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NASA NTRS2002

A Wavelet Based Suboptimal Kalman Filter for Assimilation of Stratospheric Chemical Tracer Observations

A suboptimal Kalman filter system which evolves error covariances in terms of a truncated set of wavelet coefficients has been developed for the assimilation of chemical tracer observations of CH4. The truncation is carried out in such a way that the resolution of the error covariance, is reduced only in the zonal direction, where gradients are smaller. Assimilation experiments which last 24 days, and used different degrees of truncation were carried out. These reduced the covariance, by 90, 97 and 99 % and the computational cost of covariance propagation by 80, 93 and 96 % respectively. The difference in both error covariance and the tracer field between the truncated and full systems over this period were found to be not growing in the first case, and a growing relatively slowly in the later two cases. The largest errors in the tracer fields were found to occur in regions of largest zonal gradients in the tracer field.

Auger, Ludovic↗

NASA NTRS2003

On the Use of the Log-Normal Particle Size Distribution to Characterize Global Rain

Although most parameterizations of the drop size distributions (DSD) use the gamma function, there are several advantages to the log-normal form, particularly if we want to characterize the large scale space-time variability of the DSD and rain rate. The advantages of the distribution are twofold: the logarithm of any moment can be expressed as a linear combination of the individual parameters of the distribution; the parameters of the distribution are approximately normally distributed. Since all radar and rainfall-related parameters can be written approximately as a moment of the DSD, the first property allows us to express the logarithm of any radar/rainfall variable as a linear combination of the individual DSD parameters. Another consequence is that any power law relationship between rain rate, reflectivity factor, specific attenuation or water content can be expressed in terms of the covariance matrix of the DSD parameters. The joint-normal property of the DSD parameters has applications to the description of the space-time variation of rainfall in the sense that any radar-rainfall quantity can be specified by the covariance matrix associated with the DSD parameters at two arbitrary space-time points. As such, the parameterization provides a means by which we can use the spaceborne radar-derived DSD parameters to specify in part the covariance matrices globally. However, since satellite observations have coarse temporal sampling, the specification of the temporal covariance must be derived from ancillary measurements and models. Work is presently underway to determine whether the use of instantaneous rain rate data from the TRMM Precipitation Radar can provide good estimates of the spatial correlation in rain rate from data collected in 5(sup 0)x 5(sup 0) x 1 month space-time boxes. To characterize the temporal characteristics of the DSD parameters, disdrometer data are being used from the Wallops Flight Facility site where as many as 4 disdrometers have been used to acquire data over a 2 km path. These data should help quantify the temporal form of the covariance matrix at this site.

Meneghini, Robert↗

NASA NTRS2013

Spatial Statistical Data Fusion (SSDF)

As remote sensing for scientific purposes has transitioned from an experimental technology to an operational one, the selection of instruments has become more coordinated, so that the scientific community can exploit complementary measurements. However, tech nological and scientific heterogeneity across devices means that the statistical characteristics of the data they collect are different. The challenge addressed here is how to combine heterogeneous remote sensing data sets in a way that yields optimal statistical estimates of the underlying geophysical field, and provides rigorous uncertainty measures for those estimates. Different remote sensing data sets may have different spatial resolutions, different measurement error biases and variances, and other disparate characteristics. A state-of-the-art spatial statistical model was used to relate the true, but not directly observed, geophysical field to noisy, spatial aggregates observed by remote sensing instruments. The spatial covariances of the true field and the covariances of the true field with the observations were modeled. The observations are spatial averages of the true field values, over pixels, with different measurement noise superimposed. A kriging framework is used to infer optimal (minimum mean squared error and unbiased) estimates of the true field at point locations from pixel-level, noisy observations. A key feature of the spatial statistical model is the spatial mixed effects model that underlies it. The approach models the spatial covariance function of the underlying field using linear combinations of basis functions of fixed size. Approaches based on kriging require the inversion of very large spatial covariance matrices, and this is usually done by making simplifying assumptions about spatial covariance structure that simply do not hold for geophysical variables. In contrast, this method does not require these assumptions, and is also computationally much faster. This method is fundamentally different than other approaches to data fusion for remote sensing data because it is inferential rather than merely descriptive. All approaches combine data in a way that minimizes some specified loss function. Most of these are more or less ad hoc criteria based on what looks good to the eye, or some criteria that relate only to the data at hand.

