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NASA NTRS1994

A linear method for analyzing lightning field changes

A constrained, least-squares method for analyzing multiple-station measurements of lightning field changes (delta Es) is introduced. Previous methods have attempted to fit the spatial pattern of lightning delta Es using nonlinear models, such as a point charge (Q) or a point dipole (P) model. With the linear method, the delta Es are described not by models but by a general volume charge distribution that is deposited on a large (40 x 40 x 20 cu km) Cartesian grid above the measuring network. A linear system of equations is used to relate the measured delta Es to the charges that are deposited at each grid point. With this approach, the information content of the measurements can be quantified by an eigenanalysis of the covariance matrix of the linear system. Constraints can be used to reduce the infinity of possible solutions to the linear system and also to reduce systematic biases that can be introduced by the method of solution. It is shown that a Landweber iterative method, derived from the general method of steepest descent, can be used to solve the linear system and that the resulting volume charge distributions are generally consistent with computer-simulated charge sources, when these sources are over the measuring network. The Landweber iteration has also provided solutions for natural lightning events that are consistent with Q- and P-model results.

Koshak, William J.↗
NASA NTRS1994

Calibration of the COBE FIRAS instrument

The Far-Infrared Absolute Spectrophotometer (FIRAS) instrument on the Cosmic Background Explorer (COBE) satellite was designed to accurately measure the spectrum of the cosmic microwave background radiation (CMBR) in the frequency range 1-95/cm with an angular resolution of 7 deg. We describe the calibration of this instrument, including the method of obtaining calibration data, reduction of data, the instrument model, fitting the model to the calibration data, and application of the resulting model solution to sky observations. The instrument model fits well for calibration data that resemble sky condition. The method of propagating detector noise through the calibration process to yield a covariance matrix of the calibrated sky data is described. The final uncertainties are variable both in frequency and position, but for a typical calibrated sky 2.6 deg square pixel and 0.7/cm spectral element the random detector noise limit is of order of a few times 10(exp -7) ergs/sq cm/s/sr cm for 2-20/cm, and the difference between the sky and the best-fit cosmic blackbody can be measured with a gain uncertainty of less than 3%.

Fixsen, D. J.↗
NASA NTRS1995

An approximate Kalman filter for ocean data assimilation: An example with an idealized Gulf Stream model

A practical method of data assimilation for use with large, nonlinear, ocean general circulation models is explored. A Kalman filter based on approximation of the state error covariance matrix is presented, employing a reduction of the effective model dimension, the error's asymptotic steady state limit, and a time-invariant linearization of the dynamic model for the error integration. The approximations lead to dramatic computational savings in applying estimation theory to large complex systems. We examine the utility of the approximate filter in assimilating different measurement types using a twin experiment of an idealized Gulf Stream. A nonlinear primitive equation model of an unstable east-west jet is studied with a state dimension exceeding 170,000 elements. Assimilation of various pseudomeasurements are examined, including velocity, density, and volume transport at localized arrays and realistic distributions of satellite altimetry and acoustic tomography observations. Results are compared in terms of their effects on the accuracies of the estimation. The approximate filter is shown to outperform an empirical nudging scheme used in a previous study. The examples demonstrate that useful approximate estimation errors can be computed in a practical manner for general circulation models.

Fukumori, Ichiro↗
NASA NTRS1993

Kalman filter estimation of attitude and gyro bias with the QUEST observation model

The loss function for the Wahba attitude estimation problem employs unit vector observations with scalar weights. It is usually associated with the QUEST observation model, where the actual sensor noise is assumed the same for all components of the observed vector, regardless of the position in the field of view. The QUEST model has the great advantage of being sensor-independent except for the scalar parameter that characterizes the sensor errors. Although efficient algorithms for solving the Wahba problem exist, extending these algorithms to estimate gyro biases or sensor alignments has had mixed success. However, it is straightforward to estimate bias and alignment parameters with a Kalman filter. This paper investigates the use of an extended Kalman filter for the attitude and gyro bias that incorporates the QUEST observation model, to be referred to as the Unit Vector Filter (UVF). The UVF results are compared with those from a more conventional filter, the Real-Time Sequential Filter (RTSF), for which the residual is the two-dimensional projection of the unit vector onto the plane perpendicular to the sensor boresight. The RTSF is similar in design to that used by Multimission Modular Spacecraft for onboard attitude determination. An apparent obstacle to the use of unit vectors as measurements is their singular noise covariance matrix. Shuster has shown that this problem should not affect filter performance. The UVF and RTSF are tested using actual flight data from the Extreme Ultraviolet Explorer (EUVE). It is found that these filters generate nearly identical attitude and gyro bias estimates, thus validating the use of unit vectors and the QUEST noise model.

