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NASA NTRS2021

Observing System Simulation Experiments Investigating Atmospheric Motion Vectors and Radiances from a Constellation of 4–5-μm Infrared Sounders

A set of observing system simulation experiments (OSSEs) was performed to investigate the utility of a constellation of passive infrared spectrometers, strategically designed with the aim of deriving the three-dimensional retrievals of the horizontal wind via atmospheric motion vectors (AMVs) from instruments with the spectral resolution of an infrared sounder. The instrument and constellation designs were performed in the context of the Midwave Infrared Sounding of Temperature and humidity in a Constellation for Winds (MISTiC Winds). The Global Modeling and Assimilation Office OSSE system, which includes a full suite of operational meteorological observations, served as the control. To illustrate the potential impact of this observing strategy, two experiments were performed by adding the new simulated observations to the control. First, perfect (error free) simulated AMVs and radiances were assimilated. Second, the data were made imperfect by adding realistic modeled errors to the AMVs and radiances that were assimilated. The experimentation showed beneficial impacts on both the mass and wind fields, as based on analysis verification, forecast verification, and the assessment of the observations using the forecast sensitivity to observation impact (FSOI) metric. In all variables and metrics, the impacts of the imperfect observations were smaller than those of the perfect observations, although much of the positive benefit was retained. The FSOI metric illustrated two key points. First, the largest impacts were seen in the middle troposphere AMVs, which is a targeted capability of the constellation strategy. Second, the addition of modeled errors showed that the assimilation system was unable to fully exploit the 4.3-μm carbon dioxide absorption radiances.

Will McCarty↗
DOE OSTI2017

Precise and Fast Computation of the Gravitational Field of a General Finite Body and Its Application to the Gravitational Study of Asteroid Eros

In order to obtain the gravitational field of a general finite body inside its Brillouin sphere, we developed a new method to compute the field accurately. First, the body is assumed to consist of some layers in a certain spherical polar coordinate system and the volume mass density of each layer is expanded as a Maclaurin series of the radial coordinate. Second, the line integral with respect to the radial coordinate is analytically evaluated in a closed form. Third, the resulting surface integrals are numerically integrated by the split quadrature method using the double exponential rule. Finally, the associated gravitational acceleration vector is obtained by numerically differentiating the numerically integrated potential. Numerical experiments confirmed that the new method is capable of computing the gravitational field independently of the location of the evaluation point, namely whether inside, on the surface of, or outside the body. It can also provide sufficiently precise field values, say of 14–15 digits for the potential and of 9–10 digits for the acceleration. Furthermore, its computational efficiency is better than that of the polyhedron approximation. This is because the computational error of the new method decreases much faster than that of the polyhedron models when the number of required transcendental function calls increases. As an application, we obtained the gravitational field of 433 Eros from its shape model expressed as the 24 × 24 spherical harmonic expansion by assuming homogeneity of the object.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1989

A new state reconstructor for digital controls systems using weighted-average measurements

A state reconstructor is presented for a linear continuous-time plant driven by a zero-order-hold. It takes a continuous-time output vector from the plant and convolutes it with a weighting-function matrix whose elements are time dependent. This result is integrated over T second intervals to generate weighted-averaged measurements, every T seconds, that are used in the state reconstruction process. If the plant is noise-free and can be modeled precisely, the output of this state reconstructor exactly equals the true state of the plant and accomplishes this without knowledge of the plant's initial state. If noise or modeling errors are a problem, it can be catenated with a state observer or a Kalman filter for a synergistic effect.

Polites, Michael E.↗
NASA NTRS2016

Measuring and Modeling Shared Visual Attention

Multi-person teams are sometimes responsible for critical tasks, such as flying an airliner. Here we present a method using gaze tracking data to assess shared visual attention, a term we use to describe the situation where team members are attending to a common set of elements in the environment. Gaze data are quantized with respect to a set of N areas of interest (AOIs); these are then used to construct a time series of N dimensional vectors, with each vector component representing one of the AOIs, all set to 0 except for the component corresponding to the currently fixated AOI, which is set to 1. The resulting sequence of vectors can be averaged in time, with the result that each vector component represents the proportion of time that the corresponding AOI was fixated within the given time interval. We present two methods for comparing sequences of this sort, one based on computing the time-varying correlation of the averaged vectors, and another based on a chi-square test testing the hypothesis that the observed gaze proportions are drawn from identical probability distributions.We have evaluated the method using synthetic data sets, in which the behavior was modeled as a series of activities, each of which was modeled as a first-order Markov process. By tabulating distributions for pairs of identical and disparate activities, we are able to perform a receiver operating characteristic (ROC) analysis, allowing us to choose appropriate criteria and estimate error rates.We have applied the methods to data from airline crews, collected in a high-fidelity flight simulator (Haslbeck, Gontar Schubert, 2014). We conclude by considering the problem of automatic (blind) discovery of activities, using methods developed for text analysis.

