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NASA NTRS2012

A Standard Platform for Testing and Comparison of MDAO Architectures

The Multidisciplinary Design Analysis and Optimization (MDAO) community has developed a multitude of algorithms and techniques, called architectures, for performing optimizations on complex engineering systems which involve coupling between multiple discipline analyses. These architectures seek to efficiently handle optimizations with computationally expensive analyses including multiple disciplines. We propose a new testing procedure that can provide a quantitative and qualitative means of comparison among architectures. The proposed test procedure is implemented within the open source framework, OpenMDAO, and comparative results are presented for five well-known architectures: MDF, IDF, CO, BLISS, and BLISS-2000. We also demonstrate how using open source soft- ware development methods can allow the MDAO community to submit new problems and architectures to keep the test suite relevant.

Gray, Justin S.↗

NASA NTRS2005

SOFIA'S Challenge: Scheduling Airborne Astronomy Observations

The Stratospheric Observatory for Infrared Astronomy (SOFIA) is NASA's next generation airborne astronomical observatory, and will commence operations in 2005. The facility consists of a 747-SP modified to accommodate a 2.5 meter telescope. SOFIA is expected to fly an average of 140 science flights per year over its 20 year lifetime. Depending on the nature of the instrument used during flight, 5-15 observations per flight are expected. The SOFIA telescope is mounted aft of the wings on the port side of the aircraft and is articulated through a range of 20deg to 60deg of elevation. The telescope has minimal lateral flexibility; thus, the aircraft must turn constantly to maintain the telescope's focus on an object during observations. A significant problem in future SOFIA operations is that of scheduling flights in support of observations. Investigators are expected to propose small numbers of observations, and many observations must be grouped together to make up single flights. Flight planning for the previous generation airborne observatory, the Kuiper Airborne Observatory (KAO), was done by hand; planners had to choose takeoff time, observations to perform, and decide on setup-actions (called "dead-legs") to position the aircraft prior to observing. This task frequently required between 6-8 hours to plan one flight The scope of the flight planning problem for supporting GI observations with the anticipated flight rate for SOFIA makes the manual approach for flight planning daunting. In response, we have designed an Automated Flight Planner (AFP) that accepts as input a set of requested observations, designated flight days, weather predictions and fuel limitations, and searches automatically for high-quality flight plans that satisfy all relevant aircraft and astronomer specified constraints. The AFP can generate one candidate flight plan in 5-10 minutes, of computation time, a feat beyond the capabilities of human flight planners. The rate at which the AFP can generate flights enables humans to assess and analyze complex tradeoffs between fuel consumption, estimated science quality and the percentage of scheduled observations. Due to the changing nature of SOFIA scheduling problems, this functionality will play a crucial role in optimizing science and minimizing costs during operations. In the full paper, we will summarize the technical challenges that have been met in order to build this system. These include: design of the search algorithm, design of appropriate heuristics and approximations, and reduction in the size of the search space. We will also describe technical challenges that are currently being addressed, including the extension of the existing approach to handle new solution criteria. Finally, we will describe a variety of cultural challenges that the astronomical community must address in order to successfully use SOFIA, and describe how the AFT can be used to address some of these challenges. Specifically, many of the intended science users are accustomed to using ground-based or space-based observatories; we will identify some differences that arise due to the nature of airborne observatories, and how the AFT can be extended to provide useful services to ease these cultural differences.

Frank, Jeremy↗

NASA NTRSDate not supplied

Communicating Metrics of Land Surface Temperature Variability Using Multi-sensor Machine Learning

Land surface temperature (LST) is a key climate observable used to detect changes in the Earth’s surface energy budget that influence carbon and water cycles. Land surface temperature exhibits strong diurnal variability, which geostationary satellites can observe at scale thanks to their temporal resolution. Due to anthropogenic climate and land use changes, the surface energy balance has been considerably modified and may be described by changes in diurnal temperature range and extremes. Using high performance computing and datasets from the NASA Earth Exchange, we exploit co-located, co-temporal observations from low-earth orbit (LEO) and geostationary (GEO) sensors to develop a deep learning-based method for LEO-to-GEO algorithm emulation. Our model is trained to predict MODIS Terra LST from GOES-16 thermal bands and achieves validation error <2K. Application of the model to unseen times of day (observed by MODIS Aqua) and a new GEO sensor (Himawari-8) observing an unseen spatial domain, demonstrate the generalization of the deep learning model across space, time and spectra. Further, time series clustering approaches are examined with the objective of identifying key indicators of change in diurnal cycling and extremes on a continental scale. Communicating LST variability observed by geostationary satellites can have impacts in multiple disciplines, from understanding of snow, vegetation and soil dynamics, to recognizing trends in heat events relevant to human health.

