority of the tertiary interactions are reflected in the chemical protection data as well (45). On the other hand, many of the bases not shown as paired in Fig. 1 are accessible to chemical attack (45). However, in this case a sizeable fraction of them are also protected against chemical modification (in the isolated rRNA), which suggests that considerable higher-order structure remains to be found (although all of it may not involve base-base interactions and so may not be detectable by comparative analysis). The agreement between the higher-order structure of the small-subunit rRNA and protection against chemical modification is not perfect, however; some bases shown to covary canonically are accessible to chemical modification (45).(ABSTRACT TRUNCATED AT 400 WORDS).

Review↗
NASA NTRS2007

Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets

Interpolating scattered data points is a problem of wide ranging interest. A number of approaches for interpolation have been proposed both from theoretical domains such as computational geometry and in applications' fields such as geostatistics. Our motivation arises from geological and mining applications. In many instances data can be costly to compute and are available only at nonuniformly scattered positions. Because of the high cost of collecting measurements, high accuracy is required in the interpolants. One of the most popular interpolation methods in this field is called ordinary kriging. It is popular because it is a best linear unbiased estimator. The price for its statistical optimality is that the estimator is computationally very expensive. This is because the value of each interpolant is given by the solution of a large dense linear system. In practice, kriging problems have been solved approximately by restricting the domain to a small local neighborhood of points that lie near the query point. Determining the proper size for this neighborhood is a solved by ad hoc methods, and it has been shown that this approach leads to undesirable discontinuities in the interpolant. Recently a more principled approach to approximating kriging has been proposed based on a technique called covariance tapering. This process achieves its efficiency by replacing the large dense kriging system with a much sparser linear system. This technique has been applied to a restriction of our problem, called simple kriging, which is not unbiased for general data sets. In this paper we generalize these results by showing how to apply covariance tapering to the more general problem of ordinary kriging. Through experimentation we demonstrate the space and time efficiency and accuracy of approximating ordinary kriging through the use of covariance tapering combined with iterative methods for solving large sparse systems. We demonstrate our approach on large data sizes arising both from synthetic sources and from real applications.

Memarsadeghi, Nargess↗
NASA NTRS2007

Benefits Derived From Laser Ranging Measurements for Orbit Determination of the GPS Satellite Orbit

While navigation systems for the determination of the orbit of the Global Position System (GPS) have proven to be very effective, the current research is examining methods to lower the error in the GPS satellite ephemerides below their current level. Two GPS satellites that are currently in orbit carry retro-reflectors onboard. One notion to reduce the error in the satellite ephemerides is to utilize the retro-reflectors via laser ranging measurements taken from multiple Earth ground stations. Analysis has been performed to determine the level of reduction in the semi-major axis covariance of the GPS satellites, when laser ranging measurements are supplemented to the radiometric station keeping, which the satellites undergo. Six ground tracking systems are studied to estimate the performance of the satellite. The first system is the baseline current system approach which provides pseudo-range and integrated Doppler measurements from six ground stations. The remaining five ground tracking systems utilize all measurements from the current system and laser ranging measurements from the additional ground stations utilized within those systems. Station locations for the additional ground sites were taken from a listing of laser ranging ground stations from the International Laser Ranging Service. Results show reductions in state covariance estimates when utilizing laser ranging measurements to solve for the satellite s position component of the state vector. Results also show dependency on the number of ground stations providing laser ranging measurements, orientation of the satellite to the ground stations, and the initial covariance of the satellite's state vector.

Welch, Bryan W.↗
NASA NTRS2001

Stable and verifiable state estimation methods and systems with spacecraft applications

The stability of a recursive estimator process (e.g., a Kalman filter is assured for long time periods by periodically resetting an error covariance P(t.sub.n) of the system to a predetermined reset value P.sub.r. The recursive process is thus repetitively forced to start from a selected covariance and continue for a time period that is short compared to the system's total operational time period. The time period in which the process must maintain its numerical stability is significantly reduced as is the demand on the system's numerical stability. The process stability for an extended operational time period T.sub.o is verified by performing the resetting step at the end of at least one reset time period T.sub.r whose duration is less than the operational time period T.sub.o and then confirming stability of the process over the reset time period T.sub.r. Because the recursive process starts from a selected covariance at the beginning of each reset time period T.sub.r, confirming stability of the process over at least one reset time period substantially confirms stability over the longer operational time period T.sub.o.

