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At least 163 records · Page 9
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NASA NTRS2016

Considering Combined or Separated Roughness and Vegetation Effects in Soil Moisture Retrievals

For more than six years, the Soil Moisture and Ocean Salinity (SMOS) mission has provided multi angular and full-polarization brightness temperature (TB) measurements at L-band. Geophysical products such as soil moisture (SM) and vegetation optical depth at nadir (tau(sub nad)) are retrieved by an operational algorithm using TB observations at different angles of incidence and polarizations. However, the quality of the retrievals depends on several surface effects, such as vegetation, soil roughness and texture, etc. In the microwave forward emission model used in the retrievals (L-band Microwave Emission Model, L-MEB),soil roughness is modeled with a semi-empirical equation using four main parameters (Q(sub r), H(sub r), N(sub rp), with p = H or V polarizations). At present, these parameters are calibrated with data provided by airborne studies and in situ measurements made at a local scale that is not necessarily representative of the large SMOS footprints (43 km on average) at global scale. In this study, we evaluate the impact of the calibrated values of N(sub rp) and H(sub r) on the SM and tau(sub nad) retrievals based on SMOS TB measurements (SMOS Level 3 product) over the Soil Climate Analysis Network (SCAN) network located in North America over five years (2011-2015). In this study, Qr was set equal to zero and we assumed that N(sub rH)= N(sub rV). The retrievals were performed by varying N(sub rp) from −1 to 2 by steps of 1 and H(sub r) from 0 to 0.6 by steps of 0.1. At satellite scale, the results show that combining vegetation and roughness effects in a single parameter provides the best results in terms of soil moisture retrievals, as evaluated against the in situ SM data. Even though our retrieval approach was very simplified, as we did not account for pixel heterogeneity, the accuracy we obtained in the SM retrievals was almost systematically better than those of the Level 3 product. Improved results were also obtained in terms of optical depth retrievals. These new results may have key consequences in terms of calibration of roughness effects within the algorithms of the SMOS (ESA) and the SMAP (NASA) space missions.

full-polarization↗
NASA NTRS2016

Evaluating ESA CCI Soil Moisture in East Africa

To assess growing season conditions where ground based observations are limited or unavailable, food security and agricultural drought monitoring analysts rely on publicly available remotely sensed rainfall and vegetation greenness. There are also remotely sensed soil moisture observations from missions like the European Space Agency (ESA) Soil Moisture and Ocean Salinity (SMOS) and NASAs Soil Moisture Active Passive (SMAP), however these time series are still too short to conduct studies that demonstrate the utility of these data for operational applications, or to provide historical context for extreme wet or dry events. To promote the use of remotely sensed soil moisture in agricultural drought and food security monitoring, we use East Africa as a case study to evaluate the quality of a 30+ year time series of merged active-passive microwave soil moisture from the ESA Climate Change Initiative (CCI-SM). Compared to the Normalized Difference Vegetation index (NDVI) and modeled soil moisture products, we found substantial spatial and temporal gaps in the early part of the CCI-SM record, with adequate data coverage beginning in 1992. From this point forward, growing season CCI-SM anomalies were well correlated (R greater than 0.5) with modeled, seasonal soil moisture, and in some regions, NDVI. We use correlation analysis and qualitative comparisons at seasonal time scales to show that remotely sensed soil moisture can add information to a convergence of evidence framework that traditionally relies on rainfall and NDVI in moderately vegetated regions.

Amy McNally↗
NASA NTRS2016

L-band Radiometer Calibration Consistency Assessment for the SMOS, SMAP, and Aquarius Instruments

Three L-band radiometers have been observing the Earth in order to retrieve soil moisture and ocean salinity. They use different instrument configurations and calibration and retrieval algorithms. In any case, the brightness temperature retrieved at the Earth surface should be consistent between all instruments. One reason for inconsistency would be the use of different approaches for the instrument calibration or the use of different models to retrieve surface brightness temperature. We report on the different approaches used for the SMOS, SMAP and Aquarius instruments and their impact on the observations consistency.