Kate Duffy↗

NASA NTRSDate not supplied

Shock-Layer Radiation Insights Available Through Flowfield Property Binning

This paper introduces a method for assessing similarity between available shock-layer radiation measurements and the radiative heating to a vehicle of interest. This method will be referred to as the flowfield property binning (FPB) approach. The challenge in developing this approach is that radiometer measurements from a surface normal line of sight, observed radiation measurements of the entire shock-layer from a distance, or measured shock-tube nonequilibrium profiles are notably different geometrically than the radiative flux to a vehicle surface. Therefore, the relative contributions of various regions of the shock-layer to the measurement, and how these relative contributions compare with the radiative heating to a vehicle of interest, are unclear. By tracking the radiation contribution from each flowfield location, and then grouping this contribution based on two local flowfield properties, these geometric differences are removed. This provide a two dimensional group, or bin, distribution that represents the radiation environment in the same format for various types of radiation simulations, such as observed radiation or radiative flux to a vehicle surface. In the limits of optically-thin radiation transport and thermochemical equilibrium flow, the choice of pressure and temperature as the binning flowfield properties is shown to provide a bin distribution that uniquely defines the radiation simulation. This concept is found to be sufficient for establishing similarity between two cases even when the optically-thin radiation transport and thermochemical equilibrium assumptions are not satisfied. A simple metric of comparison between bin distributions from different cases is provided by the histogram intersection algorithm. This algorithm results in a histogram intersection factor (H) between 0 and 1, which represents the fraction of the radiation that is similar between the two cases. Therefore, if H evaluated between bin distributions for two completely different radiation problems is near 1, then they represent essentially the same radiation problem. This means that they depend on the same weighting of various radiation modeling parameters evaluated at the same pressure and temperature. If one of these cases agrees well with a radiation measurement, then the matching bin distributions provide justification for applying this experimental validation to the other case. Even for H values as low as 0.2, the measurement case could be considered relevant if there are no other useful measurements. This approach allows observed radiation measurements to directly inform radiative heating simulations, which allows NASA’s past and future investments in observed radiation measurements to directly inform the radiative heating uncertainty or margins for future vehicle design. This FPB approach is also used to determine shock-tube conditions for informing backshell radiative heating simulations, where the normal shock velocity and Lagrangian time from the shock is used for the binning variables instead of pressure and temperature.

Shock Layer Radiation↗

NASA NTRS1986

Estimation of vegetation cover at subpixel resolution using LANDSAT data

The present report summarizes the various approaches relevant to estimating canopy cover at subpixel resolution. The approaches are based on physical models of radiative transfer in non-homogeneous canopies and on empirical methods. The effects of vegetation shadows and topography are examined. Simple versions of the model are tested, using the Taos, New Mexico Study Area database. Emphasis has been placed on using relatively simple models requiring only one or two bands. Although most methods require some degree of ground truth, a two-band method is investigated whereby the percent cover can be estimated without ground truth by examining the limits of the data space. Future work is proposed which will incorporate additional surface parameters into the canopy cover algorithm, such as topography, leaf area, or shadows. The method involves deriving a probability density function for the percent canopy cover based on the joint probability density function of the observed radiances.

