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NASA NTRS1993

Implementing abstract multigrid or multilevel methods

Multigrid methods can be formulated as an algorithm for an abstract problem that is independent of the partial differential equation, domain, and discretization method. In such an abstract setting, problems not arising from partial differential equations can be treated. A general theory exists for linear problems. The general theory was motivated by a series of abstract solvers (Madpack). The latest version was motivated by the theory. Madpack now allows for a wide variety of iterative and direct solvers, preconditioners, and interpolation and projection schemes, including user callback ones. It allows for sparse, dense, and stencil matrices. Mildly nonlinear problems can be handled. Also, there is a fast, multigrid Poisson solver (two and three dimensions). The type of solvers and design decisions (including language, data structures, external library support, and callbacks) are discussed. Based on the author's experiences with two versions of Madpack, a better approach is proposed. This is based on a mixed language formulation (C and FORTRAN + preprocessor). Reasons for not using FORTRAN, C, or C++ (individually) are given. Implementing the proposed strategy is not difficult.

Douglas, Craig C.↗

NASA NTRS2000

The DYNAMO Orbiter Project: High Resolution Mapping of Gravity/Magnetic Fields and In Situ Investigation of Mars Atmospheric Escape

Dynamo is a small Mars orbiter planned to be launched in 2005 or 2007, in the frame of the NASA/CNES Mars exploration program. It is aimed at improving gravity and magnetic field resolution, in order to better understand the magnetic, geologic and thermal history of Mars, and at characterizing current atmospheric escape, which is still poorly constrained. These objectives are achieved by using a low periapsis orbit, similar to the one used by the Mars Global Surveyor spacecraft during its aerobraking phases. The proposed periapsis altitude for Dynamo of 120-130 km, coupled with the global distribution of periapses to be obtained during one Martian year of operation, through about 5000 low passes, will produce a magnetic/gravity field data set with approximately five times the spatial resolution of MGS. Low periapsis provides a unique opportunity to investigate the chemical and dynamical properties of the deep ionosphere, thermosphere, and the interaction between the atmosphere and the solar wind, therefore atmospheric escape, which may have played a crucial role in removing atmosphere, and water, from the planet. There is much room for debate on the importance of current atmosphere escape processes in the evolution of the Martian atmosphere, as early "exotic" processes including hydrodynamic escape and impact erosion are traditionally invoked to explain the apparent sparse inventory of present-day volatiles. Yet, the combination of low surface gravity and the absence of a substantial internally generated magnetic field have undeniable effects on what we observe today. In addition to the current losses in the forms of Jeans and photochemical escape of neutrals, there are solar wind interaction-related erosion mechanisms because the upper atmosphere is directly exposed to the solar wind. The solar wind related loss rates, while now comparable to those of a modest comet, nonetheless occur continuously, with the intriguing possibility of important cumulative and/or enhanced effects over the several billion years of the solar system's life. If the detailed history of the Martian internal field could be traced back, and the current escape processes could be understood well enough to model the expected stronger losses under early Sun conditions, one could go a long way toward constraining this part of the mysterious history of Mars' atmosphere.

Smrekar, S.↗

NASA NTRS1982

Measurement of optical modulation functions in sparsely sampled mosaic focal plane arrays

It is pointed out that the measurement of optical modulation functions for detectors in focal plane arrays may be somewhat more difficult under 'full-up' systems conditions as compared to ideal laboratory conditions. An idealized optical modulation test arrangement is considered along with a full-up scanned system involving an earth mapper in polar orbit. In testing the system in full-up condition, a problem arises with respect to the acquisition of knife edge response data. In order to overcome this problem, a preferred method is developed for obtaining KER data on a single scan. A special 'phased edge' reticle is developed for use in the test set-up. Attention is given to aspects of knife edge reconstruction.