Li, Rongsheng↗
NASA NTRS2010

The Impact of Ocean Data Assimilation on Seasonal-to-Interannual Forecasts: A Case Study of the 2006 El Nino Event

This study investigates the impact of four different ocean analyses on coupled forecasts of the 2006 El Nino event. Forecasts initialized in June 2006 using ocean analyses from an assimilation that uses flow-dependent background error covariances are compared with those using static error covariances that are not flow dependent. The flow-dependent error covariances reflect the error structures related to the background ENSO instability and are generated by the coupled breeding method. The ocean analyses used in this study result from the assimilation of temperature and salinity, with the salinity data available from Argo floats. Of the analyses, the one using information from the coupled bred vectors (BV) replicates the observed equatorial long wave propagation best and exhibits more warming features leading to the 2006 El Nino event. The forecasts initialized from the BV-based analysis agree best with the observations in terms of the growth of the warm anomaly through two warming phases. This better performance is related to the impact of the salinity analysis on the state evolution in the equatorial thermocline. The early warming is traced back to salinity differences in the upper ocean of the equatorial central Pacific, while the second warming, corresponding to the mature phase, is associated with the effect of the salinity assimilation on the depth of the thermocline in the western equatorial Pacific. The series of forecast experiments conducted here show that the structure of the salinity in the initial conditions is important to the forecasts of the extension of the warm pool and the evolution of the 2006 El Ni o event.

Yang, Shu-Chih↗
NASA NTRS2011

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

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

Keppenne, Christian↗
NASA NTRS2012

Highlights of TOMS Version 9 Total Ozone Algorithm

The fundamental basis of TOMS total ozone algorithm was developed some 45 years ago by Dave and Mateer. It was designed to estimate total ozone from satellite measurements of the backscattered UV radiances at few discrete wavelengths in the Huggins ozone absorption band (310-340 nm). Over the years, as the need for higher accuracy in measuring total ozone from space has increased, several improvements to the basic algorithms have been made. They include: better correction for the effects of aerosols and clouds, an improved method to account for the variation in shape of ozone profiles with season, latitude, and total ozone, and a multi-wavelength correction for remaining profile shape errors. These improvements have made it possible to retrieve total ozone with just 3 spectral channels of moderate spectral resolution (approx. 1 nm) with accuracy comparable to state-of-the-art spectral fitting algorithms like DOAS that require high spectral resolution measurements at large number of wavelengths. One of the deficiencies of the TOMS algorithm has been that it doesn't provide an error estimate. This is a particular problem in high latitudes when the profile shape errors become significant and vary with latitude, season, total ozone, and instrument viewing geometry. The primary objective of the TOMS V9 algorithm is to account for these effects in estimating the error bars. This is done by a straightforward implementation of the Rodgers optimum estimation method using a priori ozone profiles and their error covariances matrices constructed using Aura MLS and ozonesonde data. The algorithm produces a vertical ozone profile that contains 1-2.5 pieces of information (degrees of freedom of signal) depending upon solar zenith angle (SZA). The profile is integrated to obtain the total column. We provide information that shows the altitude range in which the profile is best determined by the measurements. One can use this information in data assimilation and analysis. A side benefit of this algorithm is that it is considerably simpler than the present algorithm that uses a database of 1512 profiles to retrieve total ozone. These profiles are tedious to construct and modify. Though conceptually similar to the SBUV V8 algorithm that was developed about a decade ago, the SBUV and TOMS V9 algorithms differ in detail. The TOMS algorithm uses 3 wavelengths to retrieve the profile while the SBUV algorithm uses 6-9 wavelengths, so TOMS provides less profile information. However both algorithms have comparable total ozone information and TOMS V9 can be easily adapted to use additional wavelengths from instruments like GOME, OMI and OMPS to provide better profile information at smaller SZAs. The other significant difference between the two algorithms is that while the SBUV algorithm has been optimized for deriving monthly zonal means by making an appropriate choice of the a priori error covariance matrix, the TOMS algorithm has been optimized for tracking short-term variability using month and latitude dependent covariance matrices.

Bhartia, Pawan↗
NASA NTRS2013

Ensemble Data Assimilation Without Ensembles: Methodology and Application to Ocean Data Assimilation

Two methods to estimate background error covariances for data assimilation are introduced. While both share properties with the ensemble Kalman filter (EnKF), they differ from it in that they do not require the integration of multiple model trajectories. Instead, all the necessary covariance information is obtained from a single model integration. The first method is referred-to as SAFE (Space Adaptive Forecast error Estimation) because it estimates error covariances from the spatial distribution of model variables within a single state vector. It can thus be thought of as sampling an ensemble in space. The second method, named FAST (Flow Adaptive error Statistics from a Time series), constructs an ensemble sampled from a moving window along a model trajectory. The underlying assumption in these methods is that forecast errors in data assimilation are primarily phase errors in space and/or time.

