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At least 235 records · Page 13

Sources of Uncertainty in the Prediction of LAI / fPAR from MODIS

To explicate the sources of uncertainty in the prediction of biophysical variables over space, consider the general equation: where z is a variable with values on some nominal, ordinal, interval or ratio scale; y is a vector of input variables; u is the spatial support of y and z ; x and u are the spatial locations of y and z , respectively; f is a model and B is the vector of the parameters of this model. Any y or z has a value and a spatial extent which is called its support. Viewed in this way, categories of uncertainty are from variable (e.g. measurement), parameter, positional. support and model (e.g. structural) sources. The prediction of Leaf Area Index (LAI) and the fraction of absorbed photosynthetically active radiation (fPAR) are examples of z variables predicted using model(s) as a function of y variables and spatially constant parameters. The MOD15 algorithm is an example of f, called f(sub 1), with parameters including those defined by one of six biome types and solar and view angles. The Leaf Canopy Model (LCM)2, a nested model that combines leaf radiative transfer with a full canopy reflectance model through the phase function, is a simpler though similar radiative transfer approach to f(sub 1). In a previous study, MOD15 and LCM2 gave similar results for the broadleaf forest biome. Differences between these two models can be used to consider the structural uncertainty in prediction results. In an effort to quantify each of the five sources of uncertainty and rank their relative importance for the LAI/fPAR prediction problem, we used recent data for an EOS Core Validation Site in the broadleaf biome with coincident surface reflectance, vegetation index, fPAR and LAI products from the Moderate Resolution Imaging Spectrometer (MODIS). Uncertainty due to support on the input reflectance variable was characterized using Landsat ETM+ data. Input uncertainties were propagated through the LCM2 model and compared with published uncertainties from the MOD15 algorithm.

Dungan, Jennifer L.↗

Assessing within-Field Corn and Soybean Yield Variability from WorldView-3, Planet, Sentinel-2, and Landsat 8 Satellite Imagery

Crop yield monitoring is an important component in agricultural assessment. Multispectral remote sensing instruments onboard space-borne platforms such as Advanced Very High Resolution Radiometer (AVHRR), Moderate Resolution Imaging Spectroradiometer (MODIS), and Visible Infrared Imaging Radiometer Suite (VIIRS) have shown to be useful for efficiently generating timely and synoptic information on the yield status of crops across regional levels. However, the coarse spatial resolution data inherent to these sensors provides little utility at the management level. Recent satellite imagery collection advances toward finer spatial resolution (down to 1 m) alongside increased observational cadence (near daily) implies information on crops obtainable at field and within-field scales to support farming needs is now possible. To test this premise, we focus on assessing the efficiency of multiple satellite sensors, namely WorldView-3, Planet/Dove-Classic, Sentinel-2, and Landsat 8 (through Harmonized Landsat Sentinel-2 (HLS)), and investigate their spatial, spectral (surface reflectance (SR) and vegetation indices (VIs)), and temporal characteristics to estimate corn and soybean yields at sub-field scales within study sites in the US state of Iowa. Precision yield data as referenced to combine harvesters’ GPS systems were used for validation. We show that imagery spatial resolution of 3 m is critical to explaining 100% of the within-field yield variability for corn and soybean. Our simulation results show that moving to coarser resolution data of 10 m, 20 m, and 30 m reduced the explained variability to 86%, 72%, and 59%, respectively. We show that the most important spectral bands explaining yield variability were green (0.560 µm), red-edge (0.726 µm), and near-infrared (NIR - 0.865 µm). Furthermore, the high temporal frequency of Planet and a combination of Sentinel-2/Landsat 8 (HLS) data allowed for optimal date selection for yield map generation. Overall, we observed mixed performance of satellite-derived models with the coefficient of determination (R^2) varying from 0.21 to 0.88 (averaging 0.56) for the 30 m HLS and from 0.09 to 0.77 (averaging 0.30) for 3 m Planet. R^2 was lower for fields with higher yields, suggesting saturation of the satellite-collected reflectance features in those cases. Therefore, other biophysical variables, such as soil moisture and evapotranspiration, at similar fine spatial resolutions are likely needed alongside the optical imagery to fully explain the yields.

agriculture; yield; within-field; corn; soybean; r↗

A Methodology for Surface Soil Moisture and Vegetation Optical Depth Retrieval Using the Microwave Polarization Difference Index