Jasinski, Michael F.↗

NASA NTRS1990

Estimated accuracy of ground-based liquid water measurements during FIRE

Since on goal of the First ISCCP Regional Experiment (FIRE) project is to improve our understanding of the relationships between cloud microphysics and cloud reflectivity, it is important that the accuracy of remote liquid measurements by microwave radiometry be thoroughly understood. The question is particularly relevant since the uncertainty in the absolute value of the radiometric liquid measurement is greatest at low liquid water contents (less than 0.1 mm). However it should be stressed that although uncertainty exists in the absolute value of liquid, it is well known that the observed radiometric signal is proportional to the amount of liquid in the antenna beam. As a result, changes in amounts of liquid are known to greater accuracy than the absolute value, which may contain a bias. Here, an assessment of the liquid measurement accuracy attained at San Nicolas Island (SNI) is presented. The vapor and liquid water data shown were computed from the radiometric brightness temperatures using statistical retrieval algorithms. The retrieval coefficients were derived from the 69 soundings made by Colorado State University during the SNI observations. Sources of error in the vapor and liquid measurements include cross-talk in the retrieval algorithms (not a factor at low liquid contents), uncertainties in the brightness temperature measurement, and uncertainties in the vapor and liquid attenuation coefficients. The relative importance of these errors is discussed. For the retrieval of path-integrated liquid water, the greatest uncertainty is caused by the temperature dependence of the absorption at microwave frequencies. As a result, the accuracy of statistical retrieval of liquid depends to large measure upon how representative the a priori radiosonde data are of the conditions prevailing during the measurements. The microwave radiometer measurements at SNI were supplemented by an infrared (IR) radiometer modified for measurement of cloud-base temperature. Thus, the IR system provides the means to incorporate continuous measurements of the liquid temperature into the retrieval process.

Snider, Jack B.↗

NASA NTRSDate not supplied

Homogenized Ground-Based and Profile Ozone Datasets From the TOAR-II/HEGIFTOM Project: Methods and Station Trends

Within the framework of the second phase of the Tropospheric Ozone Assessment Report (TOAR-II), it was recognized that an essential first step for deriving accurate trends from the ground-based networks that monitor ozone in the free troposphere is putting the measurements on the same basis with respect to absolute references and processing methods. The relevant procedures are referred to as “harmonization” or “homogenization”. The TOAR II working group, “HEGIFTOM” (Harmonization and Evaluation of Ground-based Instruments for Free-Tropospheric Ozone Measurements), has carried out harmonization for five types of network (Figure below) instruments (mid-1990s to 2020): ozonesondes, commercial aircraft IAGOS landing/takeoff profiles, Fourier-Transform Infrared spectrometer (FTIR), tropospheric Lidar, and Brewer/Dobson Umkehr. First, we summarize the homogenization effort for each network that provides new quality-assessed ozone profile or segment (partial column) data sets, including uncertainty estimates and quality flags. Second, with the HEGIFTOM datasets forming the basis for a global assessment of tropospheric ozone column trends, results derived with various trend detection algorithms will be presented. The ultimate goal is evaluation of the consistency of the calculated trends among different techniques at selected stations and/or regions.

ozone↗

NASA NTRS2013

Summary of Data from the Fifth AIAA CFD Drag Prediction Workshop

Results from the Fifth AIAA CFD Drag Prediction Workshop (DPW-V) are presented. As with past workshops, numerical calculations are performed using industry-relevant geometry, methodology, and test cases. This workshop focused on force/moment predictions for the NASA Common Research Model wing-body configuration, including a grid refinement study and an optional buffet study. The grid refinement study used a common grid sequence derived from a multiblock topology structured grid. Six levels of refinement were created resulting in grids ranging from 0.64x10(exp 6) to 138x10(exp 6) hexahedra - a much larger range than is typically seen. The grids were then transformed into structured overset and hexahedral, prismatic, tetrahedral, and hybrid unstructured formats all using the same basic cloud of points. This unique collection of grids was designed to isolate the effects of grid type and solution algorithm by using identical point distributions. This study showed reduced scatter and standard deviation from previous workshops. The second test case studied buffet onset at M=0.85 using the Medium grid (5.1x106 nodes) from the above described sequence. The prescribed alpha sweep used finely spaced intervals through the zone where wing separation was expected to begin. Some solutions exhibited a large side of body separation bubble that was not observed in the wind tunnel results. An optional third case used three sets of geometry, grids, and conditions from the Turbulence Model Resource website prepared by the Turbulence Model Benchmarking Working Group. These simple cases were intended to help identify potential differences in turbulence model implementation. Although a few outliers and issues affecting consistency were identified, the majority of participants produced consistent results.

