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NASA NTRS2012

A Parameterized Inversion Model for Soil Moisture and Biomass from Polarimetric Backscattering Coefficients

A semi-empirical algorithm for the retrieval of soil moisture, root mean square (RMS) height and biomass from polarimetric SAR data is explained and analyzed in this paper. The algorithm is a simplification of the distorted Born model. It takes into account the physical scattering phenomenon and has three major components: volume, double-bounce and surface. This simplified model uses the three backscattering coefficients ( sigma HH, sigma HV and sigma vv) at low-frequency (P-band). The inversion process uses the Levenberg-Marquardt non-linear least-squares method to estimate the structural parameters. The estimation process is entirely explained in this paper, from initialization of the unknowns to retrievals. A sensitivity analysis is also done where the initial values in the inversion process are varying randomly. The results show that the inversion process is not really sensitive to initial values and a major part of the retrievals has a root-mean-square error lower than 5% for soil moisture, 24 Mg/ha for biomass and 0.49 cm for roughness, considering a soil moisture of 40%, roughness equal to 3cm and biomass varying from 0 to 500 Mg/ha with a mean of 161 Mg/ha

soil moisture↗

NASA NTRS2017

Validation of Globsnow-2 Snow Water Equivalent Over Eastern Canada

In Qubec, Eastern Canada, snowmelt runoff contributes more than 30% of the annual energy reserve for hydroelectricity production, and uncertainties in annual maximum snow water equivalent (SWE) over the region are one of the main constraints for improved hydrological forecasting. Current satellite-based methods for mapping SWE over Qubec's main hydropower basins do not meet Hydro-Qubec operational requirements for SWE accuracies with less than 15% error. This paper assesses the accuracy of the GlobSnow-2 (GS-2) SWE product, which combines microwave satellite data and in situ measurements, for hydrological applications in Qubec. GS-2 SWE values for a 30-year period (1980 to 2009) were compared with space- and time-matched values from a comprehensive dataset of in situ SWE measurements (a total of 38,990 observations in Eastern Canada). The root mean square error (RMSE) of the GS-2 SWE product is 94.1+/- 20.3 mm, corresponding to an overall relative percentage error (RPE) of 35.9%. The main sources of uncertainty are wet and deep snow conditions (when SWE is higher than 150 mm), and forest cover type. However, compared to a typical stand-alone brightness temperature channel difference algorithm, the assimilation of surface information in the GS-2 algorithm clearly improves SWE accuracy by reducing the RPE by about 30%. Comparison of trends in annual mean and maximum SWE between surface observations and GS-2 over 1980-2009 showed agreement for increasing trends over southern Qubec, but less agreement on the sign and magnitude of trends over northern Qubec. Extended at a continental scale, the GS-2 SWE trends highlight a strong regional variability.

Snow water equivalent↗

NASA NTRS1971

Bounding filters in the presence of inexactly known parameters.

Optimum bounding filters are derived for a specific version (steady state time-invariant with scalar observations) of the Kalman-Bucy filtering problem with inexactly known system parameters and for the Wiener filtering problem with inexactly known spectral densities. The designed filter obtains a bound on the actual error covariance which is not known, and it also prevents apparent divergence. Conditions are derived for the design of the optimum bounding filter within a permissible class of solutions; this turns out to be the min-max mean-square error filter for an extended class of solutions. The bounding filter can be of lower order than the original system, and a technique is devised for reducing the order of the filtering system and concurrently obtaining a figure of merit for its performance.

Nahi, N. E.↗

NASA NTRS2021

Asymmetry in Subseasonal Surface Air Temperature Forecast Error with Respect to Soil Moisture Initialization

Soil moisture (W) helps control evapotranspiration (ET), and ET variations can in turn have a distinct impact on 2-m air temperature (T2M), given that increases in evaporative cooling encourage reduced temperatures. Soil moisture is accordingly linked to T2M, and realistic soil moisture initialization has, in previous studies, been shown to improve the skill of subseasonal T2M forecasts. The relationship between soil moisture and evapotranspiration, however, is distinctly nonlinear, with ET tending to increase with soil moisture in drier conditions and to be insensitive to soil moisture variations in wetter conditions. Here, through an extensive analysis of subseasonal forecasts produced with a state-of-the-art seasonal forecast system, this nonlinearity is shown to imprint itself on T2M forecast error in the conterminous United States in two unique ways: (i) the T2M forecast bias (relative to independent observations) induced by a negative precipitation bias tends to be larger for dry initializations, and (ii) on average, the unbiased root-mean-square error (ubRMSE) tends to be larger for dry initializations. Such findings can aid in the identification of forecasts of opportunity; taken a step further, they suggest a pathway for improving bias correction and uncertainty estimation in subseasonal T2M forecasts by conditioning each on initial soil moisture state.