Sedlak, J.↗
NASA NTRS1996

Flight data results of estimate fusion for spacecraft rendezvous navigation from shuttle mission STS-69

A recently developed rendezvous navigation fusion filter that optimally exploits existing distributed filters for rendezvous and GPS navigation to achieve the relative and inertial state accuracies of both in a global solution is utilized here to process actual flight data. Space Shuttle Mission STS-69 was the first mission to date which gathered data from both the rendezvous and Global Positioning System filters allowing, for the first time, a test of the fusion algorithm with real flight data. Furthermore, a precise best estimate of trajectory is available for portions of STS-69, making possible a check on the performance of the fusion filter. In order to successfully carry out this experiment with flight data, two extensions to the existing scheme were necessary: a fusion edit test based on differences between the filter state vectors, and an underweighting scheme to accommodate the suboptimal perfect target assumption made by the Shuttle rendezvous filter. With these innovations, the flight data was successfully fused from playbacks of downlinked and/or recorded measurement data through ground analysis versions of the Shuttle rendezvous filter and a GPS filter developed for another experiment. The fusion results agree with the best estimate of trajectory at approximately the levels of uncertainty expected from the fusion filter's covariance matrix.

Carpenter, J. Russell↗
NASA NTRS1997

Distortion Representation of Forecast Errors for Model Skill Assessment and Objective Analysis

We proposed a novel characterization of errors for numerical weather predictions. In its simplest form we decompose the error into a part attributable to phase errors and a remainder. The phase error is represented in the same fashion as a velocity field and is required to vary slowly and smoothly with position. A general distortion representation allows for the displacement and amplification or bias correction of forecast anomalies. Characterizing and decomposing forecast error in this way has two important applications, which we term the assessment application and the objective analysis application. For the assessment application, our approach results in new objective measures of forecast skill which are more in line with subjective measures of forecast skill and which are useful in validating models and diagnosing their shortcomings. With regard to the objective analysis application, meteorological analysis schemes balance forecast error and observational error to obtain an optimal analysis. Presently, representations of the error covariance matrix used to measure the forecast error are severely limited. For the objective analysis application our approach will improve analyses by providing a more realistic measure of the forecast error. We expect, a priori, that our approach should greatly improve the utility of remotely sensed data which have relatively high horizontal resolution, but which are indirectly related to the conventional atmospheric variables. In this project, we are initially focusing on the assessment application, restricted to a realistic but univariate 2-dimensional situation. Specifically, we study the forecast errors of the sea level pressure (SLP) and 500 hPa geopotential height fields for forecasts of the short and medium range. Since the forecasts are generated by the GEOS (Goddard Earth Observing System) data assimilation system with and without ERS 1 scatterometer data, these preliminary studies serve several purposes. They (1) provide a testbed for the use of the distortion representation of forecast errors, (2) act as one means of validating the GEOS data assimilation system and (3) help to describe the impact of the ERS 1 scatterometer data.

Hoffman, Ross N.↗
NASA NTRS1996

Uncertainty Modeling for Structural Control Analysis and Synthesis

The development of an accurate model of uncertainties for the control of structures that undergo a change in operational environment, based solely on modeling and experimentation in the original environment is studied. The application used throughout this work is the development of an on-orbit uncertainty model based on ground modeling and experimentation. A ground based uncertainty model consisting of mean errors and bounds on critical structural parameters is developed. The uncertainty model is created using multiple data sets to observe all relevant uncertainties in the system. The Discrete Extended Kalman Filter is used as an identification/parameter estimation method for each data set, in addition to providing a covariance matrix which aids in the development of the uncertainty model. Once ground based modal uncertainties have been developed, they are localized to specific degrees of freedom in the form of mass and stiffness uncertainties. Two techniques are presented: a matrix method which develops the mass and stiffness uncertainties in a mathematical manner; and a sensitivity method which assumes a form for the mass and stiffness uncertainties in macroelements and scaling factors. This form allows the derivation of mass and stiffness uncertainties in a more physical manner. The mass and stiffness uncertainties of the ground based system are then mapped onto the on-orbit system, and projected to create an analogous on-orbit uncertainty model in the form of mean errors and bounds on critical parameters. The Middeck Active Control Experiment is introduced as experimental verification for the localization and projection methods developed. In addition, closed loop results from on-orbit operations of the experiment verify the use of the uncertainty model for control analysis and synthesis in space.

