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NASA NTRS1993

Attitude determination using vector observations: A fast optimal matrix algorithm

The attitude matrix minimizing Wahba's loss function is computed directly by a method that is competitive with the fastest known algorithm for finding this optimal estimate. The method also provides an estimate of the attitude error covariance matrix. Analysis of the special case of two vector observations identifies those cases for which the TRIAD or algebraic method minimizes Wahba's loss function.

Markley, F. Landis↗

NASA NTRS1993

Attitude determination using vector observations - A fast optimal matrix algorithm

Markley, F. L.↗

NASA NTRS1993

Failure modes of reduced-order orbit determination filters and their remedies

Ways in which failure can occur in reduced-order, orbit determination filter, error covariance calculations are discussed. In the context of this article, reduced-order filters denote nonoptimal filters which include fixed levels of uncertainty in some parameters of the measurement models or in the spacecraft dynamical model which are not explicitly estimated in the filter equations. Failure is defined as an increase in the orbit determination covariance with the addition of data or as an unreasonable growth in the covariance with time, i.e., nonasymptotic behavior of the covariance. Some simple, known cases of failure are discussed along with their traditional remedies. In addition, more modern remedies are discussed which are currently under development at the Jet Propulsion Laboratory. The article first describes the known problems of reduced-order filters when they are employed for orbit determination, and their traditional remedies. Then, having defined these, the relevancy and desirability of the more modern remedies are made apparent.

Scheeres, D. J.↗

NASA NTRS1993

Optimal gains for a single polar orbiting satellite

Gains are the spatial weighting of an observation in its neighborhood versus the local values of a model prediction. They are the key to data assimilation, as they are the direct measure of how the data are used to guide the model. As derived in the broad context of data assimilation by Kalman and in the context of meteorology, for example, by Rutherford, the optimal gains are functions of the prediction error covariances between the observation and analysis points. Kalman introduced a very powerful technique that allows one to calculate these optimal gains at the time of each observation. Unfortunately, this technique is both computationally expensive and often numerically unstable for dynamical systems of the magnitude of meteorological models, and thus is unsuited for use in PMIRR data assimilation. However, the optimal gains as calculated by a Kalman filter do reach a steady state for regular observing patterns like that of a satellite. In this steady state, the gains are constants in time, and thus could conceivably be computed off-line. These steady-state Kalman gains (i.e., Wiener gains) would yield optimal performance without the computational burden of true Kalman filtering. We proposed to use this type of constant-in-time Wiener gain for the assimilation of data from PMIRR and Mars Observer.

Banfield, Don↗

NASA NTRS1994

Spacecraft-spacecraft radio-metric tracking: Signal acquisition requirements and application to Mars approach navigation

Doppler and ranging measurements between spacecraft can be obtained only when the ratio of the total received signal power to noise power density (P(sub t)/N(sub 0)) at the receiving spacecraft is sufficiently large that reliable signal detection can be achieved within a reasonable time period. In this article, the requirement on P(sub t)/N(sub 0) for reliable carrier signal detection is calculated as a function of various system parameters, including characteristics of the spacecraft computing hardware and a priori uncertainty in spacecraft-spacecraft relative velocity and acceleration. Also calculated is the P(sub t)/N(sub 0) requirements for reliable detection of a ranging signal, consisting of a carrier with pseudonoise (PN) phase modulation. Once the P(sub t)/N(sub 0) requirement is determined, then for a given set of assumed spacecraft telecommunication characteristics (transmitted signal power, antenna gains, and receiver noise temperatures) it is possible to calculate the maximum range at which a carrier signal or ranging signal may be acquired. For example, if a Mars lander and a spacecraft approaching Mars are each equipped with 1-m-diameter antennas, the transmitted power is 5 W, and the receiver noise temperatures are 350 K, then S-band carrier signal acquisition can be achieved at ranges exceeding 10 million km. An error covariance analysis illustrates the utility of in situ Doppler and ranging measurements for Mars approach navigation. Covariance analysis results indicate that navigation accuracies of a few km can be achieved with either data type. The analysis also illustrates dependency of the achievable accuracy on the approach trajectory velocity.