Young, J. B.↗

NASA NTRS2017

Snow Accumulation Variability on a West Antarctic Ice Stream Observed with GPS Reflectometry, 2007-2017

Land ice loss from Antarctica is a significant and accelerating contribution to global sea level rise; however, Antarctic mass balance estimates are complicated by insufficient knowledge of surface mass balance processes such as snow accumulation. Snow accumulation is challenging to observe on a continental scale and in situ data are sparse, so we largely rely on estimates from atmospheric models. Here we employ a novel technique, GPS interferometric reflectometry (GPS-IR), to measure upper (less than 2 m) firn column thickness changes across a 23-station GPS array in West Antarctica. We compare the results with antenna heights measured in situ to establish the method's daily uncertainty (0.06 m) and with output from two atmospheric reanalysis products to categorize spatial and temporal variability of net snow accumulation. GPS-IR is an effective technique for monitoring surface mass balance processes that can be applied to both historic GPS data sets and future experiments to provide critical in situ observations of processes driving surface height evolution.

snow↗

NASA NTRS2010

Sparse Superpixel Unmixing for Hyperspectral Image Analysis

Software was developed that automatically detects minerals that are present in each pixel of a hyperspectral image. An algorithm based on sparse spectral unmixing with Bayesian Positive Source Separation is used to produce mineral abundance maps from hyperspectral images. A superpixel segmentation strategy enables efficient unmixing in an interactive session. The algorithm computes statistically likely combinations of constituents based on a set of possible constituent minerals whose abundances are uncertain. A library of source spectra from laboratory experiments or previous remote observations is used. A superpixel segmentation strategy improves analysis time by orders of magnitude, permitting incorporation into an interactive user session (see figure). Mineralogical search strategies can be categorized as supervised or unsupervised. Supervised methods use a detection function, developed on previous data by hand or statistical techniques, to identify one or more specific target signals. Purely unsupervised results are not always physically meaningful, and may ignore subtle or localized mineralogy since they aim to minimize reconstruction error over the entire image. This algorithm offers advantages of both methods, providing meaningful physical interpretations and sensitivity to subtle or unexpected minerals.

Castano, Rebecca↗

NASA NTRS2022

Midwest Water Resources: Developing an Evapotranspiration Climatology to Analyze Spatiotemporal Water Budget Patterns for Agriculture and Natural Resources Managers in the Midwest

Evapotranspiration (ET) is a climatic variable critical to the hydrologic cycle and is used to evaluate spatiotemporal trends in drought conditions. Although in-situ observations provide accurate ET information, these records are spatially sparse. The United States Department of Agriculture (USDA) Midwest Climate Hub, National Integrated Drought Information System, Minnesota Department of Agriculture, Michigan State University, and the USDA Foreign Agricultural Service have partnered with DEVELOP to gain new insights on spatiotemporal patterns of ET with NASA satellite data. This project evaluates the feasibility of using remotely sensed ET data products to understand trends from 2001 through 2020. Actual ET (aET) data were sourced from NASA’s Terra Moderate Resolution Imaging Spectroradiometer (MODIS), and reference ET (refET) and precipitation data were sourced from the Gridded Surface Meteorological Dataset (gridMET). Spatial and temporal differences in the MODIS and gridMET data sets were resolved such that gridded data could be compared. These data sets were then used to produce monthly Normals maps of mm/8-day of precipitation, aET, and refET. Using precipitation and ET Normals maps, hydrologic state maps were produced by subtracting either refET or aET from precipitation. These hydrologic state maps provide a proxy water balance by summarizing the difference between water entering and leaving the ground surface. Additionally, the ET products were compared in timeseries plots. The ET products from this project provided partners with comparable datasets to assess potential drought and flooding conditions to support Midwest agricultural and natural resource managers in decision-making.

Emma Myrick↗

NASA NTRS2011

Experiments in Reconstructing Twentieth-Century Sea Levels

One approach to reconstructing historical sea level from the relatively sparse tide-gauge network is to employ Empirical Orthogonal Functions (EOFs) as interpolatory spatial basis functions. The EOFs are determined from independent global data, generally sea-surface heights from either satellite altimetry or a numerical ocean model. The problem is revisited here for sea level since 1900. A new approach to handling the tide-gauge datum problem by direct solution offers possible advantages over the method of integrating sea-level differences, with the potential of eventually adjusting datums into the global terrestrial reference frame. The resulting time series of global mean sea levels appears fairly insensitive to the adopted set of EOFs. In contrast, charts of regional sea level anomalies and trends are very sensitive to the adopted set of EOFs, especially for the sparser network of gauges in the early 20th century. The reconstructions appear especially suspect before 1950 in the tropical Pacific. While this limits some applications of the sea-level reconstructions, the sensitivity does appear adequately captured by formal uncertainties. All our solutions show regional trends over the past five decades to be fairly uniform throughout the global ocean, in contrast to trends observed over the shorter altimeter era. Consistent with several previous estimates, the global sea-level rise since 1900 is 1.70 +/- 0.26 mm/yr. The global trend since 1995 exceeds 3 mm/yr which is consistent with altimeter measurements, but this large trend was possibly also reached between 1935 and 1950.

