Data Assimilation Applications Research Articles

Remote Sensing Data Assimilation in Crop Growth Modeling from an Agricultural Perspective: New Insights on Challenges and Prospects

The frequent occurrence of global climate change and natural disasters highlights the importance of precision agricultural monitoring, yield forecasting, and early warning systems. The data assimilation method provides a new possibility to solve the problems of low accuracy of yield prediction, strong dependence on the field, and poor adaptability of the model in traditional agricultural applications. Therefore, this study makes a systematic literature retrieval based on Web of Science, Scopus, Google Scholar, and PubMed databases, introduces in detail the assimilation strategies based on many new remote sensing data sources, such as satellite constellation, UAV, ground observation stations, and mobile platforms, and compares and analyzes the progress of assimilation models such as compulsion method, model parameter method, state update method, and Bayesian paradigm method. The results show that: (1) the new remote sensing platform data assimilation shows significant advantages in precision agriculture, especially in emerging satellite constellation remote sensing and UAV data assimilation. (2) SWAP model is the most widely used in simulating crop growth, while Aquacrop, WOFOST, and APSIM models have great potential for application. (3) Sequential assimilation strategy is the most widely used algorithm in the field of agricultural data assimilation, especially the ensemble Kalman filter algorithm, and hierarchical Bayesian assimilation strategy is considered to be a promising method. (4) Leaf area index (LAI) is considered to be the most preferred assimilation variable, and the study of soil moisture (SM) and vegetation index (VIs) has also been strengthened. In addition, the quality, resolution, and applicability of assimilation data sources are the key bottlenecks that affect the application of data assimilation in the development of precision agriculture. In the future, the development of data assimilation models tends to be more refined, diversified, and integrated. To sum up, this study can provide a comprehensive reference for agricultural monitoring, yield prediction, and crop early warning by using the data assimilation model.

MOTrainer: Distributed Measurement Operator Trainer for Data Assimilation Applications

MOTrainer: Distributed Measurement Operator Trainer for Data Assimilation Applications

Statistical Dynamics and Subgrid Modelling of Turbulence: From Isotropic to Inhomogeneous

Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with the formulation of the Eulerian direct interaction approximation (DIA) closure by Kraichnan. It was based on renormalized perturbation theory, like quantum electrodynamics, and is a bare vertex theory that is manifestly realizable. Here, we review some of the subsequent exciting achievements in closure theory and subgrid modelling. We also document in some detail the progress that has been made in extending statistical dynamical turbulence theory to the real world of interactions with mean flows, waves and inhomogeneities such as topography. This includes numerically efficient inhomogeneous closures, like the realizable quasi-diagonal direct interaction approximation (QDIA), and even more efficient Markovian Inhomogeneous Closures (MICs). Recent developments include the formulation and testing of an eddy-damped Markovian anisotropic closure (EDMAC) that is realizable in interactions with transient waves but is as efficient as the eddy-damped quasi-normal Markovian (EDQNM). As a similarly efficient closure, the realizable eddy-damped Markovian inhomogeneous closure (EDMIC) has been developed. Moreover, we present subgrid models that cater for the complex interactions that occur in geophysical flows. Recent progress includes the determination of complete sets of subgrid terms for skilful large-eddy simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows interacting with Rossby waves and topography. The success of these inhomogeneous closures has also led to further applications in data assimilation and ensemble prediction and generalization to quantum fields.

Improved Representation of Vegetation Soil Moisture Coupling Enhances Soil Moisture Data Assimilation in Water‐Limited Regimes: A Case Study Over Texas