attention↗
NASA NTRS2015

Measuring and Modeling Shared Visual Attention

Multi-person teams are sometimes responsible for critical tasks, such as flying an airliner. Here we present a method using gaze tracking data to assess shared visual attention, a term we use to describe the situation where team members are attending to a common set of elements in the environment. Gaze data are quantized with respect to a set of N areas of interest (AOIs); these are then used to construct a time series of N dimensional vectors, with each vector component representing one of the AOIs, all set to 0 except for the component corresponding to the currently fixated AOI, which is set to 1. The resulting sequence of vectors can be averaged in time, with the result that each vector component represents the proportion of time that the corresponding AOI was fixated within the given time interval.We present two methods for comparing sequences of this sort, one based on computing the time varying correlation of the averaged vectors, and another based on a chi-square test testing the hypothesis that the observed gaze proportions are drawn from identical probability distributions.We have evaluated the method using synthetic data sets, in which the behavior was modeled as a series of activities, each of which was modeled as a first-order Markov process. By tabulating distributions for pairs of identical and disparate activities, we are able to perform a receiver operating characteristic (ROC) analysis, allowing us to choose appropriate criteria and estimate error rates. Using these criteria, we have applied the methods to data from airline crews, collected in a high-fidelity flight simulator (Gontar Hoermann, 2014). We conclude by considering the problem of automatic (blind) discovery of activities, using methods developed for text analysis.

Mulligan, Jeffrey B.↗
NASA NTRSDate not supplied

Characterizing Spatiotemporal Uncertainty in Interpolated Meteorological Data

Interpolated meteorological data invariably contain errors. These errors have structure in time and space, particularly autocorrelation, which can cause the effects of errors to compound when model outputs are aggregated temporally or spatially. One way to account for this uncertainty is with a probabilistic model from which samples can be drawn that are coherent with respect to underlying spatial and temporal covariance structure. This work describes a probabilistic method for spatial interpolation of point-wise meteorological time series. Observational data from weather stations are generally sparse in space and dense in time (but sometimes missing). The method works by projecting time series onto orthogonal basis vectors and spatially interpolating each resulting component independently. Under suitable assumptions, and data transformations to better satisfy those assumptions, Gaussian process regression provides a complete description of the joint predictive distribution over a Gaussian random field. Spatiotemporally coherent realizations are generated as the sum of conditional (spatial) simulations of each orthogonal (temporal) component. Data-derived and generic orthogonal bases are considered. In addition to spatial interpolation, imputation of missing observational data is examined. The method is applied using near-surface air temperature over the Western United States and validated by comparing theoretical versus actual coverage of predictive distributions and analyzing the degree to which spatial and temporal covariance structure is reproduced. Computational considerations, relating to conditional simulation of random fields, are also addressed.

Conor T Doherty↗
DOE OSTI2014

An initial study on the estimation of time-varying volumetric treatment images and 3D tumor localization from single MV cine EPID images

Purpose: In this work the authors develop and investigate the feasibility of a method to estimate time-varying volumetric images from individual MV cine electronic portal image device (EPID) images. Methods: The authors adopt a two-step approach to time-varying volumetric image estimation from a single cine EPID image. In the first step, a patient-specific motion model is constructed from 4DCT. In the second step, parameters in the motion model are tuned according to the information in the EPID image. The patient-specific motion model is based on a compact representation of lung motion represented in displacement vector fields (DVFs). DVFs are calculated through deformable image registration (DIR) of a reference 4DCT phase image (typically peak-exhale) to a set of 4DCT images corresponding to different phases of a breathing cycle. The salient characteristics in the DVFs are captured in a compact representation through principal component analysis (PCA). PCA decouples the spatial and temporal components of the DVFs. Spatial information is represented in eigenvectors and the temporal information is represented by eigen-coefficients. To generate a new volumetric image, the eigen-coefficients are updated via cost function optimization based on digitally reconstructed radiographs and projection images. The updated eigen-coefficients are then multiplied with the eigenvectors to obtain updated DVFs that, in turn, give the volumetric image corresponding to the cine EPID image. Results: The algorithm was tested on (1) Eight digital eXtended CArdiac-Torso phantom datasets based on different irregular patient breathing patterns and (2) patient cine EPID images acquired during SBRT treatments. The root-mean-squared tumor localization error is (0.73 ± 0.63 mm) for the XCAT data and (0.90 ± 0.65 mm) for the patient data. Conclusions: The authors introduced a novel method of estimating volumetric time-varying images from single cine EPID images and a PCA-based lung motion model. This is the first method to estimate volumetric time-varying images from single MV cine EPID images, and has the potential to provide volumetric information with no additional imaging dose to the patient.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2024