Ray, Richard D.↗

NASA NTRS2006

Can MODIS Data Calibrate and Validate Coastal Sediment Transport Models? Rapid Prototyping Using 250 m Data and the ECOMSED Model for Lake Pontchartrain, LA USA

Field measurements from small boats and sparse arrays of instrumented buoys often do not provide sufficient data to capture the dynamic nature of biogeophysical parameters in may coastal aquatic environments. Several investigators have shown the MODIS 250 m images can provide daily synoptic views of suspended sediment concentration in coastal waters to determine sediment transport and fate. However, the use of MODIS for coastal environments can be limited due to a lack of cloud-free images. Sediment transport models are not constrained by sky conditions but often suffer from a lack of in situ observations for model calibration or validation. We demonstrate here the utility of MODIS 250 m to calibrate (set model parameters), validate output, and set or reset initial conditions of a hydrodynamic and sediment transport model (ECOMSED) developed for Lake Pontchartrain, LA USA. We present our approach in the context of how to quickly assess of 'prototype' an application of NASA data to support environmental managers and decision makers. The combination of daily MODIS imagery and model simulations offer a more robust monitoring and prediction system of suspended sediments than available from either system alone.

Miller, Richard L.↗

NASA NTRS2021

Dynamic Ensemble Prediction of Cognitive Performance in Space

Astronauts are exposed to a unique set of stressors in spaceflight. Microgravity, isolation, confinement, and environmental and operational hazards: all of these can impact sleep, vigilant attention, and alertness, which are critical to mission success. In this paper, we seek to understand the most important predictors of alertness over the course of a space mission, using self-reported, cognitive, and environmental data collected from 24 astronauts on 6-month missions to the International Space Station (ISS). Alertness was repeatedly and objectively assessed on the ISS with a brief 3-minute Psychomotor Vigilance Test (PVT) that is highly sensitive to sleep deprivation. To relate PVT performance to time-varying and sparsely-measured environmental, operational, and psychological covariates, we propose a n ensemble prediction model comprising of linear mixed effects regression, random forest, and functional concurrent regression models. An extensive cross-validation procedure reveals that this ensemble outperforms any one of its components alone. We also discover that a participant’s past performance, reported fatigue and stress, and temperature and radiation exposure were among the most important variables associated with alertness. This method is broadly applicable to environmental studies where the main goal is accurate, individualized prediction involving a mixture of person-level traits and irregularly measured time series.

Danni Tu↗

NASA NTRS2000

BOREAS AFM-12 1-km AVHRR Seasonal Land Cover Classification

The Boreal Ecosystem-Atmosphere Study (BOREAS) Airborne Fluxes and Meteorology (AFM)-12 team's efforts focused on regional scale Surface Vegetation and Atmosphere (SVAT) modeling to improve parameterization of the heterogeneous BOREAS landscape for use in larger scale Global Circulation Models (GCMs). This regional land cover data set was developed as part of a multitemporal one-kilometer Advanced Very High Resolution Radiometer (AVHRR) land cover analysis approach that was used as the basis for regional land cover mapping, fire disturbance-regeneration, and multiresolution land cover scaling studies in the boreal forest ecosystem of central Canada. This land cover classification was derived by using regional field observations from ground and low-level aircraft transits to analyze spectral-temporal clusters that were derived from an unsupervised cluster analysis of monthly Normalized Difference Vegetation Index (NDVI) image composites (April-September 1992). This regional data set was developed for use by BOREAS investigators, especially those involved in simulation modeling, remote sensing algorithm development, and aircraft flux studies. Based on regional field data verification, this multitemporal one-kilometer AVHRR land cover mapping approach was effective in characterizing the biome-level land cover structure, embedded spatially heterogeneous landscape patterns, and other types of key land cover information of interest to BOREAS modelers.The land cover mosaics in this classification include: (1) wet conifer mosaic (low, medium, and high tree stand density), (2) mixed coniferous-deciduous forest (80% coniferous, codominant, and 80% deciduous), (3) recent visible bum, vegetation regeneration, or rock outcrops-bare ground-sparsely vegetated slow regeneration bum (four classes), (4) open water and grassland marshes, and (5) general agricultural land use/ grasslands (three classes). This land cover mapping approach did not detect small subpixel-scale landscape features such as fens, bogs, and small water bodies. Field observations and comparisons with Landsat Thematic Mapper (TM) suggest a minimum effective resolution of these land cover classes in the range of three to four kilometers, in part, because of the daily to monthly compositing process. In general, potential accuracy limitations are mitigated by the use of conservative parameterization rules such as aggregation of predominant land cover classes within minimum horizontal grid cell sizes of ten kilometers. The AFM-12 one-kilometer AVHRR seasonal land cover classification data are available from the Earth Observing System Data and Information System (EOSDIS) Oak Ridge National Laboratory (ORNL) Distributed Active Archive Center (DAAC). The data files are available on a CD-ROM (see document number 20010000884).

