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NASA NTRS1992

Abundance recovery error analysis using simulated AVIRIS data

Measurement noise and imperfect atmospheric correction translate directly into errors in the determination of the surficial abundance of materials from imaging spectrometer data. The effects of errors on abundance recovery were investigated previously using Monte Carlo simulation methods by Sabol et. al. The drawback of the Monte Carlo approach is that thousands of trials are needed to develop good statistics on the probable error in abundance recovery. This computational burden invariably limits the number of scenarios of interest that can practically be investigated. A more efficient approach is based on covariance analysis. The covariance analysis approach expresses errors in abundance as a function of noise in the spectral measurements and provides a closed form result eliminating the need for multiple trials. Monte Carlo simulation and covariance analysis are used to predict confidence limits for abundance recovery for a scenario which is modeled as being derived from Airborne Visible/Infrared Imaging Spectrometer (AVIRIS).

Stoner, William W.↗

NASA NTRS1994

Progress in navigation filter estimate fusion and its application to spacecraft rendezvous

A new derivation of an algorithm which fuses the outputs of two Kalman filters is presented within the context of previous research in this field. Unlike other works, this derivation clearly shows the combination of estimates to be optimal, minimizing the trace of the fused covariance matrix. The algorithm assumes that the filters use identical models, and are stable and operating optimally with respect to their own local measurements. Evidence is presented which indicates that the error ellipsoid derived from the covariance of the optimally fused estimate is contained within the intersections of the error ellipsoids of the two filters being fused. Modifications which reduce the algorithm's data transmission requirements are also presented, including a scalar gain approximation, a cross-covariance update formula which employs only the two contributing filters' autocovariances, and a form of the algorithm which can be used to reinitialize the two Kalman filters. A sufficient condition for using the optimally fused estimates to periodically reinitialize the Kalman filters in this fashion is presented and proved as a theorem. When these results are applied to an optimal spacecraft rendezvous problem, simulated performance results indicate that the use of optimally fused data leads to significantly improved robustness to initial target vehicle state errors. The following applications of estimate fusion methods to spacecraft rendezvous are also described: state vector differencing, and redundancy management.

Carpenter, J. Russell↗

NASA NTRS1994

Performance of mean-frequency estimators for Doppler radar and lidar

The performance of mean-frequency estimators for Doppler radar and lidar measurements of winds is presented in terms of two basic parameters: Phi, the ratio of the average signal energy per estimate to the spectral noise level; and Omega, which is proportional to the number of independent samples per estimate. For fixed Phi and Omega, the Cramer-Rao bound (CRB) (theoretical best performance) for unbiased estimators of mean frequency (normalized by the spectral width of the signal), signal power, and spectral width are essentially independent of the number of data samples M. For large Phi, the estimators of mean frequency are unbiased and the performance is independent of M. The spectral domain estimators and covariance based estimators are bounded by the approximate period of M. The spectral domain estimators and covariance based estimators are bounded by the approximate periodogram CRB. The standard deviation of the maximum-likelihood estimator approaches the exact CRB, which can be more than a factor of 2 better than the performance of the spectral domain estimators or covariance-based estimators for typical Omega. For small Phi, the estimators are biased due to the effects of the uncorrelated noise (white noise), which results in uniformly distributed 'bad' estimates. The fraction of bad estimates is a function of Phi and M with weak dependence on the parameter Omega. Simple empirical models describe the standard deviation of the good estimates and the fraction of bad estimates. For Doppler lidar and for large Phi, better performance is obtained by using many low-energy pulses instead of one pulse with the same total energy. For small Phi, the converse is true.

Frehlich, R. G.↗

NASA NTRS1996

ILIAD Testing; and a Kalman Filter for 3-D Pose Estimation

This report presents the results of a two-part project. The first part presents results of performance assessment tests on an Internet Library Information Assembly Data Base (ILIAD). It was found that ILLAD performed best when queries were short (one-to-three keywords), and were made up of rare, unambiguous words. In such cases as many as 64% of the typically 25 returned documents were found to be relevant. It was also found that a query format that was not so rigid with respect to spelling errors and punctuation marks would be more user-friendly. The second part of the report shows the design of a Kalman Filter for estimating motion parameters of a three dimensional object from sequences of noisy data derived from two-dimensional pictures. Given six measured deviation values represendng X, Y, Z, pitch, yaw, and roll, twelve parameters were estimated comprising the six deviations and their time rate of change. Values for the state transiton matrix, the observation matrix, the system noise covariance matrix, and the observation noise covariance matrix were determined. A simple way of initilizing the error covariance matrix was pointed out.

Richardson, A. O.↗

NASA NTRS1999

Massively Parallel Assimilation of TOGA/TAO and Topex/Poseidon Measurements into a Quasi Isopycnal Ocean General Circulation Model Using an Ensemble Kalman Filter

A massively parallel ensemble Kalman filter (EnKF)is used to assimilate temperature data from the TOGA/TAO array and altimetry from TOPEX/POSEIDON into a Pacific basin version of the NASA Seasonal to Interannual Prediction Project (NSIPP)ls quasi-isopycnal ocean general circulation model. The EnKF is an approximate Kalman filter in which the error-covariance propagation step is modeled by the integration of multiple instances of a numerical model. An estimate of the true error covariances is then inferred from the distribution of the ensemble of model state vectors. This inplementation of the filter takes advantage of the inherent parallelism in the EnKF algorithm by running all the model instances concurrently. The Kalman filter update step also occurs in parallel by having each processor process the observations that occur in the region of physical space for which it is responsible. The massively parallel data assimilation system is validated by withholding some of the data and then quantifying the extent to which the withheld information can be inferred from the assimilation of the remaining data. The distributions of the forecast and analysis error covariances predicted by the ENKF are also examined.