Aquarius↗
NASA NTRS2016

Evaluation of Assimilated SMOS Soil Moisture Data for US Cropland Soil Moisture Monitoring

Remotely sensed soil moisture data can provide timely, objective and quantitative crop soil moisture information with broad geospatial coverage and sufficiently high resolution observations collected throughout the growing season. This paper evaluates the feasibility of using the assimilated ESA Soil Moisture Ocean Salinity (SMOS)Mission L-band passive microwave data for operational US cropland soil surface moisture monitoring. The assimilated SMOS soil moisture data are first categorized to match with the United States Department of Agriculture (USDA)National Agricultural Statistics Service (NASS) survey based weekly soil moisture observation data, which are ordinal. The categorized assimilated SMOS soil moisture data are compared with NASSs survey-based weekly soil moisture data for consistency and robustness using visual assessment and rank correlation. Preliminary results indicate that the assimilated SMOS soil moisture data highly co-vary with NASS field observations across a large geographic area. Therefore, SMOS data have great potential for US operational cropland soil moisture monitoring.

SMOS↗
NASA NTRS2017

Evaluating Soil Moisture Retrievals from ESA's SMOS and NASA's SMAP Brightness Temperature Datasets

Two satellites are currently monitoring surface soil moisture (SM) using L-band observations: SMOS (Soil Moisture and Ocean Salinity), a joint ESA (European Space Agency), CNES (Centre national d'tudes spatiales), and CDTI (the Spanish government agency with responsibility for space) satellite launched on November 2, 2009 and SMAP (Soil Moisture Active Passive), a National Aeronautics and Space Administration (NASA) satellite successfully launched in January 2015. In this study, we used a multilinear regression approach to retrieve SM from SMAP data to create a global dataset of SM, which is consistent with SM data retrieved from SMOS. This was achieved by calibrating coefficients of the regression model using the CATDS (Centre Aval de Traitement des Donnes) SMOS Level 3 SM and the horizontally and vertically polarized brightness temperatures (TB) at 40 deg incidence angle, over the 2013 - 2014 period. Next, this model was applied to SMAP L3 TB data from Apr 2015 to Jul 2016. The retrieved SM from SMAP (referred to here as SMAP_Reg) was compared to: (i) the operational SMAP L3 SM (SMAP_SCA), retrieved using the baseline Single Channel retrieval Algorithm (SCA); and (ii) the operational SMOSL3 SM, derived from the multiangular inversion of the L-MEB model (L-MEB algorithm) (SMOSL3). This inter-comparison was made against in situ soil moisture measurements from more than 400 sites spread over the globe, which are used here as a reference soil moisture dataset. The in situ observations were obtained from the International Soil Moisture Network (ISMN; https:ismn.geo.tuwien.ac.at) in North of America (PBO_H2O, SCAN, SNOTEL, iRON, and USCRN), in Australia (Oznet), Africa (DAHRA), and in Europe (REMEDHUS, SMOSMANIA, FMI, and RSMN). The agreement was analyzed in terms of four classical statistical criteria: Root Mean Squared Error (RMSE),Bias, Unbiased RMSE (UnbRMSE), and correlation coefficient (R). Results of the comparison of these various products with in situ observations show that the performance of both SMAP products i.e. SMAP_SCA and SMAP_Reg is 48 similar and marginally better to that of the SMOSL3 product particularly over the PBO_H2O, SCAN, and USCRN sites. However, SMOSL3 SM was closer to the in situ observations over the DAHRA and Oznet sites. We found that the correlation between all three datasets and in situ measurements is best (R 0.80) over the Oznet sites and worst (R 0.58) over the SNOTEL sites for SMAP_SCA and over the DAHRA and SMOSMANIA sites (R 0.51 and R 0.45 for SMAP_Reg and SMOSL3, respectively). The Bias values showed that all products are generally dry, except over RSMN, DAHRA, and Oznet (and FMI for SMAP_SCA). Finally, our analysis provided interesting insights that can be useful to improve the consistency between SMAP and SMOS datasets.

SMOS↗
NASA NTRS2017

Integration of SMAP and SMOS L-Band Observations

Soil Moisture Active Passive (SMAP) mission and the ESA Soil Moisture and Ocean Salinity (SMOS) missions provide brightness temperature and soil moisture estimates every 2-3 days. SMAP brightness temperature observations were compared with SMOS observations at 40 Degrees incidence angle. The brightness temperatures from the two missions are not consistent and have a bias of about 2.7K over land with respect to each other. SMAP and SMOS missions use different retrieval algorithms and ancillary datasets which result in further inconsistencies between the soil moisture products. The reprocessed constant-angle SMOS brightness temperatures were used in the SMAP soil moisture retrieval algorithm to develop a consistent multi-satellite product. The integrated product will have an increased global revisit frequency (1 day) and period of record that would be unattainable by either one of the satellites alone. Results from the development and validation of the integrated product will be presented.