Kahn, R. D.↗

NASA NTRS1995

Technical report series on global modeling and data assimilation. Volume 4: Documentation of the Goddard Earth Observing System (GEOS) data assimilation system, version 1

This report describes the analysis component of the Goddard Earth Observing System, Data Assimilation System, Version 1 (GEOS-1 DAS). The general features of the data assimilation system are outlined, followed by a thorough description of the statistical interpolation algorithm, including specification of error covariances and quality control of observations. We conclude with a discussion of the current status of development of the GEOS data assimilation system. The main components of GEOS-1 DAS are an atmospheric general circulation model and an Optimal Interpolation algorithm. The system is cycled using the Incremental Analysis Update (IAU) technique in which analysis increments are introduced as time independent forcing terms in a forecast model integration. The system is capable of producing dynamically balanced states without the explicit use of initialization, as well as a time-continuous representation of non- observables such as precipitation and radiational fluxes. This version of the data assimilation system was used in the five-year reanalysis project completed in April 1994 by Goddard's Data Assimilation Office (DAO) Data from this reanalysis are available from the Goddard Distributed Active Center (DAAC), which is part of NASA's Earth Observing System Data and Information System (EOSDIS). For information on how to obtain these data sets, contact the Goddard DAAC at (301) 286-3209, EMAIL daac@gsfc.nasa.gov.

Suarez, Max J.↗

NASA NTRS1993

Radio Doppler navigation of interplanetary spacecraft using different data processing modes

Doppler shift measurements derived from closed-loop radio tracking of distant spacecraft by ground stations of the Deep Space Network (DSN) are one of the principal means used for interplanetary navigation. This paper describes an investigation of the navigation accuracy that can be achieved with two-way coherent X-band (7.2 to 8.4 GHz) Doppler phase and frequency data; both theoretical and practical aspects of the two Doppler formulations are addressed. A new filtering strategy is also proposed, which differs from current approaches in that most of the ground system calibration errors affecting the Doppler data are represented as filter parameters, in addition to the spacecraft trajectory parameters. Error covariance computations for application of the proposed filter to a navigation scenario derived from the Mars Observer mission are provided, in order to assess the performance that might be obtained in practice. The results indicate that with a phase formulation of the Doppler data and the new filter, navigation accuracies of 15 km at Mars (0.05 microrad in an angular sense) are possible with the DSN's present X-band Doppler tracking system.

Thurman, S. W.↗

NASA NTRS1993

In-situ radio-metric tracking to support navigation for interplanetary missions with multiple spacecraft

Doppler and ranging measurements between spacecraft can be obtained only when the ratio of the total received signal power to noise power density (P(sub t/N(sub 0)) at the receiving spacecraft is sufficiently large that reliable signal detection can be achieved within a reasonable time period. In this paper, the requirements on P(sub t)/N(sub 0) for reliable carrier signal detection is calculated as a function of various system parameters, including characteristics of the spacecraft computing hardware and a priori uncertainty in spacecraft-spacecraft relative velocity and acceleration. Also calculated is the P(sub t)/N(sub 0) requirement for relaible detection of a ranging signal, consistting of a carrier with pseudo-noise phase modulation. Once the P(sub t)/N(sub 0) requirement is determined, then for a given set of assumed spacecraft telecommunication characteristics (transmitted signal power, antenna gains, receiver noise temperatures) it is possible to calculate the maximum range at which a carrier signal or ranging signal may be acquired. A brief error covariance analysis has been conducted to illustrate the utility of in situ Doppler and ranging measurements for Mars approach navigation. The results indicate that navigation accuracies of a few kilometers can be achieved with either data type. The analysis also illustrates dependency of the achievable accuracy on the approach trajectory velocity.