Steyaert, Lou↗

NASA NTRS2013

Classification of Dust Days by Satellite Remotely Sensed Aerosol Products

Considerable progress in satellite remote sensing (SRS) of dust particles has been seen in the last decade. From an environmental health perspective, such an event detection, after linking it to ground particulate matter (PM) concentrations, can proxy acute exposure to respirable particles of certain properties (i.e. size, composition, and toxicity). Being affected considerably by atmospheric dust, previous studies in the Eastern Mediterranean, and in Israel in particular, have focused on mechanistic and synoptic prediction, classification, and characterization of dust events. In particular, a scheme for identifying dust days (DD) in Israel based on ground PM10 (particulate matter of size smaller than 10 nm) measurements has been suggested, which has been validated by compositional analysis. This scheme requires information regarding ground PM10 levels, which is naturally limited in places with sparse ground-monitoring coverage. In such cases, SRS may be an efficient and cost-effective alternative to ground measurements. This work demonstrates a new model for identifying DD and non-DD (NDD) over Israel based on an integration of aerosol products from different satellite platforms (Moderate Resolution Imaging Spectroradiometer (MODIS) and Ozone Monitoring Instrument (OMI)). Analysis of ground-monitoring data from 2007 to 2008 in southern Israel revealed 67 DD, with more than 88 percent occurring during winter and spring. A Classification and Regression Tree (CART) model that was applied to a database containing ground monitoring (the dependent variable) and SRS aerosol product (the independent variables) records revealed an optimal set of binary variables for the identification of DD. These variables are combinations of the following primary variables: the calendar month, ground-level relative humidity (RH), the aerosol optical depth (AOD) from MODIS, and the aerosol absorbing index (AAI) from OMI. A logistic regression that uses these variables, coded as binary variables, demonstrated 93.2 percent correct classifications of DD and NDD. Evaluation of the combined CART-logistic regression scheme in an adjacent geographical region (Gush Dan) demonstrated good results. Using SRS aerosol products for DD and NDD, identification may enable us to distinguish between health, ecological, and environmental effects that result from exposure to these distinct particle populations.

satellite remote sensing↗

NASA NTRS2011

Isolation Effects on the Moon: High Topographic Slope Observations from the LRO and LOLA Instruments

The extremely low temperatures in the Moon's polar permanent shadow regions (PSR) has long been considered a unique factor necessary for entrapping volatile Hydrogen (H). However, recent discoveries indicate some H concentrations lie outside PSR, suggesting other geophysical factors may also influence H distributions. In this study we consider insolation and its resulting thermal effects as a loss/redistribution process influencing the Moon's near-surface < 1m volatile H budget. To isolate regional (5deg latitude band) insolation effects we correlate two data sets collected from the ongoing, 1.5 year long mapping mission of the Lunar Reconnaissance Orbiter (LRO). Epithermal neutron mapping data from the Lunar Exploration Neutron Detector (LEND) is registered and analyzed in the context of slope derivations from Lunar topography maps produced by the Lunar Observing Laser Altimeter (LOLA). Lunar epithermal neutrons are inferred to be direct geochemical evidence for near-surface H due to the correlated suppression of surface leakage fluxes of epithermal neutrons with increased H concentration. Regional suppressions of neutrons seen in LEND maps are considered localized evidence of H concentration increase in the upper 1 m of the Lunar surface. To quantify spatially localized insolation effects, LEND data are averaged from sparsely distributed pixels, classed as a function of the LOLA slope derivations.