Braverman, Amy J.↗

NASA NTRS2013

Impact of Gender on Pharmocokinetics of Intranasal Scopolamine

Introduction: An intranasal gel dosage formulation of scopolamine (INSCOP) was developed for the treatment of Space Motion Sickness (SMS), which is commonly experienced by astronauts during space missions. The bioavailability and pharmacokinetics (PK) were evaluated under IND guidelines. Since information is lacking on the effect of gender on the PK of Scopolamine, we examined gender differences in PK parameters of INSCOP at three dose levels of 0.1, 0.2 and 0.4 mg. Methods: Plasma scopolamine concentrations as a function of time data were collected from twelve normal healthy human subjects (6 male/6 female) who participated in a fully randomized double blind crossover study. The PK parameters were derived using WinNonlin. Covariate analysis of PK profiles was performed using NONMEN and statistically compared using a likelihood ratio test on the difference of objective function value (OFV). Statistical significance for covariate analysis was set at P<0.05(ΔOFV=3.84). Results: No significant difference in PK parameters between male and female subjects was observed with 0.1 and 0.2 mg doses. However, CL and Vd were significantly different between male and female subjects at the 0.4 mg dose. Results from population covariate modeling analysis indicate that a onecompartment PK model with first-order elimination rate offers best fit for describing INSCOP concentration-time profiles. The inclusion of sex as a covariate enhanced the model fitting (ΔOFV=-4.1) owing to the genderdependent CL and Vd differences after the 0.4 mg dose. Conclusion: Statistical modeling of scopolamine concentration-time data suggests gender-dependent pharmacokinetics of scopolamine at the high dose level of 0.4 mg. Clearance of the parent compound was significantly faster and the volume of distribution was significantly higher in males than in females, As a result, including gender as a covariate to the pharmacokinetic model of scopolamine offers the best fit for PK modeling of the drug at dose of 0.4 mg or higher.

Putcha, L.↗

NASA NTRS2018

The Effects of Vegetative Type, Edges, Fire History, Rainfall and Management in Fire-Maintained Habitat

The combined effects of fire history, climate, and landscape features (e.g., edges) on habitat specialists need greater focus in fire ecology studies, which usually only emphasize characteristics of the most recent fire. Florida scrub-jays are an imperiled, territorial species that prefer medium (1.2-1.7 m) shrub heights, which are dynamic because of frequent fires. We measured short, medium, and tall habitat quality states annually within 10 ha grid cells (that represented potential territories) because fires and vegetative recovery cause annual variation in habitat quality. We used multistate models and model selection to test competing hypotheses about how transition probabilities vary between states as functions of environmental covariates. Covariates included vegetative type, edges (e.g., roads, forests), precipitation, openings (gaps between shrubs), mechanical cutting, and fire characteristics. Fire characteristics not only included an annual presence/absence of fire covariate, but also fire history covariates: time since the previous fire, the longest fire-free interval, and the number of repeated fires. Statistical models with support included many covariates for each transition probability, often including fire history, interactions and nonlinear relationships. Tall territories resulted from 28 years of fire suppression and habitat fragmentation that reduced the spread of fires across landscapes. Despite 35 years of habitat restoration and prescribed fires, half the territories remained tall suggesting a regime shift to a less desirable habitat condition. Edges reduced the effectiveness of fires in setting degraded scrub and flatwoods into earlier successional states making mechanical cutting an important tool to compliment frequent prescribed fires.

Breininger, David R.↗

NASA NTRS2018

Atmosphere-Ocean Coupled Data Assimilation Using NASA GEOS: Estimation of Air-Sea Interface State Variables