Three-dimensional continuum point cloud method for large deformation and its verification

This study presents a strong form based meshfree collocation method, which is named Continuum Point Cloud Method, to solve nonlinear field equations derived from classical mechanics for deformed bodies in three-dimensional Euclidean space. The method and its implementation are benchmarked against a nonlinear vector field using manufactured solutions. The analysis of mechanical fields firstly focuses on the study of St. Venant Kirchhoff and compressible neo-Hookean materials. Results for various initial boundary value problems are presented, including benchmark cases involving unidirectional tension and simple shear. Subsequently, the study concludes with an analysis of a displacement-controlled simulation of a compressible neo-Hookean material, specifically a bar that is pulled to 50% of its original length and rotated 90°. The pure tension case yields a 1.5% error in displacement between computed and expected values and a combined tension and torsion loading case provides further insight into material behavior under complex loading conditions. The resulting normal axial and transverse stress-strain curves are also presented. Lastly, the consistency and robustness of the proposed nonlinear numerical schemes are successfully demonstrated through various numerical experiments.

Compressible neo-Hookean materials↗
DOE OSTI2011

Deformable Image Registration of Liver With Consideration of Lung Sliding Motion

Purpose: A feature based deformable registration model with sliding transformation was developed in the upper abdominal region for liver cancer. Methods: A two-step thin-plate spline (bi-TPS) algorithm was implemented to deformably register the liver organ. The first TPS registration was performed to exclusively quantify the sliding displacement component. A manual segmentation of the thoracic and abdominal cavity was performed as a priori knowledge. Tissue feature points were automatically identified inside the segmented contour on the images. The scale invariant feature transform method was utilized to match feature points that served as landmarks for the subsequent TPS registration to derive the sliding displacement vector field. To a good approximation, only motion along superior/inferior (SI) direction of voxels on each slice was averaged to obtain the sliding displacement for each slice. A second TPS transformation, as the last step, was carried out to obtain the local deformation field. Manual identification of bifurcation on liver, together with the manual segmentation of liver organ, was employed as a ''ground truth'' for assessing the algorithm's performance. Results: The proposed two-step TPS was assessed with six liver patients. The average error of liver bifurcation between manual identification and calculation for these patients was less than 1.8 mm. The residual errors between manual contour and propagated contour of liver organ using the algorithm fell in the range between 2.1 and 2.8 mm. An index of Dice similarity coefficient (DSC) between manual contour and calculated contour for liver tumor was 93.6% compared with 71.2% from the conventional TPS calculation. Conclusions: A high accuracy ({approx}2 mm) of the two-step feature based TPS registration algorithm was achievable for registering the liver organ. The discontinuous motion in the upper abdominal region was properly taken into consideration. Clinical implementation of the algorithm will find broad application in radiation therapy of liver cancer.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