McClanahan, T. P.↗

NASA NTRS1989

Very accurate upward continuation to low heights in a test of non-Newtonian theory

Recently, gravity measurements were made on a tall, very stable television transmitting tower in order to detect a non-Newtonian gravitational force. This experiment required the upward continuation of gravity from the Earth's surface to points as high as only 600 m above ground. The upward continuation was based on a set of gravity anomalies in the vicinity of the tower whose data distribution exhibits essential circular symmetry and appropriate radial attenuation. Two methods were applied to perform the upward continuation - least-squares solution of a local harmonic expansion and least-squares collocation. Both methods yield comparable results, and have estimated accuracies on the order of 50 microGal or better (1 microGal = 10(exp -8) m/sq s). This order of accuracy is commensurate with the tower gravity measurments (which have an estimated accuracy of 20 microGal), and enabled a definitive detection of non-Newtonian gravity. As expected, such precise upward continuations require very dense data near the tower. Less expected was the requirement of data (though sparse) up to 220 km away from the tower (in the case that only an ellipsoidal reference gravity is applied).

Romaides, Anestis J.↗

NASA NTRS1991

Heart-Lung Interactions in Aerospace Medicine

Few of the heart-lung interactions that are discussed have been studied in any detail in the aerospace environment, but is seems that many such interactions must occur in the setting of altered accelerative loadings and pressure breathing. That few investigations are in progress suggests that clinical and academic laboratory investigators and aerospace organizations are further apart than during the pioneering work on pressure breathing and acceleration tolerance in the 1940s. The purpose is to reintroduce some of the perennial problems of aviation physiology as well as some newer aerospace concerns that may be of interest. Many possible heart-lung interactions are pondered, by necessity often drawing on data from within the aviation field, collected before the modern understanding of these interactions developed, or on recent laboratory data that may not be strictly applicable. In the field of zero-gravity effects, speculation inevitably outruns the sparse available data.

Guy, Harold J. B.↗

NASA NTRS2019

Liftoff and Transition Database Generation for Launch Vehicles Using Data-Fusion-Based Modeling

A data fusion technique for merging multiple data sources with differing fidelity and resolution was developed to support the production of aerodynamic line load databases for the Liftoff and Transition (LOT) flight phase of the Space Launch System (SLS). The technique uses a reduced order model based on a high-fidelity line load data set from Computational Fluid Dynamics (CFD) to predict solutions for a much larger solution space. Even higher-fidelity force and moment information (from wind-tunnel tests) is then used to adjust the model. The adjustment uses constrained optimization through the method of Lagrange multipliers in order to minimize the deviation of the line load distribution from the spatially-dense CFD solution, while ensuring that the integrated force and moment values match those observed in physical wind tunnel measurements. Though the wind-tunnel data are operationally-dense (available at many flow conditions), they are spatially coarse (as only the overall forces and moments are available). Conversely, CFD for such complex configurations is expensive, and thus operationally sparse. Data fusion techniques are necessary to make the most efficient use of available information, delivering accurate results within time and resource constraints.

Wignall, T. J.↗

NASA NTRS2011

Sparse Solutions for Single Class SVMs: A Bi-Criterion Approach

In this paper we propose an innovative learning algorithm - a variation of One-class nu Support Vector Machines (SVMs) learning algorithm to produce sparser solutions with much reduced computational complexities. The proposed technique returns an approximate solution, nearly as good as the solution set obtained by the classical approach, by minimizing the original risk function along with a regularization term. We introduce a bi-criterion optimization that helps guide the search towards the optimal set in much reduced time. The outcome of the proposed learning technique was compared with the benchmark one-class Support Vector machines algorithm which more often leads to solutions with redundant support vectors. Through out the analysis, the problem size for both optimization routines was kept consistent. We have tested the proposed algorithm on a variety of data sources under different conditions to demonstrate the effectiveness. In all cases the proposed algorithm closely preserves the accuracy of standard one-class nu SVMs while reducing both training time and test time by several factors.

Das, Santanu↗

NASA NTRS2017

Daily Ambient Air Pollution Metrics for Five Cities: Evaluation of Data Fusion-Based Estimates and Uncertainties