Air-sea interface variables, such as the skin Sea Surface Temperature (SST) are essential for atmosphere-ocean coupling. In the NASA GMAO Data Assimilation System (DAS), the skin SST and 3-D atmospheric state are jointly estimated [1]. This presentation is focused on the prior or background error covariance that is used in this analysis. The GEOS DAS uses an ensemble-variational assimilation strategy. In that, specification of a climatological background (CB) error covariance for SST relies on the NOAA's OI SST, with estimates of standard deviation and correlation length scales based on weekly analyses of the bulk SST at 1 degree resolution. However, present analysis system is striving to resolve SST diurnal variability with six hourly analyses and assimilates a vast number of in-situ and satellite observations. The first part of this presentation re-derives the CB error covariance using OSTIA SST analyses and illustrates the impact of this update on assimilating satellite observations. In a hybrid assimilation system the CB error covariances are appended with a flow-dependent background error covariance estimate implied by the underlying ensemble. The second part of this presentation refers to: a. treatment of the skin SST in the ensemble members, b. corresponding ensemble spread, and c. impact of these additions on the data assimilation system. [1] S. Akella, et al. (2017), doi:10.1002/qj.2988

Akella, Santha↗

NASA NTRS2020

Untangling causality in midlatitude aerosol–cloud adjustments

Aerosol–cloud interactions represent the leading uncertainty in our ability to infer climate sensitivity from the observational record. The forcing from changes in cloud albedo driven by increases in cloud droplet number (Nd) (the first indirect effect) is confidently negative and has narrowed its probable range in the last decade, but the sign and strength of forcing associated with changes in cloud macrophysics in response to aerosol (aerosol–cloud adjustments) remain uncertain. This uncertainty reflects our inability to accurately quantify variability not associated with a causal link flowing from the cloud microphysical state to the cloud macrophysical state. Once variability associated with meteorology has been removed, covariance between the liquid water path (LWP) averaged across cloudy and clear regions (here characterizing the macrophysical state) and Nd (characterizing the microphysical) is the sum of two causal pathways linking Nd to LWP: Nd altering LWP (adjustments) and precipitation scavenging aerosol and thus depleting Nd. Only the former term is relevant to constraining adjustments, but disentangling these terms in observations is challenging. We hypothesize that the diversity of constraints on aerosol–cloud adjustments in the literature may be partly due to not explicitly characterizing covariance flowing from cloud to aerosol and aerosol to cloud. Here, we restrict our analysis to the regime of extratropical clouds outside of low-pressure centers associated with cyclonic activity. Observations from MAC-LWP (Multisensor Advanced Climatology of Liquid Water Path) and MODIS are compared to simulations in the Met Office Unified Model (UM) GA7.1 (the atmosphere model of HadGEM3-GC3.1 and UKESM1). The meteorological predictors of LWP are found to be similar between the model and observations. There is also agreement with previous literature on cloud-controlling factors finding that increasing stability, moisture, and sensible heat flux enhance LWP, while increasing subsidence and sea surface temperature decrease it. A simulation where cloud microphysics are insensitive to changes in Nd is used to characterize covariance between Nd and LWP that is induced by factors other than aerosol–cloud adjustments. By removing variability associated with meteorology and scavenging, we infer the sensitivity of LWP to changes in Nd. Application of this technique to UM GA7.1 simulations reproduces the true model adjustment strength. Observational constraints developed using simulated covariability not induced by adjustments and observed covariability between Nd and LWP predict a 25 %–30 % overestimate by the UM GA7.1 in LWP change and a 30 %–35 % overestimate in associated radiative forcing.

Daniel T. McCoy↗

NASA NTRSDate not supplied

The Relationship Between Two Methods for Estimating Uncertainties in Data Assimilation

This note examines the relationship between two apparently unrelated methods for estimating error statistics or uncertainties of relevance to data assimilation. The first method is due to (Desroziers et al., 2005, Q. J. R. Meteorol. Soc., 131, 3385–3396; referred to as DBCP hereafter) and relies on residual statistics readily available from data assimilation applications. The second method, the three- cornered hat (3CH) developed by Gray and Allan (1974, IEEE 28th Annual Symp. Freq. Control, 243–246), only recently applied to atmospheric sciences, uses three data sets and can derive estimates of relevant error uncertainties as well. The usefulness of both methods lies in them not requiring knowledge of the true value of the quantities at play. DBCP derives its results by relying explicitly on the constraints as- sociated with the data assimilation minimization problem; 3CH is general and its estimates hold as long as errors in the three data sets of choice are uncorrelated. Establishing the relationship between the methods requires applying the 3CH approach to the same observation, background, and analysis data sets used by DBCP. In this case, the same assumptions of DBCP on residual errors allow for cancellation of error cross–covariance terms in 3CH such that two of its corners derive identical estimates for observation and background error covariances as those of DBCP. The error cross–covariance terms associated with the third corner are shown to add up to twice the analysis error covariance so that the 3CH estimate for the third corner recovers the negative of the analysis error covariance. Illustrations of these findings are provided by deriving uncertainties for radio occultation bending angles.