A methodology for retrieving surface soil moisture and vegetation optical depth from satellite microwave radiometer data is presented. The procedure is tested with historical 6.6 GHz brightness temperature observations from the Scanning Multichannel Microwave Radiometer over several test sites in Illinois. Results using only nighttime data are presented at this time, due to the greater stability of nighttime surface temperature estimation. The methodology uses a radiative transfer model to solve for surface soil moisture and vegetation optical depth simultaneously using a non-linear iterative optimization procedure. It assumes known constant values for the scattering albedo and roughness. Surface temperature is derived by a procedure using high frequency vertically polarized brightness temperatures. The methodology does not require any field observations of soil moisture or canopy biophysical properties for calibration purposes and is totally independent of wavelength. Results compare well with field observations of soil moisture and satellite-derived vegetation index data from optical sensors.

Owe, Manfred↗

Heavy fragment production cross sections from 1.05 GeV/nucleon 56Fe in C, Al, Cu, Pb, and CH2 targets

We have obtained charge-changing cross sections and partial cross sections for fragmentation of 1.05 GeV/nucleon Fe projectiles incident on H, C, Al, Cu, and Pb nuclei. The energy region covered by this experiment is critical for an understanding of galactic cosmic ray propagation and space radiation biophysics. Surviving primary beam particles and fragments with charges from 12 to 25 produced within a forward cone of half-angle 61 mrad were detected using a silicon detector telescope to identify their charge and the cross sections were calculated after correction of the measured yields for finite target thickness effects. The cross sections are compared to model calculations and to previous measurements. Cross sections for the production of fragments with even-numbered nuclear charges are seen to be enhanced in almost all cases.

NASA Center HQS↗

The role of the EOS SAR in Mission to Planet Earth

The Earth Observing System (EOS) synthetic aperture radar (SAR) is a multifrequency, multipolarization radar which will be capable of the global monitoring of key quantitative measurements of geophysical and biophysical parameters. Scheduled to fly on its own dedicated spacecraft, it is one of three platforms that will make up the space-based component to EOS. The role of the SAR in the EOS mission is discussed, wtih emphasis on the geophysical product variables which will be generated with the SAR and used in global hydrologic, biogeochemical and energy cycle models. The latest concepts for processing the data, including calibration and geophysical processing strategies, are presented.

Holt, Benjamin↗

Biophysical aspects of heavy ion interactions in matter

The biological effects of high energy, high charge nuclei (HZE particles) occupy a central role in the management of space radiation hazards due to galactic cosmic rays. For the energy range of interest, the mean free path for nuclear interactions of these heavy ions is comparable to the thickness of the material traversed, and a significant fraction of stopping particles will undergo a nuclear reaction with the nuclei of the stopping material. Transport methods for HZE particles are dependent on models of the interaction of man-made systems with the space environment to an even greater extent than methods used for other types of radiation. Hence, there is a major need to validate these transport codes by comparison with experimental data. The basic physical properties of HZE particles will be reviewed and illustrated with the results of nuclear fragmentation experiments performed with 670A MeV neon ions incident on a water absorber and with measurements of multiple Coulomb scattering of uranium beams in copper. Finally, the extent to which physical measurements yield radiobiological predictions is illustrated for the example of neon.

Schimmerling, Walter↗

Picturing Data With Uncertainty

NASA is in the business of creating maps for scientific purposes to represent important biophysical or geophysical quantities over space and time. For example, maps of surface temperature over the globe tell scientists where and when the Earth is heating up; regional maps of the greenness of vegetation tell scientists where and when plants are photosynthesizing. There is always uncertainty associated with each value in any such map due to various factors. When uncertainty is fully modeled, instead of a single value at each map location, there is a distribution expressing a set of possible outcomes at each location. We consider such distribution data as multi-valued data since it consists of a collection of values about a single variable. Thus, a multi-valued data represents both the map and its uncertainty. We have been working on ways to visualize spatial multi-valued data sets effectively for fields with regularly spaced units or grid cells such as those in NASA's Earth science applications. A new way to display distributions at multiple grid locations is to project the distributions from an individual row, column or other user-selectable straight transect from the 2D domain. First at each grid cell in a given slice (row, column or transect), we compute a smooth density estimate from the underlying data. Such a density estimate for the probability density function (PDF) is generally more useful than a histogram, which is a classic density estimate. Then, the collection of PDFs along a given slice are presented vertically above the slice and form a wall. To minimize occlusion of intersecting slices, the corresponding walls are positioned at the far edges of the boundary. The PDF wall depicts the shapes of the distributions very dearly since peaks represent the modes (or bumps) in the PDFs. We've defined roughness as the number of peaks in the distribution. Roughness is another useful summary information for multimodal distributions. The uncertainty of the multi-valued data can also be interpreted by the number of peaks and the widths of the peaks as shown by the PDF walls.