Campbell, Mark E.↗
NASA NTRS1999

Improved Stratospheric Temperature Retrievals for Climate Reanalysis

The Data Assimilation Office (DAO) is embarking on plans to generate a twenty year reanalysis data set of climatic atmospheric variables. One of the focus points will be in the evaluation of the dynamics of the stratosphere. The Stratospheric Sounding Unit (SSU), flown as part of the TIROS Operational Vertical Sounder (TOVS), is one of the primary stratospheric temperature sensors flown consistently throughout the reanalysis period. Seven unique sensors made the measurements over time, with individual instrument characteristics that need to be addressed. The stratospheric temperatures being assimilated across satellite platforms will profoundly impact the reanalysis dynamical fields. To attempt to quantify aspects of instrument and retrieval bias we are carefully collecting and analyzing all available information on the sensors, their instrument anomalies, forward model errors and retrieval biases. For the retrieval of stratospheric temperatures, we adapted the minimum variance approach of Jazwinski (1970) and Rodgers (1976) and applied it to the SSU soundings. In our algorithm, the state vector contains an initial guess of temperature from a model six hour forecast provided by the Goddard EOS Data Assimilation System (GEOS/DAS). This is combined with an a priori covariance matrix, a forward model parameterization, and specifications of instrument noise characteristics. A quasi-Newtonian iteration is used to obtain convergence of the retrieved state to the measurement vector. This algorithm also enables us to analyze and address the systematic errors associated with the unique characteristics of the cell pressures on the individual SSU instruments and the resolving power of the instruments to vertical gradients in the stratosphere. The preliminary results of the improved retrievals and their assimilation as well as baseline calculations of bias and rms error between the NESDIS operational product and col-located ground measurements will be presented.

Rokke, L.↗
NASA NTRS1999

Assimilation of Satellite Altimeter Data With a Four Dimensional Model of the Japan Sea

A data assimilation is carried out to detect the variability of the Japan Sea circulation in the range from a few days to several years and from eddy to basin scale. The model applied in this study is the same 1/6 degree GFDL MOM1 as Kim and Yoon (1999) but is driven by ECMWF daily wind stress, heat and fresh water fluxes. The satellite altimeter data of TOPEX/POSEIDON, ERS-1 (phase C and G) and -2 are assimilated by an approximate Kalman filter (Fukumori and M.Rizzoli, 1995). The approximation is made by seeking an asymptotic steady error covariance matrix (Fukumori et al., 1993) and by introducing a coarser grid model for the innovation (data-model misfit). The coarse grid model is defined on 1/2 degree horizontal resolution and consists of the barotropic stream function, first baroclinic displacement and velocity amplitudes. The assimilated estimates explain about 6cm sea level variability of the data (approximately 12cm in the southern part), which is much larger than the previous reduced-gravity model and TOPEX/POSEIDON altimeter data assimilation (Hirose et al., 1999). The impacts of the T/P and ERS data on the filtered estimates are comparable. The result also shows high correlation to subsurface water temperatures measured by CTD. Many of the mesoscale eddies/disturbances travel east-northeastward with the advection speed of 1-3cm/s though most of them generated in the western region can not pass over the Oki Spur. The quasi-biennial variability found by Hirose and Ostrovskii (1999) did not show clear propagation pattern. The shallow Oki Spur may work as a "western boundary" to this signal. This is more plausible estimation than by the R-G model which has no bottom topography.