Spatiotemporal characterization of ambient air pollutant concentrations is increasingly relying on the combination of observations and air quality models to provide well-constrained, spatially and temporally complete pollutant concentration fields. Air quality models, in particular, are attractive, as they characterize the emissions, meteorological, and physiochemical process linkages explicitly while providing continuous spatial structure. However, such modeling is computationally intensive and has biases. The limitations of spatially sparse and temporally incomplete observations can be overcome by blending the data with estimates from a physically and chemically coherent model, driven by emissions and meteorological inputs. We recently developed a data fusion method that blends ambient ground observations and chemical transport-modeled (CTM) data to estimate daily, spatially resolved pollutant concentrations and associatedcorrelations. In this study, we assess the ability of the data fusion method to produce daily metrics (i.e., 1-hr max, 8-hr max, and 24-hr average) of ambient air pollution that capture spatiotemporal air pollution trends for 12 pollutants (CO, NO2, NOx, O3, SO2, PM (sub10), PM (sub 2.5), and five PM (sub 2.5) components) across five metropolitan areas (Atlanta, Birmingham, Dallas, Pittsburgh, and St. Louis), from 2002 to 2008. Three sets of comparisons are performed: (1) the CTM concentrations are evaluated for each pollutant and metropolitan domain, (2) the data fusion concentrations are compared with the monitor data, (3) a comprehensive cross-validation analysis against observed data evaluates the quality of the data fusion model simulations across multiple metropolitan domains. The resulting daily spatial field estimates of air pollutant concentrations and uncertainties are not only consistent with observations, emissions, andmeteorology, but substantially improve CTM-derived results for nearly all pollutants and all cities, with the exception of NO2 for Birmingham. The greatest improvements occur for O3 and PM (sub 2.5). Squared spatiotemporal correlation coefficients range between simulations and observations determined using cross-validation across all cities for air pollutants of secondary and mixed origins are R-squared equal to 0.88-0.93 (O3), 0.81-0.89 (SO4), 0.67-0.83 (PM (sub 2.5)), 0.52-0.72 (NO3), 0.43-0.80 (NH4), 0.32-0.51 (OC), and 0.14-0.71 (PM (sub 10)). Results for relatively homogeneous pollutants of secondary origin, tend to be better than those for more spatially heterogeneous (larger spatial gradients) pollutants of primary origin (NOx, CO, SO2 and EC). Generally, background concentrations and spatial concentration gradients reflect interurban airshed complexity and the effects of regional transport, whereas daily spatial pattern variability shows intra urban consistency in the fused data. With sufficiently high CTM spatial resolution, traffic-related pollutants exhibit gradual concentration gradients that peak toward the urban centers. Ambient pollutant concentration uncertainty estimates for the fused data are both more accurate and smaller than those for either the observations or the model simulations alone.

spatiotemporal fusion↗

NASA NTRS2020

Recommendations for Using Noise Monitors to Estimate Noise Exposure During X-59 Community Tests

A low fidelity simulation approach is used to explore how to place and use noise monitors during X-59 QueSST community tests, where people’s annoyance to the noise produced by the X-59 aircraft will be gathered. Several recommendations are provided including: 1) the desired number of sparsely spaced noise monitor sites within the survey area, 2) whether to group and average measurements across multiple noise monitors located at a site, 3) what spacing should be used if grouped noise monitors are used, 4) an approach to mitigate ambient noise contamination at the measurement sites, 5) a method to combine empirical and predicted dose estimates to provide a single dose estimate for respondents, and 6) assessing how changes in turbulence intensity and array configuration affect dose uncertainty. To make these recommendations, the error that is expected when fitting contrived, smoothly varying sonic boom “reference exposure surfaces” is studied when a spatially sparse and scattered set of samples is used as responses for the fit. The reference exposure surfaces mimic the sonic boom exposure at ground level that might be expected in the X-59 survey area in the absence of atmospheric turbulence, ambient noise, and other localized effects. The spatial extent of these surfaces varies and is representative of the different survey area sizes that might be expected during future X-59 community overflight tests. These contrived reference surfaces are sampled, and those reference samples are then perturbed to mimic atmospheric turbulence, ambient noise and other localized effects that might affect noise monitor measurements within overflown communities. Two different surface fitting methods are investigated when fitting these perturbed samples to approximate the reference surface. The first method uses interpolation between the perturbed data at the scattered sites to compute the fit. The second method fits a polynomial surface model to the perturbed data using ordinary least squares regression analysis. For both fitting methods, the root mean square fit error is computed from the pointwise difference between the fit surface and the reference surface as the count and configuration of the sites is varied while also averaging the error across many different realizations of both the smooth variation of the reference exposure surface and the random, localized perturbations at the sample sites. Different site configurations are compared using this error statistic to make the recommendations noted above. Additionally, the two fitting approaches (interpolation vs linear regression) are compared based on the fit error observed in these simulations. These analyses, comparisons, and recommendations should inform future decisions on the noise monitor placement and the methods used to analyze the noise monitor data that is collected during X-59 community overflights.

sonic boom↗