Levy, David W.↗

NASA NTRS2015

How Certain are We of the Uncertainties in Recent Ozone Profile Trend Assessments of Merged Limbo Ccultation Records? Challenges and Possible Ways Forward

Most recent assessments of long-term changes in the vertical distribution of ozone (by e.g. WMO and SI2N) rely on data sets that integrate observations by multiple instruments. Several merged satellite ozone profile records have been developed over the past few years; each considers a particular set of instruments and adopts a particular merging strategy. Their intercomparison by Tummon et al. revealed that the current merging schemes are not sufficiently refined to correct for all major differences between the limb/occultation records. This shortcoming introduces uncertainties that need to be known to obtain a sound interpretation of the different satellite-based trend studies. In practice however, producing realistic uncertainty estimates is an intricate task which depends on a sufficiently detailed understanding of the characteristics of each contributing data record and on the subsequent interplay and propagation of these through the merging scheme. Our presentation discusses these challenges in the context of limb/occultation ozone profile records, but they are equally relevant for other instruments and atmospheric measurements. We start by showing how the NDACC and GAW-affiliated ground-based networks of ozonesonde and lidar instruments allowed us to characterize fourteen limb/occultation ozone profile records, together providing a global view over the last three decades. Our prime focus will be on techniques to estimate long-term drift since our results suggest this is the main driver of the major trend differences between the merged data sets. The single-instrument drift estimates are then used for a tentative estimate of the systematic uncertainty in the profile trends from merged data records. We conclude by reflecting on possible further steps needed to improve the merging algorithms and to obtain a better characterization of the uncertainties involved.

Merging Algorithms↗

NASA NTRSDate not supplied

Field Reconstruction from PIV Measurements Employing Bernstein Polynomial Derived Operators

A fluid-dynamic reconstruction algorithm is presented that generates a least-squares best-fit, two-dimensional density field from a prespecified two-dimensional velocity field. This method recasts the mass-conservation equation as a modified Sylvester equation employing high-order operators derived from modified Bernstein polynomial expansions. To demonstrate its practical utility, this analytic methodology is applied to two canonical cases and a Particle Image Velocimetry dataset obtained from a Mach-2, mechanically back-pressured, isolator experiment. This methodology is envisioned to be used in conjunction with hypersonic-diagnostic techniques to aid in the quantification of isolator flow fields. However, also note that this reconstruction technique is well suited to other applications relevant to fluid dynamics, such as obtaining three-dimensional flow field reconstructions.

Bernstein Polynomials↗

NASA NTRS2015

Coastal and Inland Aquatic Data Products for the Hyperspectral Infrared Imager (HyspIRI)

The HyspIRI Aquatic Studies Group (HASG) has developed a conceptual list of data products for the HyspIRI mission to support aquatic remote sensing of coastal and inland waters. These data products were based on mission capabilities, characteristics, and expected performance. The topic of coastal and inland water remote sensing is very broad. Thus, this report focuses on aquatic data products to keep the scope of this document manageable. The HyspIRI mission requirements already include the global production of surface reflectance and temperature. Atmospheric correction and surface temperature algorithms, which are critical to aquatic remote sensing, are covered in other mission documents. Hence, these algorithms and their products were not evaluated in this report. In addition, terrestrial products (e.g., land use land cover, dune vegetation, and beach replenishment) were not considered. It is recognized that coastal studies are inherently interdisciplinary across aquatic and terrestrial disciplines. However, products supporting the latter are expected to already be evaluated by other components of the mission. The coastal and inland water data products that were identified by the HASG, covered six major environmental and ecological areas for scientific research and applications: wetlands, shoreline processes, the water surface, the water column, bathymetry and benthic cover types. Accordingly, each candidate product was evaluated for feasibility based on the HyspIRI mission characteristics and whether it was unique and relevant to the HyspIRI science objectives.