RELAP5-3D Developer Guidelines and Programming Practices

Our ultimate goal is to create and maintain RELAP5-3D as the best software tool available to analyze nuclear power plants. This begins with writing excellent programming and requires thorough testing. This document covers development of RELAP5-3D software, the behavior of the RELAP5-3D program that must be maintained, and code testing. RELAP5-3D must perform in a manner consistent with previous code versions with backward compatibility for the sake of the users. Thus file operations, code termination, input and output must remain consistent in form and content while adding appropriate new files, input and output as new features are developed. As computer hardware, operating systems, and other software change, RELAP5-3D must adapt and maintain performance. The code must be thoroughly tested to ensure that it continues to perform robustly on the supported platforms. The coding must be written in a consistent manner that makes the program easy to read to reduce the time and cost of development, maintenance and error resolution. The programming guidelines presented her are intended to institutionalize a consistent way of writing FORTRAN code for the RELAP5-3D computer program that will minimize errors and rework. A common format and organization of program units creates a unifying look and feel to the code. This in turn increases readability and reduces time required for maintenance, development and debugging. It also aids new programmers in reading and understanding the program. Therefore, when undertaking development of the RELAP5-3D computer program, the programmer must write computer code that follows these guidelines. This set of programming guidelines creates a framework of good programming practices, such as initialization, structured programming, and vector-friendly coding. It sets out formatting rules for lines of code, such as indentation, capitalization, spacing, etc. It creates limits on program units, such as subprograms, functions, and modules. It establishes documentation guidance on internal comments. The guidelines apply to both existing and new subprograms. They are written for both FORTRAN 77 and FORTRAN 95. The guidelines are not so rigorous as to inhibit a programmer’s unique style, but do restrict the variations in acceptable coding to create sufficient commonality that new readers will find the coding in each new subroutine familiar. It is recognized that this is a “living” document and must be updated as languages, compilers, and computer hardware and software evolve.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1981

MAGSAT data processing: A report for investigators

The in-flight attitude and vector magnetometer data bias recovery techniques and results are described. The attitude bias recoveries are based on comparisons with a magnetic field model and are thought to be accurate to 20 arcsec. The vector magnetometer bias recoveries are based on comparisons with the scalar magnetometer data and are thought to be accurate to 3 nT or better. The MAGSAT position accuracy goals of 60 m radially and 300 m horizontally were achieved for all but the last 3 weeks of Magsat lifetime. This claim is supported by ephemeris overlap statistics and by comparisons with ephemerides computed with an independent orbit program using data from an independent tracking network. MAGSAT time determination accuracy is estimated at 1 ms. Several errors in prelaunch assumptions regarding data time tags, which escaped detection in prelaunch data tests, and were discovered and corrected postlaunch are described. Data formats and products, especially the Investigator-B tapes, which contain auxiliary parameters in addition to the basic magnetometer and ephemeris data, are described.

Langel, R. A.↗
NASA NTRS1994

Improvements to TOVS retrievals over sea ice and applications to estimating Arctic energy fluxes

Modeling studies suggest that polar regions play a major role in modulating the Earth's climate and that they may be more sensitive than lower latitudes to climate change. Until recently, however, data from meteorological stations poleward of 70 degs have been sparse, and consequently, our understanding of air-sea-ice interaction processes is relatively poor. Satellite-borne sensors now offer a promising opportunity to observe polar regions and ultimately to improve parameterizations of energy transfer processes in climate models. This study focuses on the application of the TIROS-N operational vertical sounder (TOVS) to sea-ice-covered regions in the nonmelt season. TOVS radiances are processed with the improved initialization inversion ('3I') algorithm, providng estimates of layer-average temperature and moisture, cloud conditions, and surface characteristics at a horizontal resolution of approximately 100 km x 100 km. Although TOVS has flown continuously on polar-orbiting satellites since 1978, its potential has not been realized in high latitudes because the quality of retrievals is often significantly lower over sea ice and snow than over the surfaces. The recent availability of three Arctic data sets has provided an opportunity to validate TOVS retrievals: the first from the Coordinated Eastern Arctic Experiment (CEAREX) in winter 1988/1989, the second from the LeadEx field program in spring 1992, and the third from Russian drifting ice stations. Comparisons with these data reveal deficiencies in TOVS retrievals over sea ice during the cold season; e.g., ice surface temperature is often 5 to 15 K too warm, microwave emissivity is approximately 15% too low at large view angles, clear/cloudy scenes are sometimes misidentified, and low-level inversions are often not captured. In this study, methods to reduce these errors are investigated. Improvements to the ice surface temperature retrieval have reduced rms errors from approximately 7 K to 3 K; correction of microwaves (50 GHz) brightness temperatures for view angle dependence now allows the surface type (open water versus sea ice) to be determined; modifications to the clear/cloud tests have improved cloud detection over sea ice, especially where it is inhomogeneous; and improved surface temperature estimates have resulted in more successful diagnoses of low-level stratification. Applications of improved retrievals to studies of the Arctic energy budget are encouraging. Preliminary calculations of air-ice stress vectors, 10-m wind speed, and horizontal advection of heat and moisture are presented.