Kao, David↗

Quantifying Grassland-to-Woodland Transitions and the Implications for Carbon and Nitrogen Dynamics in the Southwest United States

Replacement of grasslands and savannas by shrublands and woodlands has been widely reported in tropical, temperate and high-latitude rangelands worldwide (Archer 1994). These changes in vegetation structure may reflect historical shifts in climate and land use; and are likely to influence biodiversity, productivity, above- and below ground carbon and nitrogen sequestration and biophysical aspects of land surface-atmosphere interactions. The goal of our proposed research is to investigate how changes in the relative abundance of herbaceous and woody vegetation affect carbon and nitrogen dynamics across heterogeneous savannas and shrub/woodlands. By linking actual land-cover composition (derived through spectral mixture analysis of AVIRIS, TM, and AVHRR imagery) with a process-based ecosystem model, we will generate explicit predictions of the C and N storage in plants and soils resulting from changes in vegetation structure. Our specific objectives will be to (1) continue development and test applications of spectral mixture analysis across grassland-to-woodland transitions; (2) quantify temporal changes in plant and soil C and N storage and turnover for remote sensing and process model parameterization and verification; and (3) couple landscape fraction maps to an ecosystem simulation model to observe biogeochemical dynamics under changing landscape structure and climatological forcings.

Wessman, Carol A.↗

Spectral characterization of biophysical characteristics in a boreal forest: Relationship between Thematic Mapper band reflectance and leaf area index for Aspen

Results from analysis of a data set of simultaneous measurements of Thematic Mapper band reflectance and leaf area index are presented. The measurements were made over pure stands of Aspen in the Superior National Forest of northern Minnesota. The analysis indicates that the reflectance may be sensitive to the leaf area index of the Aspen early in the season. The sensitivity disappears as the season progresses. Based on the results of model calculations, an explanation for the observed relationship is developed. The model calculations indicate that the sensitivity of the reflectance to the Aspen overstory depends on the amount of understory present.

Badhwar, G.↗

Spectral characterization of biophysical characteristics in a boreal forest - Relationship between Thematic Mapper band reflectance and leaf area index for Aspen

Results from analysis of a data set of simultaneous measurements of Thematic Mapper band reflectance and leaf area index are presented. The measurements were made over pure stands of Aspen in the Superior National Forest of northern Minnesota. The analysis indicates that the reflectance may be sensitive to the leaf area index of the Aspen early in the season. The sensitivity disappears as the season progresses. Based on the results of model calculations, an explanation for the observed relationship is developed. The model calculations indicate that the sensitivity of the reflectance to the Aspen overstory depends on the amount of understory present.

Badhwar, G. D.↗

Sensitivity of June Near-Surface Temperatures and Precipitation in the Eastern United States to Historical Land Cover Changes Since European Settlement

Land cover changes alter the near surface weather and climate. Changes in land surface properties such as albedo, roughness length, stomatal resistance, and leaf area index alter the surface energy balance, leading to differences in near surface temperatures. This study utilized a newly developed land cover data set for the eastern United States to examine the influence of historical land cover change on June temperatures and precipitation. The new data set contains representations of the land cover and associated biophysical parameters for 1650, 1850, 1920, and 1992, capturing the clearing of the forest and the expansion of agriculture over the eastern United States from 1650 to the early twentieth century and the subsequent forest regrowth. The data set also includes the inferred distribution of potentially water-saturated soils at each time slice for use in the sensitivity tests. The Regional Atmospheric Modeling System, equipped with the Land Ecosystem-Atmosphere Feedback (LEAF-2) land surface parameterization, was used to simulate the weather of June 1996 using the 1992, 1920, 1850, and 1650 land cover representations. The results suggest that changes in surface roughness and stomatal resistance have caused present-day maximum and minimum temperatures in the eastern United States to warm by about 0.3 C and 0.4 C, respectively, when compared to values in 1650. In contrast, the maximum temperatures have remained about the same, while the minimums have cooled by about 0.1 C when compared to 1920. Little change in precipitation was found.