SMOS↗
NASA NTRS2016

Benefit of Modeling the Observation Error in a Data Assimilation Framework Using Vegetation Information Obtained From Passive Based Microwave Data

A primary operational goal of the United States Department of Agriculture (USDA) is to improve foreign market access for U.S. agricultural products. A large fraction of this crop condition assessment is based on satellite imagery and ground data analysis. The baseline soil moisture estimates that are currently used for this analysis are based on output from the modified Palmer two-layer soil moisture model, updated to assimilate near-real time observations derived from the Soil Moisture Ocean Salinity (SMOS) satellite. The current data assimilation system is based on a 1-D Ensemble Kalman Filter approach, where the observation error is modeled as a function of vegetation density. This allows for offsetting errors in the soil moisture retrievals. The observation error is currently adjusted using Normalized Difference Vegetation Index (NDVI) climatology. In this paper we explore the possibility of utilizing microwave-based vegetation optical depth instead.

Vegetation↗
NASA NTRS2017

Localization of L-Band RFI Sources from SMAP Data

RFI (Radio-Frequency Interference) in the 1400-1427 MHz band degrades the quality of measurements made by satellite missions such as SMAP (Soil Moisture Active/Passive), Aquarius and SMOS (Soil Moisture and Ocean Salinity). A technique is presented here to estimate the location on the ground of RFI sources using SMAP measurements. The results of this technique have been validated against data derived by other means.

Soldo, Yan↗
NASA NTRS2018

Global-scale Evaluation of SMAP, SMOS and ASCAT Soil Moisture Products Using Triple Collocation

Global-scale surface soil moisture products are currently available from multiple remote sensing platforms. Footprint-scale assessments of these products are generally restricted to limited number of densely-instrumented validation sites. However, by taking active and passive soil moisture products together with a third independent soil moisture estimates via land surface modeling, triple collocation (TC) can be applied to estimate the correlation metric of satellite soil moisture products (versus an unknown ground truth) over a quasi-global domain. Here, an assessment of Soil Moisture Active Passive (SMAP), Soil Moisture Ocean Salinity (SMOS) and Advanced SCATterometer (ASCAT) surface soil moisture retrievals via TC is presented. Considering the potential violation of TC error assumptions, the impact of active-passive and satellite-model error cross correlations on the TC-derived inter-comparison results is examined at in situ sites using quadruple collocation analysis. In addition, confidence intervals for the TC-estimated correlation metric are constructed from moving-block bootstrap sampling designed to preserve the temporal persistence of the original (unevenly-sampled) soil moisture time-series. This study is the first to apply TC to obtain a robust global-scale cross-assessment of SMAP, SMOS and ASCAT soil moisture retrieval accuracy in terms of anomaly temporal correlation. Our results confirm the overall advantage of SMAP (with a global average anomaly correlation of 0.76) over SMOS (0.66) and ASCAT (0.63) that has been established in several recent regional, ground-based studies. SMAP is also the best-performing product over the majority of applicable land pixels (52%), although SMOS and ASCAT each shows advantage in distinct geographic regions.

global-scale↗
NASA NTRS2018

Integration of SMAP and SMOS Observations

Soil Moisture Active Passive (SMAP) mission and the Soil Moisture and Ocean Salinity (SMOS) missions provide brightness temperature and soil moisture estimates every 2-3 days. SMAP brightness temperature observations were compared with SMOS observations at 40o incidence angle. The brightness temperatures from the two missions are not consistent. SMAP observations show a warmer TB bias (about 1.27 K: V pol and 0.62 K: H pol) as compared to SMOS. SMAP and SMOS missions use different retrieval algorithms and ancillary datasets which result in further inconsistencies between their soil moisture products. The reprocessed constant-angle SMOS brightness temperatures were used in the SMAP soil moisture retrieval algorithm to develop a consistent multi-satellite product. The integrated product has an increased global revisit frequency (1 day) and period of record that is unattainable by either one of the satellites alone. Results from the development and validation of the integrated soil moisture product will be presented.