Ricardo Todling↗

NASA NTRS2017

Filter Tuning Using the Chi-Squared Statistic

This paper examines the use of the Chi-square statistic as a means of evaluating filter performance. The goal of the process is to characterize the filter performance in the metric of covariance realism. The Chi-squared statistic is the value calculated to determine the realism of a covariance based on the prediction accuracy and the covariance values at a given point in time. Once calculated, it is the distribution of this statistic that provides insight on the accuracy of the covariance. The process of tuning an Extended Kalman Filter (EKF) for Aqua and Aura support is described, including examination of the measurement errors of available observation types, and methods of dealing with potentially volatile atmospheric drag modeling. Predictive accuracy and the distribution of the Chi-squared statistic, calculated from EKF solutions, are assessed.

Filter Tuning↗

NASA NTRS1971

State estimation with small nonlinearities

A variety of techniques is available for estimating the states of nonlinear dynamic systems from noisy data. The differences among several of these procedures in the presence of small dynamic and observational nonlinearities are investigated. Four discrete estimation algorithms are analyzed. The first is a strictly least square estimator, while the others are recursive algorithms similar to the Kalman filter used for estimating the states of linear systems. A group of analytic expressions is developed for the mean and covariance of the error in each of these estimators so that they may be compared without lengthy Monte Carlo simulations. The covariance expressions show that, to first order, all the estimators have the same covariance. Expressions for the means show that each estimator has a different bias. Several examples are carried out demonstrating that the relative magnitudes of the bias errors in the various estimators can be a strong function of such parameters as initial covariances and number of data points. Under some circumstances, more complicated algorithms can have larger biases than smaller ones.

Conrad, B.↗

NASA NTRS1972

Decision-directed adaptive recursive estimators - Divergence prevention.

A method is proposed for dealing with the divergence phenomenon problem encountered in applications of minimum-variance recursive estimators when the error covariance calculated by the estimator becomes inconsistent with the actual error covariance. The proposed method differs from previous ones in that the state model and statistics are accepted as given. The form of the optimal estimator is used, but a constant check is made on the consistency of the calculated and actual error covariances. The method is independent of the source of error, whether it be inaccuracies in the system model, incorrect values of the a priori and random process statistics, approximations required in the case of nonlinear systems, or computational roundoff. Several simulated examples, in which inconsistencies in the calculated and actual error covariances exist, show a significant improvement in the performance of the estimator when the given procedure is applied.

Nahi, N. E.↗

NASA NTRS1986

System identification for modeling for control of flexible structures

The major components of a design and operational flight strategy for flexible structure control systems are presented. In this strategy an initial distributed parameter control design is developed and implemented from available ground test data and on-orbit identification using sophisticated modeling and synthesis techniques. The reliability of this high performance controller is directly linked to the accuracy of the parameters on which the design is based. Because uncertainties inevitably grow without system monitoring, maintaining the control system requires an active on-line system identification function to supply parameter updates and covariance information. Control laws can then be modified to improve performance when the error envelopes are decreased. In terms of system safety and stability the covariance information is of equal importance as the parameter values themselves. If the on-line system ID function detects an increase in parameter error covariances, then corresponding adjustments must be made in the control laws to increase robustness. If the error covariances exceed some threshold, an autonomous calibration sequence could be initiated to restore the error enveloped to an acceptable level.

Mettler, Edward↗

NASA NTRS1985

A Kalman filter for a two-dimensional shallow-water model

A two-dimensional Kalman filter is described for data assimilation for making weather forecasts. The filter is regarded as superior to the optimal interpolation method because the filter determines the forecast error covariance matrix exactly instead of using an approximation. A generalized time step is defined which includes expressions for one time step of the forecast model, the error covariance matrix, the gain matrix, and the evolution of the covariance matrix. Subsequent time steps are achieved by quantifying the forecast variables or employing a linear extrapolation from a current variable set, assuming the forecast dynamics are linear. Calculations for the evolution of the error covariance matrix are banded, i.e., are performed only with the elements significantly different from zero. Experimental results are provided from an application of the filter to a shallow-water simulation covering a 6000 x 6000 km grid.