Hirose, Naoki↗
NASA NTRS2000

Effect of Medium Symmetries in Limiting the Number of Parameters Estimated with Polarimetric Interferometry

The addition of interferometric backscattering pairs to the conventional polarimetric SAR data over forests and other vegetated areas increases the dimensionality of the data space, in principle enabling the estimation of a larger number of vegetation parameters. Without regard to the sensitivity of these data to vegetation scattering parameters, this paper poses the question: Will increasing the data channels as such result in a one-to-one increase in the number of parameters that can be estimated, or do vegetation and data properties inherently limit that number otherwise? In this paper, the complete polarimetric interferometric covariance matrix is considered and various symmetry properties of the scattering medium are used to study whether any of the correlation pairs can be eliminated. The number of independent pairs has direct consequences in their utility in parameter estimation schemes, since the maximum number of parameters that can be estimated cannot exceed the number of unique measurements. The independent components of the polarimetric interferometric SAR (POL/INSAR) data are derived for media with reflection, rotation, and azimuth symmetries, which are often encountered in vegetated surfaces. Similar derivations have been carried out before for simple polarimetry, i.e., zero baseline. This paper extends those to the interferometric case of general nonzero baselines. It is shown that depending on the type of symmetries present, the number of independent available measurements that can be used to estimate medium parameters will vary. In particular, whereas in the general case there are 27 mathematically independent measurements possible from a polarimetric interferometer, this number can be reduced to 15, 9, and 6 if the medium has reflection, rotation, or azimuthal symmetries, respectively. The results can be used in several ways in the interpretation of SAR data and the development of parameter estimation schemes, which will be discussed at the presentation. Recent POL/INSAR data from the JPL AIRSAR over a forested area will be used to demonstrate the results of this derivation. This work was performed by the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, under contract from the National Aeronautics and Space Administration.

Moghaddam, Mahta↗
NASA NTRS2001

Principal Components Analysis of Triaxial Vibration Data From Helicopter Transmissions

Research on the nature of the vibration data collected from helicopter transmissions during flight experiments has led to several crucial observations believed to be responsible for the high rates of false alarms and missed detections in aircraft vibration monitoring systems. This work focuses on one such finding, namely, the need to consider additional sources of information about system vibrations. In this light, helicopter transmission vibration data, collected using triaxial accelerometers, were explored in three different directions, analyzed for content, and then combined using Principal Components Analysis (PCA) to analyze changes in directionality. In this paper, the PCA transformation is applied to 176 test conditions/data sets collected from an OH58C helicopter to derive the overall experiment-wide covariance matrix and its principal eigenvectors. The experiment-wide eigenvectors. are then projected onto the individual test conditions to evaluate changes and similarities in their directionality based on the various experimental factors. The paper will present the foundations of the proposed approach, addressing the question of whether experiment-wide eigenvectors accurately model the vibration modes in individual test conditions. The results will further determine the value of using directionality and triaxial accelerometers for vibration monitoring and anomaly detection.

Tumer, Irem Y.↗
NASA NTRS2002

Retrieval of Atmospheric and Surface Parameters from AIRS/AMSU/HSB Data Under Cloudy Conditions

New state of the art methodology is described to analyze AIRS/AMSU/HSB data in the presence of multiple cloud formations. The methodology forms the basis for the AIRS Science Team algorithm which will be used to analyze AIRS/AMSU/HSB data on EOS Aqua. The cloud clearing methodology requires no knowledge of the spectral properties of the clouds. The basic retrieval methodology is general and extracts the maximum information from the radiances, consistent with the channel noise covariance matrix. The retrieval methodology minimizes the dependence of the solution on the first guess field and the first guess error characteristics. Results are shown for AIRS Science Team simulation studies with multiple cloud formations. These simulation studies imply that clear column radiances can be reconstructed under partial cloud cover with an accuracy comparable to single spot channel noise in the temperature and water vapor sounding regions, temperature soundings can be produced under partial cloud cover with RMS errors on the order of, or better than, 1deg K in 1 km thick layers from the surface to 700 mb, 1 km layers from 700 mb to 300 mb, 3 km layers from 300 mb to 30 mb, and 5 km layers from 30 mb to 1 mb, and moisture profiles can be obtained with an accuracy better than 20% absolute errors in 1 km layers from the surface to nearly 200 mb.