Air Temperature↗

NASA NTRSDate not supplied

T2M Forecasts at Subseasonal Leads: Do Different Soil Moisture Initial States Have Different Impacts?

2-meter Temperature↗

NASA NTRS2019

Global Validation of Columnar Water Vapor Derived from EOS MODIS-MAIAC Algorithm Against the Ground-Based AERONET Observations

The water vapor is a relevant greenhouse gas in the Earth's climate system, and satellite products become one of the most effective way to characterize and monitor the columnar water vapor (CWV) content at global scale. Recently, a new product (MCD19) was released as part of MODIS (Moderate Resolution Imaging Spectroradiometer) Collection 6 (C6). This operational product from the Multi-Angle Implementation for Atmospheric Correction (MAIAC) algorithm includes a high 1-kilometer resolution CWV retrievals. This study presents the first global validation of MAIAC C6 CWV obtained from MODIS MCD19A2 product. This evaluation was performed using Aerosol Robotic Network (AERONET) observations at 265 sites (2000-2017). Overall, the results show a good agreement between MAIAC/AERONET CWV retrievals, with correlation coefficient higher than 0.95 and RMS (Root Mean Square) error lower than 0.250 centimeters. The binned error analysis revealed an underestimation (approximately 10 percent) of Aqua CWV retrievals with negative bias for CWV higher than 3.0 centimeters. In contrast, Terra CWV retrievals show a slope of regression close to unity and a low mean bias of 0.075 centimeters. While the accuracy is relatively similar between 1.0 and 5.0 centimeters for both sensor products, Terra dataset is more reliable for applications in humid tropical areas (less than 5.0 centimeters). The expected error was defined as plus or minus 15 percent, with less than 68 percent of retrievals falling within this envelope. However, the accuracy is regionally dependent, and lower error should be expected in some regions, such as South America and Oceania. Since MODIS instruments have exceeded their design lifetime, time series analysis was also presented for both sensor products. The temporal analysis revealed a systematic offset of global average between Terra and Aqua CWV records. We also found an upward trend (approximately 0.2 centimeters per decade) in Terra CWV retrievals, while Aqua CWV retrievals remain stable over time. The sensor degradation influences the ability to detect climate signals, and this study indicates the need for revisiting calibration of the MODIS bands 17-19, mainly for Terra instrument, to assure the quality of the MODIS water vapor product. Finally, this study presents a comprehensive validation analysis of MAIAC CWV over land, raising the understanding of its overall quality.

MAIAC↗

NASA NTRS2011

Space-Time Data Fusion

Space-time Data Fusion (STDF) is a methodology for combing heterogeneous remote sensing data to optimally estimate the true values of a geophysical field of interest, and obtain uncertainties for those estimates. The input data sets may have different observing characteristics including different footprints, spatial resolutions and fields of view, orbit cycles, biases, and noise characteristics. Despite these differences all observed data can be linked to the underlying field, and therefore the each other, by a statistical model. Differences in footprints and other geometric characteristics are accounted for by parameterizing pixel-level remote sensing observations as spatial integrals of true field values lying within pixel boundaries, plus measurement error. Both spatial and temporal correlations in the true field and in the observations are estimated and incorporated through the use of a space-time random effects (STRE) model. Once the models parameters are estimated, we use it to derive expressions for optimal (minimum mean squared error and unbiased) estimates of the true field at any arbitrary location of interest, computed from the observations. Standard errors of these estimates are also produced, allowing confidence intervals to be constructed. The procedure is carried out on a fine spatial grid to approximate a continuous field. We demonstrate STDF by applying it to the problem of estimating CO2 concentration in the lower-atmosphere using data from the Atmospheric Infrared Sounder (AIRS) and the Japanese Greenhouse Gasses Observing Satellite (GOSAT) over one year for the continental US.