Parrish, D. F.↗

NASA NTRS1986

Satellite rainfall retrieval by logistic regression

The potential use of logistic regression in rainfall estimation from satellite measurements is investigated. Satellite measurements provide covariate information in terms of radiances from different remote sensors.The logistic regression technique can effectively accommodate many covariates and test their significance in the estimation. The outcome from the logistical model is the probability that the rainrate of a satellite pixel is above a certain threshold. By varying the thresholds, a rainrate histogram can be obtained, from which the mean and the variant can be estimated. A logistical model is developed and applied to rainfall data collected during GATE, using as covariates the fractional rain area and a radiance measurement which is deduced from a microwave temperature-rainrate relation. It is demonstrated that the fractional rain area is an important covariate in the model, consistent with the use of the so-called Area Time Integral in estimating total rain volume in other studies. To calibrate the logistical model, simulated rain fields generated by rainfield models with prescribed parameters are needed. A stringent test of the logistical model is its ability to recover the prescribed parameters of simulated rain fields. A rain field simulation model which preserves the fractional rain area and lognormality of rainrates as found in GATE is developed. A stochastic regression model of branching and immigration whose solutions are lognormally distributed in some asymptotic limits has also been developed.

Chiu, Long S.↗

NASA NTRS1989

Extensions of output variance constrained controllers to hard constraints

Covariance Controllers assign specified matrix values to the state covariance. A number of robustness results are directly related to the covariance matrix. The conservatism in known upperbounds on the H infinity, L infinity, and L (sub 2) norms for stability and disturbance robustness of linear uncertain systems using covariance controllers is illustrated with examples. These results are illustrated for continuous and discrete time systems. **** ONLY 2 BLOCK MARKERS FOUND -- RETRY *****

Skelton, R.↗

NASA NTRS1991

Simplification of the Kalman filter for meteorological data assimilation

The paper proposes a new statistical method of data assimilation that is based on a simplification of the Kalman filter equations. The forecast error covariance evolution is approximated simply by advecting the mass-error covariance field, deriving the remaining covariances geostrophically, and accounting for external model-error forcing only at the end of each forecast cycle. This greatly reduces the cost of computation of the forecast error covariance. In simulations with a linear, one-dimensional shallow-water model and data generated artificially, the performance of the simplified filter is compared with that of the Kalman filter and the optimal interpolation (OI) method. The simplified filter produces analyses that are nearly optimal, and represents a significant improvement over OI.

Dee, Dick P.↗

NASA NTRS1991

Analysis of altimetry over the Aral Sea

Satellite based altimetric data taken by GEOS-3, SEASAT, and GEOSAT over the Aral Sea are analyzed and a least squares collocation technique is used to predict the geoid undulations on a 0.25 by 0.25 degree grid and to transform these geoid undulations to free air gravity anomalies. Rapp's 180 by 180 geopotential model is taken as the empirical model. The collocation procedure is performed with a set of local residual empirical covariance functions. For comparison, Rapp's global covariance functions and Jordan's self-consistent theoretical covariance functions based on Jordan's formulation and on locally derived parameters are also used to grid geoid undulations and to predict gravity anomalies. The sensitivity of the collocation results to the choice of covariance functions is discussed.

Au, Andrew Y.↗

NASA NTRS1992

Incorporation of a priori gravity field information in satellite orbit determination using bin parameters

A method to determine satellite orbits using tracking data and a priori gravitational field is described. The a priori constraint on the orbit dynamics is determined by the covariance matrix of the spherical harmonic coefficients for the gravity model, so that the optimal combination of the measurements and gravitational field is achieved. A set of bin parameters is introduced to represent the perturbation of the gravitational field on the position of the satellite orbit. The covariance matrix of a conventional gravity model is transformed into that for the bin parameters by the variational partial derivatives. The covariance matrices of the bin parameters and the epoch state are combined to form the covariance matrix of the satellite positions at the measurement times. The combined matrix is used as the a priori information to estimate the satellite positions with measurements.

Wu, Jiun-Tsong↗