Data Assimilation↗
NASA NTRS2013

Predicting Transition from Laminar to Turbulent Flow over a Surface

A prediction of whether a point on a computer-generated surface is adjacent to laminar or turbulent flow is made using a transition prediction technique. A plurality of boundary-layer properties at the point are obtained from a steady-state solution of a fluid flow in a region adjacent to the point. A plurality of instability modes are obtained, each defined by one or more mode parameters. A vector of regressor weights is obtained for the known instability growth rates in a training dataset. For each instability mode in the plurality of instability modes, a covariance vector is determined, which is the covariance of a predicted local growth rate with the known instability growth rates. Each covariance vector is used with the vector of regressor weights to determine a predicted local growth rate at the point. Based on the predicted local growth rates, an n-factor envelope at the point is determined.

Rajnarayan, Dev↗
NASA NTRS2018

Intercomparison of Evapotranspiration Measurement Methods for Vegetable Crops in California

Recent drought events in California and legislation passed with the goal of increasing the sustainability of groundwater supplies have led to increased interest in tools to optimize irrigation schedules and increase on-farm water used efficiency. With more than 400 different crops produced in California, evapotranspiration-based irrigation scheduling is a promising and well-established approach. However, there is a need for accurate methods to estimate crop evapotranspiration (ET(sub c)) across the diverse range of crops grown, coupled with cost-effective methods for quantifying the accuracy of these tools. In this study, we evaluated remotely sensed estimates of ET(sub c) and associated crop water requirements from NASA's Satellite Irrigation Support (SIMS) system for two vegetable crops and measured crop evapotranspiration ET(sub c) using multiple methods, including weighing lysimeters, eddy covariance towers (EC), and surface renewal stations. We compared ET(sub c) data from these measurements with remotely sensed basal crop evapotranspiration (ET(sub cb)) data from SIMS as well as ET(sub c) data from a standard FAO-56 crop coefficient approach. Studies were conducted for sugar beets in Five Points, CA from 2014 to 2015 and studies are ongoing for fresh market tomatoes in Firebaugh, CA. We present results from these intercomparison studies and describe implications for future studies to quantify the accuracy of remotely sensed measures of ET(sub c). Highlights from results to date include strong correlations between ET measured with both surface renewal instrumentation and eddy covariance calculations using a 3D sonic anemometer and ET(sub c) data measured with the weighing lysimeter, with respective R2 values of 0.7964 (surface renewal) and 0.8034 (eddy covariance). This study provides insights into agreement between different approaches for monitoring evapotranspiration and provides another reference point for the community working to develop accurate and cost-effective tools that support growers in optimizing irrigation management.

Measurement↗
NASA NTRS2021

Integrating continuous atmospheric boundary layer and tower-based flux measurements to advance understanding of land-atmosphere interactions

The atmospheric boundary layer mediates the exchange of energy, matter, and momentum between the land surface and the free troposphere, integrating a range of physical, chemical, and biological processes and is defined as the lowest layer of the atmosphere (ranging from a few meters to 3 km). In this review, we investigate how continuous, automated observations of the atmospheric boundary layer can enhance the scientific value of co-located eddy covariance measurements of land-atmosphere fluxes of carbon, water, and energy, as are being made at FLUXNET sites worldwide. We highlight four key opportunities to integrate tower-based flux measurements with continuous, long-term atmospheric boundary layer measurements: (1) to interpret surface flux and atmospheric boundary layer exchange dynamics and feedbacks at flux tower sites, (2) to support flux footprint modelling, the interpretation of surface fluxes in heterogeneous and mountainous terrain, and quality control of eddy covariance flux measurements, (3) to support regional-scale modeling and upscaling of surface fluxes to continental scales, and (4) to quantify land-atmosphere coupling and validate its representation in Earth system models. Adding a suite of atmospheric boundary layer measurements to eddy covariance flux tower sites, and supporting the sharing of these data to tower networks, would allow the Earth science community to address new emerging research questions, better interpret ongoing flux tower measurements, and would present novel opportunities for collaborations between FLUXNET scientists and atmospheric and remote sensing scientists.

Manuel Helbig↗
NASA NTRSDate not supplied

Is There A Relationship Between Cornered-Hat Methods and A Residual Approach to Estimate System Uncertainty?

Recently a relationship has been established between a now traditional residual diagnostic used to estimate observation, background and analysis error covariances of interest to data assimilation and the three-cornered hat (3CH) method. An existing extension of the traditional residual diagnostic uses residuals from a fixed lag-1 Kalman smoother to retrieve system (model) error covariance. It is thus natural to ask if an additional relationship can be established between this extended residual method and some form of cornered--hat method. The answer might seem straightforward. Unlike in the standard residual estimation case, attempting to estimate model error amounts to estimating the statistics of a residual quantity itself. As shown in this presentation, in such cases the 3CH method becomes trivial: the sought out uncertainty can be derived directly from the covariance of the residual quantity at hand. In the particular case of estimating model error, one of the corners would have to be composed of vectors providing estimates of model error; these are not typically available in practice. Therefore, at first glance the present work finds no relationship between the lag-1 smoother residual diagnostic for system error estimation and cornered--hat methods. However, the work suggests that a two-tiered 3CH might be all that is necessary for system uncertainty to be obtained with 3CH.