Greenhouse Gases Observing Satellite (GOSAT)↗

NASA NTRS2021

Assimilation of SMAP Brightness Temperature Observations in the GEOS Land–Atmosphere Data Assimilation System

Errors in soil moisture adversely impact the modeling of land–atmosphere water and energy fluxes and, consequently, near-surface atmospheric conditions in atmospheric data assimilation systems (ADAS). To mitigate such errors, a land surface analysis is included in many such systems, although not yet in the currently operational NASA Goddard Earth Observing System (GEOS) ADAS. This article investigates the assimilation of L-band brightness temperature (Tb) observations from the Soil Moisture Active Passive (SMAP) mission in the GEOS weakly coupled land–atmosphere data assimilation system (LADAS) during boreal summer 2017. The SMAP Tb analysis improves the correlation of LADAS surface and root-zone soil moisture versus in situ measurements by ∼0.1–0.26 over that of ADAS estimates; the unbiased root-mean-square error of LADAS soil moisture is reduced by 0.002–0.008 m^3 /m^3 from that of ADAS. Furthermore, the global land average RMSE versus in situ measurements of screen-level air specific humidity (q2m) and daily maximum temperature (T2m_max ) is reduced by 0.05 g/kg and 0.04 K, respectively, for LADAS compared to ADAS estimates. Regionally, the RMSE of LADAS q2m and T2m_max is improved by up to 0.4 g/kg and 0.3 K, respectively. Improvement in LADAS specific humidity extends into the lower troposphere (below ∼700 mb), with relative improvements in bias of 15–25%, although LADAS air temperature bias slightly increases relative to that of ADAS. Finally, the root mean square of the LADAS Tb observation-minus-forecast residuals is smaller by up to ∼0.1 K than in a land-only assimilation system, corroborating the positive impact of the Tb analysis on the modeled land–atmosphere coupling.

microwave remote sensing↗

NASA NTRS2019

Quantitative Comparison of Proprietary and Open-Source Georeferencing Tools for Use with Astronaut Photography

The Crew Earth Observations (CEO) Facility within the Earth Science and Remote Sensing Unit at NASA’s Johnson Space Center supports the acquisition, analysis, and curation of astronaut photography of Earth’s surface and atmosphere. Astronauts on the International Space Station (ISS) respond to requests from CEO to acquire imagery of scientific and education targets, to include high profile targets in response to activations from the International Charter for Space & Major Disasters (also known as the International Disaster Charter, or IDC) and NASA’s Disasters Program. CEO facilitates the acquisition of astronaut photography in response to IDC events and delivers georeferenced data products to the United States Geological Survey (USGS) for distribution to the disaster community. Using GeoRef, an internal web-based tool developed in collaboration with NASA’s Ames Research Center, CEO generates data packages of georeferenced imagery, uncertainty images for assessing control and tie point accuracy, and metadata documenting raw and processed data. Operational experience with the Georef software identified vulnerabilities to internal code and server errors that can significantly increase time of data production. As such, CEO developed a backup procedure in case the GeoRef software experiences front-end or back-end errors. A system using OSGEO’s open-source QGIS software combined with a semi-automated pipeline using the object-oriented Python language and the Geospatial Abstract Library for generating metadata is quantitatively compared to GeoRef’s data package for quality and productivity. Root Mean Square Error (RMSE) provides a standard measurement of data quality as it relates to ground error. Assessing RMSE measurements generated from georeferenced astronaut photographs acquired with different obliquity and focal length offers a comprehensive accuracy assessment of the software’s transformation algorithms. This assessment will indicate the software's ability to produce data products with the least ground-error or highest data quality regarding ground accuracy. In addition, a comparison of the software’s efficiency in generating a data package that includes georeferenced images, metadata, and uncertainty images for measuring tie/ground point error was performed. Initial results, based on the comparison of three nadir-facing astronaut photographs acquired with a 95mm focal length, reveal the QGIS-based system's average RMSE is 2.36 (pixels) suggesting its georectification system produces data products that meet and perhaps improve upon Georef solution's average RMSE of 32.99 (pixels). However, the QGIS system was unable to reproduce two unique Georef data products, uncertainty images for measuring tie and control point errors and a translated unwrapped image. In addition, the Georef software is designed to accept handheld camera pose information from a hardware component (Geosens) scheduled for deployment on the ISS in late 2018; this information is intended to provide increased accuracy and auto-registration capability for astronaut photographs. Future work is expected to determine the QGIS-based georectification system’s potential as an open-source alternative (and operational backup) to Georef for georeferencing the full range of resolutions and viewing angles unique to handheld digital camera imagery in support of ISS disaster response activities.