Susskind, Joel↗
NASA NTRS2001

Maximally Informative Statistics for Localization and Mapping

This paper presents an algorithm for localization and mapping for a mobile robot using monocular vision and odometry as its means of sensing. The approach uses the Variable State Dimension filtering (VSDF) framework to combine aspects of Extended Kalman filtering and nonlinear batch optimization. This paper describes two primary improvements to the VSDF. The first is to use an interpolation scheme based on Gaussian quadrature to linearize measurements rather than relying on analytic Jacobians. The second is to replace the inverse covariance matrix in the VSDF with its Cholesky factor to improve the computational complexity. Results of applying the filter to the problem of localization and mapping with omnidirectional vision are presented.

Deans, Matthew C.↗
NASA NTRS2003

An optimal modification of a Kalman filter for time scales

The Kalman filter in question, which was implemented in the time scale algorithm TA(NIST), produces time scales with poor short-term stability. A simple modification of the error covariance matrix allows the filter to produce time scales with good stability at all averaging times, as verified by simulations of clock ensembles.

time scale Kalman filter↗
NASA NTRS1995

A Reduced Dimension Static, Linearized Kalman Filter and Smoother

An approximate Kalman filter and smoother, based on approximations of the state estimation error covariance matrix, is described. Approximations include a reduction of the effective state dimension, use of a static asymptotic error limit, and a time-invariant linearization of the dynamic model for error integration. The approximations lead to dramatic computational savings in applying estimation theory to large complex systems. Examples of use come from TOPEX/POSEIDON.

Kalman filter smoother modeling ocean simulation↗
NASA NTRS2007

Non-Parametric Collision Probability for Low-Velocity Encounters

An implicit, but not necessarily obvious, assumption in all of the current techniques for assessing satellite collision probability is that the relative position uncertainty is perfectly correlated in time. If there is any mis-modeling of the dynamics in the propagation of the relative position error covariance matrix, time-wise de-correlation of the uncertainty will increase the probability of collision over a given time interval. The paper gives some examples that illustrate this point. This paper argues that, for the present, Monte Carlo analysis is the best available tool for handling low-velocity encounters, and suggests some techniques for addressing the issues just described. One proposal is for the use of a non-parametric technique that is widely used in actuarial and medical studies. The other suggestion is that accurate process noise models be used in the Monte Carlo trials to which the non-parametric estimate is applied. A further contribution of this paper is a description of how the time-wise decorrelation of uncertainty increases the probability of collision.

Carpenter, J. Russell↗
NASA NTRS2004

Three dimensional empirical mode decomposition analysis apparatus, method and article manufacture

An apparatus and method of analysis for three-dimensional (3D) physical phenomena. The physical phenomena may include any varying 3D phenomena such as time varying polar ice flows. A repesentation of the 3D phenomena is passed through a Hilbert transform to convert the data into complex form. A spatial variable is separated from the complex representation by producing a time based covariance matrix. The temporal parts of the principal components are produced by applying Singular Value Decomposition (SVD). Based on the rapidity with which the eigenvalues decay, the first 3-10 complex principal components (CPC) are selected for Empirical Mode Decomposition into intrinsic modes. The intrinsic modes produced are filtered in order to reconstruct the spatial part of the CPC. Finally, a filtered time series may be reconstructed from the first 3-10 filtered complex principal components.

Gloersen, Per↗
NASA NTRS2008

Comparison of Nonlinear Filtering Techniques for Lunar Surface Roving Navigation

Leading up to the Apollo missions the Extended Kalman Filter, a modified version of the Kalman Filter, was developed to estimate the state of a nonlinear system. Throughout the Apollo missions, Potter's Square Root Filter was used for lunar navigation. Now that NASA is returning to the Moon, the filters used during the Apollo missions must be compared to the filters that have been developed since that time, the Bierman-Thornton Filter (UD) and the Unscented Kalman Filter (UKF). The UD Filter involves factoring the covariance matrix into UDUT and has similar accuracy to the Square Root Filter; however it requires less computation time. Conversely, the UKF, which uses sigma points, is much more computationally intensive than any of the filters; however it produces the most accurate results. The Extended Kalman Filter, Potter's Square Root Filter, the Bierman-Thornton UD Filter, and the Unscented Kalman Filter each prove to be the most accurate filter depending on the specific conditions of the navigation system.

Kimber, Lemon↗