Strack, John E.↗

Section on Observed Impacts on El Nino

Agricultural applications of El Nino forecasts are already underway in some countries and need to be evaluated or re-evaluated. For example, in Peru, El Nino forecasts have been incorporated into national planning for the agricultural sector, and areas planted with rice and cotton (cotton being the more drought-tolerant crop) are adjusted accordingly. How well are this and other such programs working? Such evaluations will contribute to the governmental and intergovernmental institutions, including the Inter-American Institute for Global Change Research and the US National Ocean and Atmospheric Agency that are fostering programs to aid the effective use of forecasts. As El Nino climate forecasting grows out of the research mode into operational mode, the research focus shifts to include the design of appropriate modes of utilization. Awareness of and sensitivity to the costs of prediction errors also grow. For example, one major forecasting model failed to predict the very large El Nino event of 1997, when Pacific sea-surface temperatures were the highest on record. Although simple correlations between El Nino events and crop yields may be suggestive, more sophisticated work is needed to understand the subtleties of the interplay among the global climate system, regional climate patterns, and local agricultural systems. Honesty about the limitations of an forecast is essential, especially when human livelihoods are at stake. An end-to-end analysis links tools and expertise from the full sequence of ENSO cause-and-effect processes. Representatives from many disciplines are needed to achieve insights, e.g, oceanographers and atmospheric scientists who predict El Nino events, climatologists who drive global climate models with sea-surface temperature predictions, agronomists who translate regional climate connections in to crop yield forecasts, and economists who analyze market adjustments to the vagaries of climate and determine the value of climate forecasts. Methods include historical studies to understand past patterns and to test hindcasts of the prediction tools, crop modeling, spatial analysis and remote sensing. This research involves expanding, deepening, and applying the understanding of physical climate to the fields of agronomy and social science; and the reciprocal understanding of crop growth and farm economics to climatology. Delivery of a regional climate forecast with no information about how the climate forecast was derived limits its effectiveness. Explanation of a region's major climate driving forces helps to place a seasonal forecast in context. Then, a useful approach is to show historical responses to previous El Nino events, and projections, with uncertainty intervals, of crop response from dynamic process crop growth models. Regional ID forecasts should be updated with real-time weather conditions. Since every El Nino event is different, it is important to track, report and advise on each new event as it unfolds. The stability of human enterprises depends on understanding both the potentialities and the limits of predictability. Farmers rely on past experience to anticipate and respond to fluctuations in the biophysical systems on which their livelihoods depend. Now scientists are improving their ability to predict some major elements of climate variability. The improvements in the reliability of El Nino forecasts are encouraging, but seasonal forecasts for agriculture are not, and will probably never be completely infallible, due to the chaotic nature of the climate system. Uncertainties proliferate as we extend beyond Pacific sea-surface temperatures to climate teleconnections and agricultural outcomes. The goal of this research is to shed as a clear light as possible on these inherent uncertainties and thus to contribute to the development of appropriate responses to El Nino and other seasonal forecasts for a range of stakeholders, which, ultimately, includes food consumers everywhere.

Rosenzweig, Cynthia↗

Deriving the intermediate spectra and photocycle kinetics from time-resolved difference spectra of bacteriorhodopsin. The simpler case of the recombinant D96N protein

The bacteriorhodopsin photocycle contains more than five spectrally distinct intermediates, and the complexity of their interconversions has precluded a rigorous solution of the kinetics. A representation of the photocycle of mutated D96N bacteriorhodopsin near neutral pH was given earlier (Varo, G., and J. K. Lanyi. 1991. Biochemistry. 30:5008-5015) as BRhv-->K<==>L<==>M1-->M2--> BR. Here we have reduced a set of time-resolved difference spectra for this simpler system to three base spectra, each assumed to consist of an unknown mixture of the pure K, L, and M difference spectra represented by a 3 x 3 matrix of concentration values between 0 and 1. After generating all allowed sets of spectra for K, L, and M (i.e., M1 + M2) at a 1:50 resolution of the matrix elements, invalid solutions were eliminated progressively in a search based on what is expected, empirically and from the theory of polyene excited states, for rhodopsin spectra. Significantly, the average matrix values changed little after the first and simplest of the search criteria that disallowed negative absorptions and more than one maximum for the M intermediate. We conclude from the statistics that during the search the solutions strongly converged into a narrow region of the multidimensional space of the concentration matrix. The data at three temperatures between 5 and 25 degrees C yielded a single set of spectra for K, L, and M; their fits are consistent with the earlier derived photocycle model for the D96N protein.