Infrared↗

NASA NTRS2004

Adjoint Algorithm for CAD-Based Shape Optimization Using a Cartesian Method

Adjoint solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape optimization. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (geometric parameters that control the shape). More recently, emerging adjoint applications focus on the analysis problem, where the adjoint solution is used to drive mesh adaptation, as well as to provide estimates of functional error bounds and corrections. The attractive feature of this approach is that the mesh-adaptation procedure targets a specific functional, thereby localizing the mesh refinement and reducing computational cost. Our focus is on the development of adjoint-based optimization techniques for a Cartesian method with embedded boundaries.12 In contrast t o implementations on structured and unstructured grids, Cartesian methods decouple the surface discretization from the volume mesh. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin et developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation. Central to automated shape optimization algorithms is the issue of geometry modeling and control. The need to optimize complex, "real-life" geometry provides a strong incentive for the use of parametric-CAD systems within the optimization procedure. In previous work, we presented an effective optimization framework that incorporates a direct-CAD interface. In this work, we enhance the capabilities of this framework with efficient gradient computations using the discrete adjoint method. We present details of the adjoint numerical implementation, which reuses the domain decomposition, multigrid, and time-marching schemes of the flow solver. Furthermore, we explain and demonstrate the use of CAD in conjunction with the Cartesian adjoint approach. The final paper will contain a number of complex geometry, industrially relevant examples with many design variables to demonstrate the effectiveness of the adjoint method on Cartesian meshes.

Nemec, Marian↗

NASA NTRS1986

Further development of an improved altimeter wind speed algorithm

A previous altimeter wind speed retrieval algorithm was developed on the basis of wind speeds in the limited range from about 4 to 14 m/s. In this paper, a new approach which gives a wind speed model function applicable over the range 0 to 21 m/s is used. The method is based on comparing 50 km along-track averages of the altimeter normalized radar cross section measurements with neighboring off-nadir scatterometer wind speed measurements. The scatterometer winds are constructed from 100 km binned measurements of radar cross section and are located approximately 200 km from the satellite subtrack. The new model function agrees very well with earlier versions up to wind speeds of 14 m/s, but differs significantly at higher wind speeds. The relevance of these results to the Geosat altimeter launched in March 1985 is discussed.

Chelton, Dudley B.↗

NASA NTRS2011

Distributed Monitoring of the R(sup 2) Statistic for Linear Regression

The problem of monitoring a multivariate linear regression model is relevant in studying the evolving relationship between a set of input variables (features) and one or more dependent target variables. This problem becomes challenging for large scale data in a distributed computing environment when only a subset of instances is available at individual nodes and the local data changes frequently. Data centralization and periodic model recomputation can add high overhead to tasks like anomaly detection in such dynamic settings. Therefore, the goal is to develop techniques for monitoring and updating the model over the union of all nodes data in a communication-efficient fashion. Correctness guarantees on such techniques are also often highly desirable, especially in safety-critical application scenarios. In this paper we develop DReMo a distributed algorithm with very low resource overhead, for monitoring the quality of a regression model in terms of its coefficient of determination (R2 statistic). When the nodes collectively determine that R2 has dropped below a fixed threshold, the linear regression model is recomputed via a network-wide convergecast and the updated model is broadcast back to all nodes. We show empirically, using both synthetic and real data, that our proposed method is highly communication-efficient and scalable, and also provide theoretical guarantees on correctness.

Bhaduri, Kanishka↗

NASA NTRSDate not supplied

RadLab Platform: Investigating Space Radiation

The RadLab Project, initiated by the Open Science Data Repository (OSDR) for Space Biology at NASA Ames Research Center (osdr.nasa.gov), aims to be an ongoing compilation of radiation data relevant to human space flight. Sponsored by the NASA Human Research Program, RadLab serves as the latest database in this domain. RadLab is intended to serve the needs of both the space radiation detector and space radiation biology communities. The RadLab architecture is being designed to accommodate data both from low Earth orbit (LEO) and beyond low Earth orbit (BLEO). Our long-term vision entails the establishment of a sustainable database receiving continuous updates through APIs connecting to other databases, as well as individual investigator contributions via a RadLab submission portal, modeled on the OSDR (https://osdr.nasa.gov/bio/submission-sso-login.html). In addition to serving as a resource for space biologists and space radiation physicists, such a database would facilitate the deployment of AI algorithms to study the impact of location in space and within spacecraft on the ambient space radiation field. The ultimate goal is to develop predictive dosimetry algorithms for future BLEO missions.