passive microwave↗
NASA NTRS2018

Assimilation of Satellite Derived Soil Moisture Profiles into a Crop Modeling System for Robust Yield Estimates

Soil Moisture Measurement - Remote Sensing. - Microwave (MW) Remote Sensing: Physically based and quantitative in nature; Based on difference in dielectric constant; Coarse spatial resolution 25-40 km; Shallow SM estimation depth 0-5 cm (approx.); All weather capabilities (e.g. Advanced Microwave Scanning Radiometer - Earth Observing System (AMSR-E), Soil Moisture and Ocean Salinity (SMOS), Soil Moisture Active Passive (SMAP) etc.) - Thermal Infrared (TIR) Remote Sensing: Indirect SM retrieval through energy flux estimations; Relatively better spatial resolution 1-10 km; Root-zone moisture retrieval capability; Can not penetrate through clouds, hence data gaps (e.g. Surface Energy Balance Algorithm for Land (SEBAL), Atmospheric Land Exchange Inverse (ALEXI) etc.)

SERVIR↗
NASA NTRS2019

Assimilation of Satellite Soil Moisture for Improved Atmospheric Reanalyses

A newly developed, weakly coupled land and atmosphere data assimilation system for NASA’s Global Earth Observing System model is presented, and used to demonstrate the benefit of assimilating satellite soil moisture into an atmospheric reanalysis. Specifically, Advanced Scatterometer and Soil Moisture Ocean Salinity soil moisture retrievals are assimilated into a system that uses the same model, atmospheric assimilation system, and atmospheric observations as the Modern-Era Retrospective analysis for Research and Applications, version 2 (MERRA-2). The atmosphere is sensitive to soil moisture only under certain conditions. Hence, while the globally averaged model improvements were small, regionally, the soil moisture assimilation induced some substantial improvements. For example, in a large region spanning from western Europe across southern Russia, the soil moisture assimilation decreased the RMSE against independent station observations of daily maximum 2-m temperature (T(sup 2m, sub max)) by up to 0.4 K, and of 2-m specific humidity (q(exp 2m)) by up to 0.5 g/kg. Over all available stations, the mean T(sup 2m, sub max) RMSE was reduced from 2.82 to 2.79 K, while the mean q(exp 2m) RMSE was reduced from 1.25 to 1.20 g/kg. The soil moisture assimilation also reduced the mean RMSE across 29 flux tower sites from 34.2 to 32.6 Wm(exp −2) for latent heating, and from 37.7 to 36.5 Wm(exp −2) for sensible heating. For all variables evaluated, the soil moisture assimilation improved the model at monthly to seasonal, rather than daily, time scales. Based on the above experiments, it is recommended that satellite soil moisture be assimilated into future reanalyses, including the follow-on to MERRA-2.

Clara Draper↗
NASA NTRS2019

Uncertainty in Soil Moisture Retrievals: an Ensemble Approach Using SMOS L-Band Microwave Data

The uncertainty of soil moisture (SM) retrievals from satellite brightness temperature (TB) observations depends primarily on the choice of radiative transfer model (RTM) parameters, prior SM information and TB inputs. This paper studies the sensitivity of several (quasi-)operational and experimental SM retrieval products from the Soil Moisture Ocean Salinity (SMOS) mission to these choices at 11 reference sites, located in 7 watersheds across the United States (US). Different literature-based RTM parameter sets cause large biases between retrievals. Whereas typical RTM parameter sets are calibrated for SM retrievals, it is shown that a parameter set carefully optimized for TB forward modeling can also be used for retrieving SM. It is also shown that the inclusion of dynamic prior SM estimates in a Bayesian retrieval scheme can strongly improve SM retrievals, regardless of the choice of RTM parameters, and that the use of multi-angular and multi-polarization TB does not necessarily lead to superior retrievals compared to retrievals based on TB data at a single incidence angle and polarization. The second part of this paper evaluates ensemble uncertainty metrics for SM retrievals obtained by propagating a wide range of RTM parameters through the RTM. As expected for bounded variables, the spread in the ensemble SM retrievals is smallest for wet and dry SM values and highest for intermediate SM values. After removal of the strong long-term mean bias associated with the RTM parameter values for individual ensemble members, the remaining anomaly ensemble SM spread of 0.037 cu m/cu m approximates the actual time series unbiased root-mean-square-difference of 0.042 cu m/cu m between ensemble mean retrievals and in situ reference data across the reference sites. However, the temporal variability in the anomaly ensemble spread reveals higher-order biases in the retrieval error, which should be accounted for when characterizing retrieval error.