Jagge, Amy M.↗

NASA NTRS1993

A critical analysis of the accuracy of several numerical techniques for combustion kinetic rate equations

A detailed analysis of the accuracy of several techniques recently developed for integrating stiff ordinary differential equations is presented. The techniques include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP4 developed specifically to solve chemical kinetic rate equations. The accuracy study is made by application of these codes to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. To illustrate the error variation in the different combustion regimes the species are divided into three types (reactants, intermediates, and products), and error versus time plots are presented for each species type and the temperature. These plots show that CHEMEQ is the most accurate code during induction and early heat release. During late heat release and equilibration, however, the other codes are more accurate. A single global quantity, a mean integrated root-mean-square error, that measures the average error incurred in solving the complete problem is used to compare the accuracy of the codes. Among the codes examined, LSODE is the most accurate for solving chemical kinetics problems. It is also the most efficient code, in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that use of the algebraic enthalpy conservation equation to compute the temperature can be more accurate and efficient than integrating the temperature differential equation.

Radhadrishnan, Krishnan↗

NASA NTRS2017

Monitoring Land Surface Albedo and Vegetation Dynamics Using High Spatial and Temporal Resolution Synthetic Time Series from Landsat and the MODIS BRDF/NBAR/Albedo Product

Seasonal vegetation phenology can significantly alter surface albedo which in turn affects the global energy balance and the albedo warmingcooling feedbacks that impact climate change. To monitor and quantify the surface dynamics of heterogeneous landscapes, high temporal and spatial resolution synthetic time series of albedo and the enhanced vegetation index (EVI) were generated from the 500-meter Moderate Resolution Imaging Spectroradiometer (MODIS) operational Collection V006 daily BRDF (Bidirectional Reflectance Distribution Function) / NBAR (Nadir BRDF-Adjusted Reflectance) / albedo products and 30-meter Landsat 5 albedo and near-nadir reflectance data through the use of the Spatial and Temporal Adaptive Reflectance Fusion Model (STARFM). The traditional Landsat Albedo (Shuai et al., 2011) makes use of the MODIS BRDFAlbedo products (MCD43) by assigning appropriate BRDFs from coincident MODIS products to each Landsat image to generate a 30-meter Landsat albedo product for that acquisition date. The available cloud free Landsat 5 albedos (due to clouds, generated every 16 days at best) were used in conjunction with the daily MODIS albedos to determine the appropriate 30-meter albedos for the intervening daily time steps in this study. These enhanced daily 30-meter spatial resolution synthetic time series were then used to track albedo and vegetation phenology dynamics over three Ameriflux tower sites (Harvard Forest in 2007, Santa Rita in 2011 and Walker Branch in 2005). These Ameriflux sites were chosen as they are all quite nearby new towers coming on line for the National Ecological Observatory Network (NEON), and thus represent locations which will be served by spatially paired albedo measures in the near future. The availability of data from the NEON towers will greatly expand the sources of tower albedometer data available for evaluation of satellite products. At these three Ameriflux tower sites the synthetic time series of broadband shortwave albedos were evaluated using the tower albedo measurements with a Root Mean Square Error (RMSE) less than 0.013 and a bias within the range of 0.006. These synthetic time series provide much greater spatial detail than the 500 meter gridded MODIS data, especially over more heterogeneous surfaces, which improves the efforts to characterize and monitor the spatial variation across species and communities. The mean of the difference between maximum and minimum synthetic time series of albedo within the MODIS pixels over a subset of satellite data of Harvard Forest (16 kilometers by 14 kilometers) was as high as 0.2 during the snow-covered period and reduced to around 0.1 during the snow-free period. Similarly, we have used STARFM to also couple MODIS Nadir BRDF-Adjusted Reflectances (NBAR) values with Landsat 5 reflectances to generate daily synthetic times series of NBAR and thus Enhanced Vegetation Index (NBAR-EVI) at a 30-meter resolution. While normally STARFM is used with directional reflectances, the use of the view angle corrected daily MODIS NBAR values will provide more consistent time series. These synthetic times series of EVI are shown to capture seasonal vegetation dynamics with finer spatial and temporal details, especially over heterogeneous land surfaces.

Vegetation Index↗

NASA NTRS1994

Comparison of transform coding methods with an optimal predictor for the data compression of digital elevation models

Statistical encoding techniques enable the reduction of the number of bits required to encode a set of symbols, and are derived from their probabilities. Huffman encoding is an example of statistical encoding that has been used for error-free data compression. The degree of compression given by Huffman encoding in this application can be improved by the use of prediction methods. These replace the set of elevations by a set of corrections that have a more advantageous probability distribution. In particular, the method of Lagrange Multipliers for minimization of the mean square error has been applied to local geometrical predictors. Using this technique, an 8-point predictor achieved about a 7 percent improvement over an existing simple triangular predictor.