Sylvain V Costes↗

NASA NTRS1988

Spaceborne meteorological radar studies

Various radar designs and methods are studied for the estimation of rainfall parameters from space. An immediate goal is to support the development of the spaceborne radar that has been proposed for the Tropical Rain Measuring Mission (TRMM). The effort is divided into two activities: a cooperative airborne rain measuring experiment with the Radio Research Laboratory of Japan (RRL), and the modelling of spaceborne weather radars. An airborne rain measuring experiment was conducted at Wallops Flight Facility in 1985 to 1986 using the dual-wavelength radar/radiometer developed by RRL. The data are presently being used to test a number of methods that are relevant to spaceborne weather radars. An example is shown of path-averaged rain rates as estimated from three methods: the standard reflectivity rain rate method (Z-R), a dual-wavelength method, and a surface reference method. The results from the experiment shows for the first time the feasibility of using attenuation methods from space. The purposes of the modelling are twofold: to understand in a quantitative manner the relationships between a particular radar design and its capability for estimating precipitation parameters and to help devise and test new methods. The models are being used to study the impact of various TRMM radar designs on the accuracy of rain rate estimation as well as to test the performance of range-profiling algorithms, the mirror-image method, and some recently devised graphical methods for the estimation of the drop size distribution.

Meneghini, R.↗

NASA NTRS2003

Numerical Models of Human Circulatory System under Altered Gravity: Brain Circulation

A computational fluid dynamics (CFD) approach is presented to model the blood flow through the human circulatory system under altered gravity conditions. Models required for CFD simulation relevant to major hemodynamic issues are introduced such as non-Newtonian flow models governed by red blood cells, a model for arterial wall motion due to fluid-wall interactions, a vascular bed model for outflow boundary conditions, and a model for auto-regulation mechanism. The three-dimensional unsteady incompressible Navier-Stokes equations coupled with these models are solved iteratively using the pseudocompressibility method and dual time stepping. Moving wall boundary conditions from the first-order fluid-wall interaction model are used to study the influence of arterial wall distensibility on flow patterns and wall shear stresses during the heart pulse. A vascular bed modeling utilizing the analogy with electric circuits is coupled with an auto-regulation algorithm for multiple outflow boundaries. For the treatment of complex geometry, a chimera overset grid technique is adopted to obtain connectivity between arterial branches. For code validation, computed results are compared with experimental data for steady and unsteady non-Newtonian flows. Good agreement is obtained for both cases. In sin-type Gravity Benchmark Problems, gravity source terms are added to the Navier-Stokes equations to study the effect of gravitational variation on the human circulatory system. This computational approach is then applied to localized blood flows through a realistic carotid bifurcation and two Circle of Willis models, one using an idealized geometry and the other model using an anatomical data set. A three- dimensional anatomical Circle of Willis configuration is reconstructed from human-specific magnetic resonance images using an image segmentation method. The blood flow through these Circle of Willis models is simulated to provide means for studying gravitational effects on the brain circulation under auto-regulation.

Kim, Chang Sung↗

NASA NTRS1984

Center of mass estimation in closed vortices - A verification in principle and practice

The problem of tracking closed mesoscale vortices using center of mass estimation techniques is studied. Three estimators are evaluated using data from a warm core Gulf Stream ring. The comparisons show that a method based on the intersection of perpendicular bisectors and one using a least-squares fit of a conic section perform comparably. The perpendicular bisector algorithm is used in conjunction with a Gaussian ring model and a star-shaped survey pattern to produce an expected error curve as a function of vortex translation, survey speed and vortex size. For typical ring parameters, center estimation is usually possible to within + or - 5 km. The feasibility of using differing data sets to construct a history of ring motion based on a coordinate system moving with the ring is also investigated. In this way, the validity of using satellite-derived data and drifter trajectories to estimate the center of mass of a mesoscale feature is assessed. The results of the analysis demonstrate that the location of the deeper structure of the ring and the surface expression are sufficiently well correlated to permit dynamically relevant calculations based on surface measurements. It is shown that satellite-derived data can be used to approximate the center of mass trajectory to within the error in the individual center estimates for the period analyzed. The Lagrangian-drifter-derived centers are offset from the center of mass trajectory in a manner consistent with kinematic arguments.

Hooker, S. B.↗