Jan Quets↗
NASA NTRS2019

Improving Surface Soil Moisture Estimates in West Africa Through GRACE Data Assimilation

Recent annual trends of precipitation and terrestrial water storage (TWS) in West Africa have been increasing over the past decade. Despite a significant impact of soil moisture on the TWS in West Africa, there is little research on the recent spatial and temporal behaviors of surface soil moisture (SSM) along with the hydrological trends and variability in West Africa. In this study, we assimilate TWS estimates from the Gravity Recovery and Climate Experiment (GRACE) mission into the Catchment Land Surface Model (CLSM) and evaluate its impacts on SSM simulations for the years, 2002–2017. The evaluation is performed using reference datasets: the African Monsoon Multidisciplinary Analysis (AMMA) in situ soil moisture observations, three currently available microwave satellite SSM observations from the Advanced Scatterometer (ASCAT), the Soil Moisture Ocean Salinity (SMOS), and the Soil Moisture Active Passive (SMAP) satellites and also the triple collocation analysis (TCA). Overall, modeled SSM shows good agreement with the reference datasets in terms of the anomaly SSM correlations. However, both modeled and ASCAT SSM are limited in their representation of the drying rates, as observed by ground observations, SMOS and SMAP estimates. Further, GRACE data assimilation results in improved SSM simulations in the humid regions with large annual TWS variability. This study demonstrates the utility of land data assimilation to inform hydrological conditions in West Africa, where soil moisture monitoring is necessary for water resource and livestock management.

West Africa↗
NASA NTRS2016

Managing and Servicing Physical Oceanographic Data at a NASA Distributed Active Archive Center

The NASA Earth Science Data Information Systems Project funds and operates 12 Distributed Active Archive Center(s) (DAAC) throughout the United States. Of these 12 centers, the Physical Oceanography DAAC (PO.DAAC) is committed to providing long term archival, distribution and stewardship for NASA physical oceanographic data, primarily derived from space-born satellite systems, but also including a growing set of recent and future in situ observations from the SPURS-1 and SPURS-2 campaigns. Notable NASA missions supported include: Seasat, TOPEX/Poseidon, NSCAT, QuikSCAT, ISS-RapidScat, Jason-1, Jason-2/OSTM, GRACE, Aquarius, GHRSST, and MODIS. The following interagency and international missions are also supported by PO.DAAC: AVHRR, Coriolis, DMSP, MetOp-A, MetOp-B, Oceansat-2. The PO.DAAC currently holds 525 datasets in public distribution, spanning the following observational parameters: sea surface temperature, sea surface salinity, ocean color, ocean surface currents, ocean surface wind speed, ocean surface wind direction, sea surface height, significant wave height, ocean water mass/thickness, and sea ice age. A hundred of these datasets are available in near-real-time. Datasets are distributed through a variety of open-source access protocols including FTP, OPeNDAP, and THREDDS. FTP will soon be phased out in favor of a recently introduced HTTPS PO.DAAC Drive interface that supports WebDAV and interoperable machine-to-machine communication. OPeNDAP supports remote data/metadata query, subset, and download. THREDDS provides the features of OPeNDAP with the additional feature of temporal aggregation. PO.DAAC also offers proprietary tools and services to further enhance the data discovery, visualization and analysis experience, including but not limited to: State of the Ocean, Web Services (data/metadata discovery and extraction), HiTIDE Level-2 subsetter, Live Access Server (LAS), Webification (w10nsci), and Rich Site Summary (RSS) Datacasting. To assist with provenance of datasets, PO.DAAC has implemented DOIs for the data it distributes so that they can be properly cited. There is a user forum and helpdesk that contains data recipes and via which users can get guidance. In summary, this presentation aims to provide a general overview of PO.DAAC’s web portal and data holdings along with a set of illustrative examples leading prospective data users into the practical utility of its tools and services.