Kahn, Robert D.↗

NASA NTRS1996

Conflict Probability Estimation for Free Flight

The safety and efficiency of free flight will benefit from automated conflict prediction and resolution advisories. Conflict prediction is based on trajectory prediction and is less certain the farther in advance the prediction, however. An estimate is therefore needed of the probability that a conflict will occur, given a pair of predicted trajectories and their levels of uncertainty. A method is developed in this paper to estimate that conflict probability. The trajectory prediction errors are modeled as normally distributed, and the two error covariances for an aircraft pair are combined into a single equivalent covariance of the relative position. A coordinate transformation is then used to derive an analytical solution. Numerical examples and Monte Carlo validation are presented.

Paielli, Russell A.↗

NASA NTRS1998

Implementation of a Parallel Kalman Filter for Stratospheric Chemical Tracer Assimilation

A Kalman filter for the assimilation of long-lived atmospheric chemical constituents has been developed for two-dimensional transport models on isentropic surfaces over the globe. An important attribute of the Kalman filter is that it calculates error covariances of the constituent fields using the tracer dynamics. Consequently, the current Kalman-filter assimilation is a five-dimensional problem (coordinates of two points and time), and it can only be handled on computers with large memory and high floating point speed. In this paper, an implementation of the Kalman filter for distributed-memory, message-passing parallel computers is discussed. Two approaches were studied: an operator decomposition and a covariance decomposition. The latter was found to be more scalable than the former, and it possesses the property that the dynamical model does not need to be parallelized, which is of considerable practical advantage. This code is currently used to assimilate constituent data retrieved by limb sounders on the Upper Atmosphere Research Satellite. Tests of the code examined the variance transport and observability properties. Aspects of the parallel implementation, some timing results, and a brief discussion of the physical results will be presented.

Chang, Lang-Ping↗

NASA NTRS1995

Assimilation of TOPEX Sea Level Measurements with a Reduced-Gravity, Shallow Water Model of the Tropical Pacific Ocean

Sea surface height variability measured by TOPEX is analyzed in the tropical Pacific Ocean by way of assimilation into a wind-driven, reduced-gravity, shallow water model using an approximate Kalman filter and smoother. The analysis results in an optimal fit of the dynamic model to the observations, providing it dynamically consistent interpolation of sea level and estimation of the circulation. Nearly 80% of the expected signal variance is accounted for by the model within 20 deg of the equator, and estimation uncertainty is substantially reduced by the voluminous observation. Notable features resolved by the analysis include seasonal changes associated with the North Equatorial Countercurrent and equatorial Kelvin and Rossby waves. Significant discrepancies are also found between the estimate and TOPEX measurements, especially near the eastern boundary. Improvements in the estimate made by the assimilation are validated by comparisons with independent tide gauge and current meter observations. The employed filter and smoother are based on approximately computed estimation error covariance matrices, utilizing a spatial transformation and an symptotic approximation. The analysis demonstrates the practical utility of a quasi-optimal filter and smoother.

Fukumori, Ichiro↗

NASA NTRS1999

Improving Reanalyses Using TRMM and SSM/I-Derived Precipitation and Total Precipitable Water Observations