Francis, Jennifer A.↗
NASA NTRS2016

Measuring and Modeling Shared Visual Attention

Multi-person teams are sometimes responsible for critical tasks, such as flying an airliner. Here we present a method using gaze tracking data to assess shared visual attention, a term we use to describe the situation where team members are attending to a common set of elements in the environment. Gaze data are quantized with respect to a set of N areas of interest (AOIs); these are then used to construct a time series of N dimensional vectors, with each vector component representing one of the AOIs, all set to 0 except for the component corresponding to the currently fixated AOI, which is set to 1. The resulting sequence of vectors can be averaged in time, with the result that each vector component represents the proportion of time that the corresponding AOI was fixated within the given time interval. We present two methods for comparing sequences of this sort, one based on computing the time-varying correlation of the averaged vectors, and another based on a chi-square test testing the hypothesis that the observed gaze proportions are drawn from identical probability distributions. We have evaluated the method using synthetic data sets, in which the behavior was modeled as a series of "activities," each of which was modeled as a first-order Markov process. By tabulating distributions for pairs of identical and disparate activities, we are able to perform a receiver operating characteristic (ROC) analysis, allowing us to choose appropriate criteria and estimate error rates. We have applied the methods to data from airline crews, collected in a high-fidelity flight simulator (Haslbeck, Gontar & Schubert, 2014). We conclude by considering the problem of automatic (blind) discovery of activities, using methods developed for text analysis.

attention↗
NASA NTRS2020

Observing System Simulation Experiments Investigating Atmospheric Motion Vectors and Radiances from a Constellation of 4-5 μm Infrared Sounders

A set of Observing System Simulation Experiments (OSSEs) was performed to investigate the utility of a constellation of passive infrared spectrometers, strategically designed with the aim of deriving the three-dimensional retrievals of the horizontal wind via atmospheric motion vectors (AMVs) from instruments with the spectral resolution of an infrared sounder. The instrument and constellation designs were performed in the context of the Midwave Infrared Sounding of Temperature and humidity in a Constellation for Winds, or MISTiC Winds. The Global Modeling and Assimilation Office OSSE system, which includes a full suite of operational meteorological observations, served as the control. To illustrate the potential impact of this observing strategy, two experiments were performed by adding the new simulated observations to the control. First, perfect (error-free) simulated AMVs and radiances were assimilated. Second, the data were made imperfect by adding realistic modeled errors to the AMVs and radiances that were assimilated. The experimentation showed beneficial impacts on both the mass and wind fields, as based on analysis verification, forecast verification, and the assessment of the observations using the Forecast Sensitivity to Observation Impact (FSOI) metric. In all variables and metrics, the impacts of the imperfect observations were smaller than those of the perfect observations, though much of the positive benefit was retained. The FSOI metric illustrated two key points. First, the largest impacts were seen in the middle troposphere AMVs, which is a targeted capability of the constellation strategy. Second, the addition of modeled errors showed that the assimilation system was unable to fully exploit the 4.3 μm carbon dioxide absorption radiances.

Will McCarty↗
DOE OSTI2018

Assessing Alignment Between a Top-Down LCA Model and a Bottom-Up Supply Chain Model for Data Gap Identification and Prioritization

This work presents a method for evaluating the alignment between the results of bottom-up and top-down models and using the evaluation to enhance the bottom-up model and underlying data set. Disaggregation is used to enhance the embodied energy results of top-down models, which are then compared to bottom-up model results. Using the disaggregated top-down results makes it possible to identify specific data gaps and deficiencies within the bottom-up model and its data set. The top-down results will also provide information around which data gaps in the bottom-up model are contributing the most error to the results; in this way, data gaps can be identified as either critical or non-critical, and the effort needed to acquire and add data can be prioritized. Although the proposed method could be applied to any physical quantity or environmental impact that can be quantified with both bottom-up models (typically in physical units) and top-down models (typically in monetary units), embodied energy is the focus of this work. The disaggregated energy baseline is calculated from the throughput vector x, which gives the total economic activity required from each sector to provide a final demand. Using the direct requirements matrix, each element in x is disaggregated into a vector of inputs to the corresponding sector. Energy data from the interventions matrix B is then used to convert the monetary flows of the disaggregated x into physical energy flows. For each energy flow, the origin sector, the destination sector, and the energy carrier type will all be known. These energetic flows are analogous but not identical to direct energy inputs that would appear in a unit process used in life cycle analysis and other bottom-up methods, and thus can be used to evaluate the alignment between the top-down results and the results of a bottom-up model. To demonstrate and further develop the proposed method, embodied energy for chemicals assessed in the 2015 U.S. DOE bandwidth study on energy use within the chemical sector will be calculated using a top-down and a bottom-up model. The Materials Flows through Industry (MFI) modeling tool developed at the National Renewable Energy Laboratory is used as the bottom-up model, and the U.S. EEIO model, recently published by the U.S. Environmental Protection Agency, is used as the top-down model. U.S. EEIO and MFI results will be evaluated to determine how well the model results align. Where the two models are not in good alignment, the disaggregation method described above will be applied to the top-down model to generate a more detailed set of results that will aid in identifying data gaps in the MFI database. Using only MFI, data gaps can be identified in terms of which sectors have few or no unit processes in the database, but no information is available regarding the impact of the missing data on MFI results. The disaggregated top-down results will provide this information on which gaps in the MFI data are critical and have a significant impact on the results, and which are less important.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2024