Ricardo Todling↗
NASA NTRS2024

Evaluation of Sex Differences in Physiologic Responses to Submaximal Cycling Under Normoxic and Hypoxic Conditions

BACKGROUND: During Lunar missions, astronauts may live and operate in conditions where altered atmospheric pressure and oxygen concentrations may result in a mildly hypoxic environment. While the compensatory hemodynamic mechanisms ensuring adequate oxygen delivery to contracting muscles during exercise in hypoxic conditions are well-studied, less research has focused on potential sex differences in the responses to hypoxia exposure during exercise. As female astronauts make up half of the Artemis astronaut corps, understanding whether physiologic responses in a hypoxic environment differ between sexes may inform recommendations for exercising safely in exploration environments. METHODS: Fourteen subjects (7M/7F) from NASA’s Exploration Atmosphere Study performed two submaximal cycle exercise trials (10 min of exercise at 40% peak aerobic capacity [VO 2 pk]) ergometer under normobaric normoxic gas (21% O 2 ) and normobaric hypoxic (18% O 2 and balanced N 2 ) conditions in randomized order. Linear mixed models with Bonferroni post hoc corrections (fixed effects: condition, sex, VO 2 pk [covariate], body mass [BM, covariate]; random effects: subject, mission) were performed to evaluate the effect of condition and sex on physiologic responses to exercise (oxygen uptake [VO 2 ], carbon dioxide production [VCO 2 ], ventilation [VE], oxygen saturation [SpO 2 ], and heart rate [HR]). RESULTS: Males were comparable to females for age (36.6±4.7 vs 36.4±9.3 yrs; p>0.05) but had greater BM (87.3±10.9 vs 64.5±7.4 kg; p<0.001) and absolute VO2pk (3.5±0.6 vs 2.4±4.7 L/min; p<0.001). Additionally, males had higher VO2 (p=0.005), VCO2 (p=0.02), and VE (p=0.01) during exercise trials, independent of condition; however, when VO 2 pk was added as a covariate, the effect of sex was no longer significant. SpO 2 was reduced during hypoxic exercise compared to the normoxic condition (p<0.001), but neither sex nor environmental condition impacted HR. CONCLUSIONS: Though most physiologic responses to submaximal, short-duration exercise between normoxia and mild hypoxia were similar, females exhibited lower VO 2 , VCO 2 , and VE during both conditions, likely driven by lower aerobic capacity. Future research is needed to determine whether similar findings result from multiday hypobaric hypoxia experienced during missions.

N C Strock↗
NASA NTRSDate not supplied

NASA CARA Tools

The NASA Conjunction Assessment Risk Analysis (CARA) team has created and posted to GitHub a set of MATLAB software tools for collision assessment (CA) analysis. The software is provided to the public under a NASA Open-Source Software Agreement and is maintained/updated by the CARA Analysis team. Capabilities provided within the toolset include algorithms in assessing probability of collision (Pc), collision consequence, covariance realism, Orbit Determination (OD) quality assessment, and single covariance max Pc. Additionally, a set of utilities are provided for astrodynamics topics closely related to CA, such as: Conjunction Data Message (CDM) parsing, covariance transformations, coordinate system transformations, etc. The repository is regularly updated on a monthly basis and new functionality will be added as algorithms are approved for public release. This presentation will provide a brief introduction to the toolset, highlight some of the key components which can be used in operations, and will present information on algorithms that are in the release pipeline and can be expected soon. This software will enable operators to independently perform CA calculations using the same core algorithms that CARA uses in operations.

L. Baars↗
NASA NTRS2022

Deep Interacting Multiple Model Filtering

In this paper, a deep learning-based multiple model estimation framework is presented for the state estimation of hybrid dynamical systems from high dimensional observations such as camera images. A low dimensional vector which represents the measurement of the latent dynamical system and its corresponding variance are learned using a deep encoder neural network. An Interacting Multiple Model (IMM) filter is used to generate the latent state estimates and covariances using multiple dynamical models, which can be learned using backpropagation through time. The state estimates of the dynamical system and the corresponding covariance matrix are generated from the latent state estimates and covariance using a deep decoder neural network. The whole network is trained in an end-to-end manner using a loss function which minimizes the negative log-likelihood of the neural network parameters. Simulation results are presented using a 2D bouncing ball example and estimation error statistics are computed which demonstrates the accuracy and consistency of the estimation.

Ghananeel Rotithor↗