Lewis, Michael↗

NASA NTRS2022

The Joint Assimilation of Remotely Sensed Leaf Area Index and Surface Soil Moisture into a Land Surface Model

This work tests the hypothesis that jointly assimilating satellite observations of leaf area index and surface soil moisture into a land surface model improves the estimation of land vegetation and water variables. An Ensemble Kalman Filter is used to test such hypothesis across the Contiguous United States during April 2015 – December 2018. The performance of the proposed methodology is assessed for several modeled vegetation and water variables (evapotranspiration, net ecosystem exchange, and soil moisture) in terms of random errors and anomaly correlation coefficients against a set of independent validation datasets (i.e., Global Land Evaporation Amsterdam Model, FLUXCOM, and International Soil Moisture Network). Results show that the assimilation of leaf area index mostly improves the estimation of evapotranspiration and net ecosystem exchange, whereas the assimilation of surface soil moisture alone improves surface soil moisture content, especially in the western US, in terms of both root mean squared error and anomaly correlation coefficient. The joint assimilation of vegetation and soil moisture information combines the results of individual vegetation and soil moisture assimilations and reduces errors (and increases correlations with the reference datasets) in evapotranspiration, net ecosystem exchange, and surface soil moisture simulated by the land surface model. However, because soil moisture satellite observations only provide information on the water content in the top 5 cm of the soil column, the impact of the proposed data assimilation technique on root zone soil moisture is limited. This work moves one step forward in the direction of improving our estimation and understanding of land surface interactions using a multi-variate data assimilation approach, which can be particularly useful in regions of the world where ground observations are sparse or missing altogether.

Data assimilation↗

NASA NTRS1981

End-to-end quality measure for transmission of compressed imagery over a noisy coded channel

For the transmission of imagery at high data rates over large distances with limited power and system gain, it is usually necessary to compress the data before transmitting it over a noisy channel that uses channel coding to reduce the effect of noise introduced errors. Both compression and channel noise introduce distortion into the imagery. In order to design a communication link that provides adequate quality of received images, it is necessary first to define some suitable distortion measure that accounts for both these kinds of distortion and then to perform various tradeoffs to arrive at system parameter values that will provide a sufficiently low level of received image distortion. The overall mean square error is used as the distortion measure and a description of how to perform these tradeoffs are included.

Korwar, V. N.↗

NASA NTRS2019

Multilevel Monte Carlo Estimation of Unbiased Expectation via Sample Reuse and the Low Variance Estimation of Asymptotic Rates

A new variant of the multilevel Monte Carlo estimator [5, 3, 9, 12] is presented for the estimation of expectation statistics that utilizes sample reuse in specified levels, explicitly removes approximation error bias associated with numerically computed output quantities of interest that have an asymptotic limit behavior, and permits a low variance estimate of the asymptotic rate of convergence to that limit. In addition, it is shown that this new multilevel Monte Carlo variant can yield a computational cost savings. A review of Monte Carlo and multilevel Monte Carlo estimators is presented that includes analysis of expected value, expected mean squared error, and the calculation of optimized multilevel sample size parameters. The multilevel Monte Carlo estimator produces estimates of expectation for numerically approximated output quantities of interest that are biased by approximation error. When the quantity of interest can be modeled as the asymptotic limit of numerically approximated output quantities of interest, it is theoretically possible to remove this approximation error bias in the multilevel Monte Carlo estimator. In actual implementations, however, this procedure is unreliable due to statistical variability and inaccuracy in estimating the needed asymptotic limit. Analysis and numerical experiment show that the proposed variant of the multilevel Monte Carlo method greatly reduces (in some cases eliminates) the statistical variability in this limit estimation.

Barth, Timothy↗

NASA NTRS1991

Estimation of time averages from irregularly spaced observations - With application to coastal zone color scanner estimates of chlorophyll concentration

The sampling error of an arbitrary linear estimate of a time-averaged quantity constructed from a time series of irregularly spaced observations at a fixed located is quantified through a formalism. The method is applied to satellite observations of chlorophyll from the coastal zone color scanner. The two specific linear estimates under consideration are the composite average formed from the simple average of all observations within the averaging period and the optimal estimate formed by minimizing the mean squared error of the temporal average based on all the observations in the time series. The resulting suboptimal estimates are shown to be more accurate than composite averages. Suboptimal estimates are also found to be nearly as accurate as optimal estimates using the correct signal and measurement error variances and correlation functions for realistic ranges of these parameters, which makes it a viable practical alternative to the composite average method generally employed at present.