Keppenne, Christian L.↗

NASA NTRS1997

The Influence of Correlated Crustal Signals in Modelling the Main Geomagnetic Field

Algorithms used in geomagnetic main-field modelling have for the most part treated the noise in the field measurements as if it were white. A major component of the noise consists of the field due to magnetization in the crust and it has been realized for some time that such signals are highly correlated at satellite altitude. Hence approximation by white noise, while of undoubted utility, is of unknown validity. In this paper we study two plausible statistical models for the crustal magnetization, in which the magnetization is a realization of a stationary, isotropic, random process. At a typical satellite altitude the associated fields exhibit significant correlation over ranges as great as 15 deg. or more, which introduces off-diagonal elements into the covariance matrix, elements that have usually been neglected in modelling procedures. Dealing with a full covariance matrix for a large data set would present a formidable computational challenge, but fortunately most of the entries in the covariance matrix are so small that they can be replaced by zeros. The resultant matrix comprises only about 3 per cent non-zero entries and thus we can take advantage of efficient sparse matrix techniques to solve the numerical system. We construct several main-field models based on vertical-component data from a selected 5 deg. by 5 deg. data set derived from the Magsat mission. Models with and without off-diagonal terms are compared.

Rygaard-Hjalsted, C.↗

NASA NTRS2000

The GEOS Ozone Data Assimilation System: Specification of Error Statistics

A global three-dimensional ozone data assimilation system has been developed at the Data Assimilation Office of the NASA/Goddard Space Flight Center. The Total Ozone Mapping Spectrometer (TOMS) total ozone and the Solar Backscatter Ultraviolet (SBUV) or (SBUV/2) partial ozone profile observations are assimilated. The assimilation, into an off-line ozone transport model, is done using the global Physical-space Statistical Analysis Scheme (PSAS). This system became operational in December 1999. A detailed description of the statistical analysis scheme, and in particular, the forecast and observation error covariance models is given. A new global anisotropic horizontal forecast error correlation model accounts for a varying distribution of observations with latitude. Correlations are largest in the zonal direction in the tropics where data is sparse. Forecast error variance model is proportional to the ozone field. The forecast error covariance parameters were determined by maximum likelihood estimation. The error covariance models are validated using x squared statistics. The analyzed ozone fields in the winter 1992 are validated against independent observations from ozone sondes and HALOE. There is better than 10% agreement between mean Halogen Occultation Experiment (HALOE) and analysis fields between 70 and 0.2 hPa. The global root-mean-square (RMS) difference between TOMS observed and forecast values is less than 4%. The global RMS difference between SBUV observed and analyzed ozone between 50 and 3 hPa is less than 15%.

Stajner, Ivanka↗

NASA NTRS1999

Measuring Greenland Ice Mass Variation With Gravity Recovery and the Climate Experiment Gravity and GPS

The response of the Greenland ice sheet to climate change could significantly alter sea level. The ice sheet was much thicker at the last glacial maximum. To gain insight into the global change process and the future trend, it is important to evaluate the ice mass variation as a function of time and space. The Gravity Recovery and Climate Experiment (GRACE) mission to fly in 2001 for 5 years will measure gravity changes associated with the current ice variation and the solid earth's response to past variations. Our objective is to assess the separability of different change sources, accuracy and resolution in the mass variation determination by the new gravity data and possible Global Positioning System (GPS) bedrock uplift measurements. We use a reference parameter state that follows a dynamic ice model for current mass variation and a variant of the Tushingham and Peltier ICE-3G deglaciation model for historical deglaciation. The current linear trend is also assumed to have started 5 kyr ago. The Earth model is fixed as preliminary reference Earth model (PREM) with four viscoelastic layers. A discrete Bayesian inverse algorithm is developed employing an isotropic Gaussian a priori covariance function over the ice sheet and time. We use data noise predicted by the University of Texas and JPL for major GRACE error sources. A 2 mm/yr uplift uncertainty is assumed for GPS occupation time of 5 years. We then carry out covariance analysis and inverse simulation using GRACE geoid coefficients up to degree 180 in conjunction with a number of GPS uplift rates. Present-day ice mass variation and historical deglaciation are solved simultaneously over 146 grids of roughly 110 km x 110 km and with 6 time increments of 3 kyr each, along with a common starting epoch of the current trend. For present-day ice thickness change, the covariance analysis using GRACE geoid data alone results in a root mean square (RMS) posterior root variance of 2.6 cm/yr, with fairly large a priori uncertainties in the parameters and a Gaussian correlation length of 350 km. Simulated inverse can successfully recover most features in the reference present-day change. The RMS difference between them over the grids is 2.8 cm/yr. The RMS difference becomes 1.1 cm/yr when both are averaged with a half Gaussian wavelength of 150 km. With a fixed Earth model, GRACE alone can separate the geoid signals due to past and current load fairly well. Shown are the reference geoid signatures of direct and elastic effects of the current trend, the viscoelastic effect of the same trend starting from 5 kyr ago, the Post Glacial Rebound (PGR), and the predicted GRACE geoid error. The difference between the reference and inverse modeled total viscoelastic signatures is also shown. Although past and current ice mass variations are allowed the same spatial scale, their geoid signals have different spatial patterns. GPS data can contribute to the ice mass determination as well. Additional information is contained in the original.

Wu, Xiao-Ping↗

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↗