NASA Program Exobiology↗

Quantum Hardware-Enabled Molecular Dynamics via Transfer Learning

The ability to perform ab initio molecular dynamics simulations using potential energy surfaces provided by quantum computers would open the door to virtually exact dynamics for a variety of chemical and biochemical systems, with impacts on catalysis and biophysics. Nonetheless, performing molecular dynamics on surfaces produced by quantum hardware has been hampered by the noisy energies typically produced by quantum computers and challenges associated with computing gradients and scaling to large systems interest. A recent set of advances in machine learning, known as transfer learning, provides a new path forward for molecular dynamics simulations on quantum hardware. Transfer learning offers a workaround, where one first trains models on larger, less accurate classical datasets and then refines them on smaller, more accurate quantum datasets. We explore this approach by training machine learning models to predict a molecule's potential energy based on its geometric structure using Behler-Parrinello neural networks. When successfully trained, the model enables energy gradient predictions necessary for dynamic simulations. To reduce the quantum resources needed, the model is initially trained with data derived from classical density functional theory and subsequently refined with a smaller dataset obtained from a variational quantum eigensolver optimization of the unitary coupled cluster ansatz. We show that this approach significantly reduces the size of the needed quantum training dataset while capturing the high accuracies needed within quantum chemistry simulations. The success of this two-step training method opens more opportunities to apply machine learning models on quantum data, a significant stride towards efficient quantum-classical hybrid computational models.

quantum computing↗

Space-Time Controls on Carbon Sequestration Over Large-Scale Amazon Basin

A major research focus of the LBA Ecology Program is an assessment of the carbon budget and the carbon sequestering capacity of the large scale forest-pasture system that dominates the Amazonia landscape, and its time-space heterogeneity manifest in carbon fluxes across the large scale Amazon basin ecosystem. Quantification of these processes requires a combination of in situ measurements, remotely sensed measurements from space, and a realistically forced hydrometeorological model coupled to a carbon assimilation model, capable of simulating details within the surface energy and water budgets along with the principle modes of photosynthesis and respiration. Here we describe the results of an investigation concerning the space-time controls of carbon sources and sinks distributed over the large scale Amazon basin. The results are derived from a carbon-water-energy budget retrieval system for the large scale Amazon basin, which uses a coupled carbon assimilation-hydrometeorological model as an integrating system, forced by both in situ meteorological measurements and remotely sensed radiation fluxes and precipitation retrieval retrieved from a combination of GOES, SSM/I, TOMS, and TRMM satellite measurements. Brief discussion concerning validation of (a) retrieved surface radiation fluxes and precipitation based on 30-min averaged surface measurements taken at Ji-Parana in Rondonia and Manaus in Amazonas, and (b) modeled carbon fluxes based on tower CO2 flux measurements taken at Reserva Jaru, Manaus and Fazenda Nossa Senhora. The space-time controls on carbon sequestration are partitioned into sets of factors classified by: (1) above canopy meteorology, (2) incoming surface radiation, (3) precipitation interception, and (4) indigenous stomatal processes varied over the different land covers of pristine rainforest, partially, and fully logged rainforests, and pasture lands. These are the principle meteorological, thermodynamical, hydrological, and biophysical control paths which perturb net carbon fluxes and sequestration, produce time-space switching of carbon sources and sinks, undergo modulation through atmospheric boundary layer feedbacks, and respond to any discontinuous intervention on the landscape itself such as produced by human intervention in converting rainforest to pasture or conducting selective/clearcut logging operations.