Global reanalyses currently contain significant errors in the primary fields of the hydrological cycle such as precipitation, evaporation, moisture, and the related cloud fields, especially in the tropics. The Data Assimilation Office (DAO) at the NASA Goddard Space Flight Center has been exploring the use of rainfall and total precipitable water (TPW) observations from the Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) and the Special Sensor Microwave/ Imager (SSM/I) instruments to improve these fields in reanalyses. The DAO has developed a "1+1"D procedure to assimilate 6-hr averaged rainfall and TPW into the Goddard Earth Observing System (GEOS) Data Assimilation System (DAS). The algorithm is based on a 6-hr time integration of a column version of the GEOS DAS. The "1+1" designation refers to one spatial dimension plus one temporal dimension. The scheme minimizes the least-square differences between the satellite-retrieved rain rates and those produced by the column model over the 6-hr analysis window. The control variables are analysis increments of moisture within the Incremental Analysis Update (IAU) framework of the GEOS DAS. This 1+1D scheme, in its generalization to four dimensions, is related to the standard 4D variational assimilation but differs in its choice of the control variable. Instead of estimating the initial condition at the beginning of the assimilation cycle, it estimates the constant IAU forcing applied over a 6-hr assimilation cycle. In doing so, it imposes the forecast model as a weak constraint in a manner similar to the variational continuous assimilation techniques. We present results from an experiment in which the observed rain rate and TPW are assumed to be "perfect". They show that assimilating the TMI and SSM/I-derived surface precipitation and TPW observations improves not only the precipitation and moisture fields but also key climate parameters directly linked to convective activities such as clouds, the outgoing longwave radiation, and the large-scale circulation in the tropics. In particular, assimilating these data types reduce the state-dependent systematic errors in the assimilated products. The improved analysis also leads to a better short-range forecast, but the impact is modest compared with improvements in the time-averaged fields. These results suggest that, in the presence of biases and other errors of the forecast model, it is possible to improve the time-averaged "climate content" in the assimilated data without comparable improvements in the short-range forecast skill. Results of this experiment provide a useful benchmark for evaluating error covariance models for optimal use of these data types.

Hou, Arthur Y.↗

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

A Global Data Assimilation System for Atmospheric Aerosol

We will give an overview of an aerosol data assimilation system which combines advances in remote sensing of atmospheric aerosols, aerosol modeling and data assimilation methodology to produce high spatial and temporal resolution 3D aerosol fields. Initially, the Goddard Aerosol Assimilation System (GAAS) will assimilate TOMS, AVHRR and AERONET observations; later we will include MODIS and MISR. This data assimilation capability will allows us to integrate complementing aerosol observations from these platforms, enabling the development of an assimilated aerosol climatology as well as a global aerosol forecasting system in support of field campaigns. Furthermore, this system provides an interactive retrieval framework for each aerosol observing satellites, in particular TOMS and AVHRR. The Goddard Aerosol Assimilation System (GAAS) takes advantage of recent advances in constituent data assimilation at DAO, including flow dependent parameterizations of error covariances and the proper consideration of model bias. For its prognostic transport model, GAAS will utilize the Goddard Ozone, Chemistry, Aerosol, Radiation and Transport (GOCART) model developed at NASA/GSFC Codes 916 and 910.3. GOCART includes the Lin-Rood flux-form, semi-Langrangian transport model with parameterized aerosol chemistry and physical processes for absorbing (dust and black carbon) and non-absorbing aerosols (sulfate and organic carbon). Observations and model fields are combined using a constituent version of DAO's Physical-space Statistical Analysis System (PSAS), including its adaptive quality control system. In this talk we describe the main components of this assimilation system and present preliminary results obtained by assimilating TOMS data.

daSilva, Arlindo↗

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

Comparison of On-Orbit Performance of Rate Sensing Gyroscopes

This work presents results from the study of a large volume of spacecraft flight data pertaining to gyroscope performance. We have examined long and short term trends of gyroscope biase of first, an exact solution for the time-dependence of the attitude part of the state error covariance, averaged over the three spacecraft axes, is given. This solution is more complete than the usual cubic polynomial for the variance in that it includes coupling among the three gyroscope axes and includes an important term arising from the initial correlations. Second, several continuous 24-hour spans of gyroscope data are examined to verify the short-term statistical model. This analysis demonstrates that in-flight data can be used to determine the strength of the white noise driving the random walk of the gyroscope bias. This may be useful for postlaunch improvement to the noise model and for diagnosing the health of the gyroscope. Third, the long-term trends in gyroscope biases show a nearly linear systematic variation over time scales of years. This has been found on three different missions. While the random walk model is adequate as a basis for onboard Kalman filters or for state estimation using relatively short time spans, these trends indicate that some applications could benefit by accounting for the secular changes in the biases. One example is a new gyroscope calibration method that is under development that allows for multi-epoch bias solutions.