Moroni, David F.↗
NASA NTRS2019

Integrated SMAP and SMOS Soil Moisture Observations

Soil Moisture Active Passive (SMAP) mission and the Soil Moisture and Ocean Salinity (SMOS) missions provide brightness temperature and soil moisture estimates every 2-3 days. SMAP brightness temperature observations were compared with SMOS observations at 40o incidence angle. The brightness temperatures from the two missions are close to each other but SMAP observations show a warmer TB bias (about 0.64 K: V pol and 1.14 K: H pol) as compared to SMOS. SMAP and SMOS missions use different retrieval algorithms and ancillary datasets which result in further inconsistencies between their soil moisture products. The reprocessed constant-angle SMOS brightness temperatures (SMOS-SMAP) were used in the SMAP soil moisture retrieval algorithm to develop a consistent multi-satellite product. The integrated product has an increased global revisit frequency (1 day) and period of record that is unattainable by either one of the satellites alone. The SMOS-SMAP soil moisture retrievals compared with in situ observations show a retrieval accuracy of less than 0.04 m3/m3. Results from the development and validation of the integrated soil moisture product will be presented.

SMOS↗
NASA NTRS2019

Assimilation of Satellite Soil Moisture for Improved Atmospheric Reanalyses

Satellite soil moisture observations are assimilated into NASA's Global Earth Observing System using a weakly coupled land/atmosphere data assimilation system. Specifically, Advanced Scatterometer and Soil Moisture Ocean Salinity soil moisture retrievals are assimilated with an Ensemble Kalman Filter into a system that uses the same model, atmospheric assimilation system (3DVar), and atmospheric observations as the Modern-Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2). The first set of experiments investigate the impact of the soil moisture assimilation on the model states and fluxes within the cycling DA system. Comparison to independent ground-based observations from global networks of soil moisture, latent and sensible heating, and 2m temperature and specific humidity showed small, but positive, improvements in the global mean statistics. Additionally, in some regions, the soil moisture assimilation induced more substantial improvements. For all variables evaluated, the soil moisture assimilation improved the model at monthly to seasonal, rather than daily, time scales. Based on the above experiments, it is recommended that satellite soil moisture be assimilated into future reanalyses, including the follow-on to MERRA-2.

Draper, Clara↗
NASA NTRS2019

Global Assimilation of L-Band Brightness Temperature Observations from SMAP and SMOS into the Catchment Land Surface Model and Contribution to the Skill of Soil Moisture Estimates

The Soil Moisture Active Passive (SMAP) and Soil Moisture Ocean Salinity (SMOS) missions provide global observations of L-band (1.4 GHz) passive microwave brightness temperature (Tb) observations at a resolution of ~40 km every 2-3 days. These observations have been available since 2015 from SMAP and since 2010 from SMOS.Soil moisture estimates from the separate assimilation of SMAP and SMOS Tb observations into land surface models were previously shown to improve over model-only estimates, thereby demonstrating the value of assimilating L-band Tb observations for soil moisture estimation. The assimilation experiments documented in the literature do not, however, establish whether the joint assimilation of SMAP and SMOS Tbs improves the skill of the resulting soil moisture estimates beyond what can be achieved with either set of observations alone. Moreover, because the published SMAP and SMOS results used different assimilation systems and configurations and applied different evaluation data and procedures, it is unclear whether the assimilation of SMAP-only or SMOS-only Tbs results in better soil moisture skill. In this presentation, we compare the results of three separate assimilation experiments for the period from April 2015 to present. The experiments utilize the SMAP Level-4 Soil Moisture (L4_SM) algorithm, which assimilates L-band Tb observations into the NASA Catchment land surface model using a spatially distributed ensemble Kalman filter. Specifically, the three experiments presented here assimilate (i) SMAP Tbs only, as in the L4_SM product, (ii) SMOS Tbs only, after interpolation of 40° incidence angle, and (iii) both SMAP and SMOS Tbs. In all other respects, the configuration of three experiments is identical, as is the validation of the output. Preliminary results suggest that the joint assimilation of SMAP and SMOS Tbs yields the most skillful soil moisture estimates when compared to independent in situ measurements. Moreover, the skill of the SMAP-only assimilation exceeds that of the SMOS-only assimilation. The presentation provides an in-depth evaluation of the skill of the assimilation estimates vs. independent in situ and satellite measurements and in terms of statistics of the observation-minus-forecast residuals and analysis increments.

Reichle, Rolf↗