Smith, Eric A.↗

BOREAS Level-2 MAS Surface Reflectance and Temperature Images in BSQ Format

The BOReal Ecosystem-Atmosphere Study (BOREAS) Staff Science Aircraft Data Acquisition Program focused on providing the research teams with the remotely sensed aircraft data products they needed to compare and spatially extend point results. The MODIS Airborne Simulator (MAS) images, along with other remotely sensed data, were collected to provide spatially extensive information over the primary study areas. This information includes biophysical parameter maps such as surface reflectance and temperature. Collection of the MAS images occurred over the study areas during the 1994 field campaigns. The level-2 MAS data cover the dates of 21-Jul-1994, 24-Jul-1994, 04-Aug-1994, and 08-Aug-1994. The data are not geographically/geometrically corrected; however, files of relative X and Y coordinates for each image pixel were derived by using the C130 navigation data in a MAS scan model. The data are provided in binary image format files.

Hall, Forrest G.↗

Controls on Variations of Surface Energy, Water, and Carbon Budgets within Large-Scale Amazon Basin

A key research focus of the LBA Research Program is understanding the space-time variations in interlinked surface energy, water, and carbon budgets, the controls on these variations, and the implications of these controls on the carbon sequestering capacity of the large scale forest-pasture system that dominates the Amaz6nia landscape. Quantification of these variations and controls are investigated by a combination of in situ measurements, remotely sensed measurements from space, and a realistically forced hydrometeorological model coupled to a carbon assimilation model, capable of simulating details within the surface energy and water budgets along with the principle processes of photosynthesis and respiration. Herein we describe the results of an investigation concerning the space-time controls of carbon sources and sinks distributed over the large scale Amazon basin. The results are derived from a carbon-water-energy budget retrieval system for the large scale Amazon basin, which uses a coupled carbon assimilation-hydrometeorological model as an integrating system, forced by both in situ meteorological measurements and remotely sensed radiation and precipitation fluxes obtained from a combination of GOES, SSM/I, TOMS, and TRh4M satellite measurements. Results include validation of (a) retrieved surface radiation and precipitation fluxes based on 30-min averaged surface measurements taken at Ji-Parani in Rondania and Manaus in Amazonas, and (b) modeled sensible, latent, and C02 fluxes based on tower measurements taken at Reserva Jaru, Manaus and Fazenda Nossa Senhora. The space-time controls on carbon sequestration are partitioned into sets of factors classified by: (1) above canopy meteorology, (2) incoming surface radiation, (3) precipitation interception, and (4) indigenous stomatal processes varied over the different land covers of pristine rainforest, partially, and fully logged rainforests, and pasture lands. These are the principle meteorological, thermodynamical, hydrological, and biophysical control paths which perturb net carbon fluxes and sequestration, produce time-space switching of carbon sources and sinks, undergo modulation through atmospheric boundary layer feedbacks, and respond to any discontinuous intervention on the landscape itself such as produced by human intervention in converting rainforest to pasture or conducting selective/clearcut logging operations. The results demonstrate how relative carbon sequestration capacity of the Amazonian ecosystem responds to these controls, and how interpretation of space-time heterogeneities in carbon sequestration depends on a fairly exact quantification of the interacting non-linear properties of photosynthesis in response to incoming solar flux, air-canopy temperatures, and leaf water interception -- and soil respiration in response to upper layer soil temperature and water content. The results also show how the interpretation of the control processes is highly sensitive to the scales at which the surface fluxes are analyzed.

Smith, Eric A.↗

Visualizing 2D Probability Distributions from Satellite Image-Derived Data

Creating maps of biophysical and geophysical variables using Earth Observing System (EOS) satellite image data is an important component of Earth science. These 2D maps have a single value at every location and standard techniques are used to visualize them. Current tools fall short, however, when it is necessary to describe a distribution of values at each location. Distributions may represent a frequency of occurrence over time, frequency of occurrence from multiple runs of an ensemble forecast or possible values from an uncertainty model. 'Distribution data sets' are described, then a case study is presented to visualize such 2D distributions. Distribution data sets are different from multivariate data sets in the sense that the values are for a single variable instead of multiple variables. Our case study data consists of multiple realizations of percent forest cover, generated using a geostatistical technique that combines ground measurements and satellite imagery to model uncertainty about forest cover. We present several approaches for analyzing and visualizing such data sets. The first is a pixel-wise analysis of the probability density functions for the 2D image while the second is an analysis of features identified within the image. Such pixel-wise and feature-wise views will give Earth scientists a more complete understanding of distribution data sets.

Kao, David↗