Sedlak, Joseph↗

NASA NTRS2000

Preliminary Results from an Assimilation of Saharan Dust Using TOMS Radiances and the GOCART Model

At NASA Goddard we are developing a global aerosol data assimilation system that combines advances in remote sensing and modeling of atmospheric aerosols. The goal is to provide high resolution, 3-D aerosol distributions to the research community. Our first step is to develop a simple assimilation system for Saharan mineral aerosol. The Goddard Chemistry and Aerosol Radiation model (GOCART) provides accurate 3-D mineral aerosol size distributions. Surface mobilization, wet and dry deposition, convective and long-range transport are all driven by assimilated fields from the Goddard Earth Observing System Data Assimilation System, GEOS-DAS. Our version of GOCART transports sizes from .08-10 microns and only simulates Saharan dust. We draw the assimilation to two observables in this study: the TOMS aerosol index (Al) which is directly related to the ratio of the 340 and 380 radiances and the 380 radiance alone. The forward model that simulates the observables requires the aerosol optical thickness, the single scattering albedo and the height of the aerosol layer from the GOCART fields. The forward model also requires a refractive index for the dust. We test three index values to see which best fits the TOMS observables. These are 1) for Saharan dust reported by Patterson, 2) for a mixture of Saharan dust and a highly reflective material (sea salt or sulfate) and 3) for pure illite. The assimilation works best assuming either pure illite or the dust mixture. Our assimilation cycle first determines values of the aerosol index (Al) and the radiance at 380 nm based on the GOCART aerosol fields. Differences between the observed and GOCART model calculated Al and 380 nm radiance are first analyzed horizontally using the Physical-space Statistical Analysis System (PSAS). A quasi-Newton iteration is then performed to produce analyzed 3D aerosol fields according to parameterized background and observation error covariances. We only assimilate observations into the the GOCART model over regions of Africa and the Atlantic where mineral aerosols are dominant and carbonaceous aerosols are minimal.

Weaver, C. J.↗

NASA NTRS2001

DAO's Next Generation Physical-Space/Finite-Volume Data Assimilation System: Formulation and Initial Evaluation

The Physical-space/Finite-volume Data Assimilation System (fvDAS) is the next generation global atmospheric data assimilation system in development at the Data Assimilation Office at NASA's Goddard Space Flight Center. It is based on a new finite-volume general circulation model jointly developed by NASA and NCAR, and on the Physical-Space Statistical Analysis System (PSAS) developed at the DAO. In this talk we will describe the general system formulation, the adaptive quality control and general aspects of the error covariance modeling. The NASA-NCAR GCM is a completely new model which replaces the CEOs GCM used in the previous GEOS-1/2/3 Data Assimilation systems. A particular configuration of adaptive Statistical Quality Control and the Physical-space Statistical Analysis System (PSAS) are currently implemented in DAO's operational Data Assimilation System. However, the unique finite-volume formulation of the NASA-NCAR GCM, combined with the generality of the observation-space formulation of PSAS, provides for a very simple and accurate model-analysis interface. The system assimilates a variety of conventional and satellite observations. In particular, TOVS Level 1B radiances are assimilated using a 1-D variational scheme, both in clear sky and cloudy conditions. Computationally, the fvDAS runs approximately 10 times faster than the operational GEOS-Terra system. We will show that the next-generation fvDAS has much improved observation-minus-6hr forecast (O-F) statistics, as well as 5-day forecast skills. Top of the atmosphere radiation fields are in closer agreement with CERES measurements, with realistic precipitation and moisture fields. We will also show that the finite-volume formulation of the fvDAS produce assimilated fields which are more suitable for driving constituent transport models.

daSilva, A.↗