Isovector Axial Charge and Form Factors of Nucleons from Lattice QCD

A survey of the calculations of the isovector axial vector form factor of the nucleon using lattice QCD is presented. Attention is paid to statistical and systematic uncertainties, in particular those due to excited state contributions. Based on a comparison of results from various collaborations, a case is made that lattice results are consistent within 10%. A similar level of uncertainty is in the axial charge gAu−d, the mean squared axial charge radius ⟨rA2⟩, the induced pseudoscalar charge gP∗, and the pion–nucleon coupling gπNN. Even with the current methodology, a significant reduction in errors is expected over the next few years with higher statistics data on more ensembles closer to the physical point. Lattice QCD results for the form factor GA(Q2) are compatible with those obtained from the recent MINERνA experiment but lie 2–3σ higher than the phenomenological extraction from the old ν–deuterium bubble chamber scattering data for Q2>0.3 GeV2. Current data show that the dipole ansatz does not have enough parameters to fit the form factor over the range 0≤Q2≤1 GeV2, whereas even a z2 truncation of the z expansion or a low order Padé are sufficient. Looking ahead, lattice QCD calculations will provide increasingly precise results over the range 0≤Q2≤1 GeV2, and MINERνA-like experiments will extend the range to Q2∼2 GeV2 or higher. Nevertheless, improvements in lattice methods to (i) further control excited state contributions and (ii) extend the range of Q2 are needed.

Astronomy & Astrophysics↗
NASA NTRS1980

Modal methods in optimal control synthesis

Efficient algorithms for solving linear smoother-follower problems with quadratic criteria are presented. For time-invariant systems, the algorithm consists of one backward integration of a linear vector equation and one forward integration of another linear vector equation. Furthermore, the backward and forward Riccati matrices can be expressed in terms of the eigenvalues and eigenvectors of the Euler-Lagrange equations. Hence, the gains of the forward and backward Kalman-Bucy filters and of the optimal state-feedback regulator can be determined without integration of matrix Riccati equations. A computer program has been developed, based on this method of determining the gains, to synthesize the optimal time-invariant compensator in the presence of random disturbance inputs and random measurement errors. The program also computes the rms state and control variables of the optimal closed-loop system.

Bryson, A. E., Jr.↗
NASA NTRS1986

On realizations of least-squares estimation and Kalman filtering by systolic arrays

Least-squares (LS) estimation is a basic operation in many signal processing problems. Given y = Ax + v, where A is a m x n coefficient matrix, y is a m x 1 observation vector, and v is a m x 1 zero mean white noise vector, a simple least-squares solution is finding the estimated vector x which minimizes the norm of /Ax-y/. It is well known that for an ill-conditioned matrix A, solving least-squares problems by orthogonal triangular (QR) decomposition and back substitution has robust numerical properties under finite word length effect since 2-norm is preserved. Many fast algorithms have been proposed and applied to systolic arrays. Gentleman-Kung (1981) first presented the trianglular systolic array for a basic Givens reduction. McWhirter (1983) used this array structure to find the least-squares estimation errors. Then by geometric approach, several different systolic array realizations of the recursive least-squares estimation algorithms of Lee et al (1981) were derived by Kalson-Yao (1985). Basic QR decomposition algorithms are considered in this paper and it is found that under a one-row time updating situation, the Householder transformation degenerates to a simple Givens reduction. Next, an improved least-squares estimation algorithm is derived by considering a modified version of fast Givens reduction. From this approach, the basic relationship between Givens reduction and Modified-Gram-Schmidt transformation can easily be understood. This improved algorithm also has simpler computational and inter-cell connection complexities while compared with other known least-squares algorithms and is more realistic for systolic array implementation.

Chen, M. J.↗