Chelton, Dudley B.↗

NASA NTRS2017

The Use of Sun Elevation Angle for Stereogrammetric Boreal Forest Height in Open Canopies

Stereogrammetry applied to globally available high resolution spaceborne imagery (HRSI; less than 5 m spatial resolution) yields fine-scaled digital surface models (DSMs) of elevation. These DSMs may represent elevations that range from the ground to the vegetation canopy surface, are produced from stereoscopic image pairs (stereo pairs) that have a variety of acquisition characteristics, and have been coupled with lidar data of forest structure and ground surface elevation to examine forest height. This work explores surface elevations from HRSI DSMs derived from two types of acquisitions in open canopy forests. We (1) apply an automated mass-production stereogrammetry workflow to along-track HRSI stereo pairs, (2) identify multiple spatially coincident DSMs whose stereo pairs were acquired under different solar geometry, (3) vertically co-register these DSMs using coincident spaceborne lidar footprints (from ICESat-GLAS) as reference, and(4) examine differences in surface elevations between the reference lidar and the co-registered HRSI DSMs associated with two general types of acquisitions (DSM types) from different sun elevation angles. We find that these DSM types, distinguished by sun elevation angle at the time of stereo pair acquisition, are associated with different surface elevations estimated from automated stereogrammetry in open canopy forests. For DSM values with corresponding reference ground surface elevation from spaceborne lidar footprints in open canopy northern Siberian Larix forests with slopes less than10, our results show that HRSI DSM acquired with sun elevation angles greater than 35deg and less than 25deg (during snow-free conditions) produced characteristic and consistently distinct distributions of elevation differences from reference lidar. The former include DSMs of near-ground surfaces with root mean square errors less than 0.68 m relative to lidar. The latter, particularly those with angles less than 10deg, show distributions with larger differences from lidar that are associated with open canopy forests whose vegetation surface elevations are captured. Terrain aspect did not have a strong effect on the distribution of vegetation surfaces. Using the two DSM types together, the distribution of DSM-differenced heights in forests (6.0 m, sigma = 1.4 m) was consistent with the distribution of plot-level mean tree heights (6.5m, sigma = 1.2 m). We conclude that the variation in sun elevation angle at time of stereo pair acquisition can create illumination conditions conducive for capturing elevations of surfaces either near the ground or associated with vegetation canopy. Knowledge of HRSI acquisition solar geometry and snow cover can be used to understand and combine stereogrammetric surface elevation estimates to co-register rand difference overlapping DSMs, providing a means to map forest height at fine scales, resolving the vertical structure of groups of trees from spaceborne platforms in open canopy forests.

Paul M Montesano↗

NASA NTRS2013

Spatial Statistical Data Fusion (SSDF)

As remote sensing for scientific purposes has transitioned from an experimental technology to an operational one, the selection of instruments has become more coordinated, so that the scientific community can exploit complementary measurements. However, tech nological and scientific heterogeneity across devices means that the statistical characteristics of the data they collect are different. The challenge addressed here is how to combine heterogeneous remote sensing data sets in a way that yields optimal statistical estimates of the underlying geophysical field, and provides rigorous uncertainty measures for those estimates. Different remote sensing data sets may have different spatial resolutions, different measurement error biases and variances, and other disparate characteristics. A state-of-the-art spatial statistical model was used to relate the true, but not directly observed, geophysical field to noisy, spatial aggregates observed by remote sensing instruments. The spatial covariances of the true field and the covariances of the true field with the observations were modeled. The observations are spatial averages of the true field values, over pixels, with different measurement noise superimposed. A kriging framework is used to infer optimal (minimum mean squared error and unbiased) estimates of the true field at point locations from pixel-level, noisy observations. A key feature of the spatial statistical model is the spatial mixed effects model that underlies it. The approach models the spatial covariance function of the underlying field using linear combinations of basis functions of fixed size. Approaches based on kriging require the inversion of very large spatial covariance matrices, and this is usually done by making simplifying assumptions about spatial covariance structure that simply do not hold for geophysical variables. In contrast, this method does not require these assumptions, and is also computationally much faster. This method is fundamentally different than other approaches to data fusion for remote sensing data because it is inferential rather than merely descriptive. All approaches combine data in a way that minimizes some specified loss function. Most of these are more or less ad hoc criteria based on what looks good to the eye, or some criteria that relate only to the data at hand.

Braverman, Amy J.↗