AbstractAssimilating remotely sensed surface soil moisture (SSM) into land surface models (LSM) is widely used to improve model representations of soil moisture (SM). However, the efficacy of SSM data assimilation (DA) has been found to be limited, particularly in resolving root‐zone soil moisture (RZSM). This study investigates how the representation of vegetation phenology, modulates the efficacy of SSM DA in enhancing the realism of RZSM simulations. To this end, two sets of climatological leaf area index (LAI) are implemented in Noah‐MP LSM over the state of Texas: (a) Noah‐MP default based on long‐term MODIS observations, (b) an alternative LAI adapted from AVHRR products. The former are found to exhibit conspicuous phase errors whereas the latter are more consistent with observed seasonal cycle. Two sets of DA experiments were performed accordingly, wherein SMAP L3 SSM is assimilated into Noah‐MP equipped with each LAI product from 2015 to 2019, Validation of the resulting products against in‐situ data reveals that (a) using the AVHRR‐based LAI, the Noah‐MP outperforms the baseline in reproducing the dynamics of RZSM, and the outperformance is particularly evident over the warm season and water‐stressed western Texas; (b) using the alternative LAI enhances the ability of DA to improve the accuracy of Noah‐MP RZSM, and to a lesser extent, SSM; and (c) gains in SM attained through improvement of LAI and application of DA is most pronounced over regions featuring tight vertical SM coupling. Additional model mechanistic limitations that need to be overcome to improve efficacy of DA are discussed.

Application of Continuous Data Assimilation in High-Resolution Ocean Modeling

Application of Continuous Data Assimilation in High-Resolution Ocean Modeling

A variational data assimilation approach for sparse velocity reference data in coarse RANS simulations through a corrective forcing term

The Reynolds-averaged Navier–Stokes (RANS) equations provide a computationally efficient method for solving fluid flow problems in engineering applications. However, the use of closure models to represent turbulence effects can reduce their accuracy. To address this issue, recent research has explored data-driven techniques such as data assimilation and machine learning.An efficient variational data assimilation (DA) approach is presented to enhance steady-state eddy viscosity based RANS simulations. To account for model deficiencies, a corrective force term is introduced in the momentum equation. In the case of only velocity reference data, this term can be represented by a potential field and is divergence-free. The DA implementation relies on the discrete adjoint method and approximations for efficient gradient evaluation.The implementation is based on a two-dimensional coupled RANS solver in OpenFOAM, which is extended to allow the computation of the adjoint velocity and pressure as well as the adjoint gradient. A gradient-based optimizer is used to minimize the difference between the simulation results and the reference data. To evaluate this approach, it is compared with alternative data assimilation methods for canonical stationary two-dimensional turbulent flow problems. For the data assimilation, sparsely distributed reference data from averaged high-fidelity simulation results are used.The results suggest that the proposed method achieves the optimization goal more efficiently compared to applying data assimilation for obtaining the eddy viscosity, or a field modifying the eddy viscosity, directly. The method works well for different reference data configurations and runs efficiently by leveraging coarse meshes.

Quest to find compromised spatial and temporal resolutions for integrating remote sensing data with an agro-ecosystem model for grasslands

This paper addressed one of the main challenges in assimilating remote sensing derived variables into process-based crop model simulations, which is the inconsistent spatial and temporal resolution between information obtained from remote sensing and the outputs of process-based agroecosystem model. We proposed an applied method to reduce the number of required simulations by identifying (i) optimal points in time where additional information from remote sensing has the largest positive influence on the model performance and (ii) options to cluster 10 m grid cells to larger cells without compromising their information content. The MONICA (Model for Nitrogen and Carbon) model was applied to simulate above and below ground biomass in two grassland sites located in southern and eastern parts of Germany. The model was calibrated using LAI values obtained from Sentinel-2 and the sensitivity of output variables to two key root parameters (Root Form factor and specific root length) was evaluated. Our results showed that one or two satellite images covering the critical time periods right after cutting events significantly improved the predictions of grass yields produced by a mechanistic agroecosystem models (by up to 30 %). A larger number of images at other grass growth stages would not further improve the predictive power of the model. We also found that the sensitivity to these critical time periods was independent of model parameters. The mixed-resolution scheme (between 10 and 50 m) achieved better results compared with the high-resolution standalone state-updating method, yet it reduced computational costs by more than 50 %. In conclusion, we proposed a methodology to reduce the number of required simulations for data assimilation by aggregating data from fine to coarse resolutions. Our method was promising for applying data assimilation over large areas and benefiting more from satellite information for real-time prediction of agricultural productivity.

Primary Impact Evaluation of Surface Temperature Observations for Microwave Temperature Sounding Data Assimilation over Land

Observations from the Advanced Microwave Sounding Unit-A (AMSU-A) onboard polar-orbiting satellites are considered to be the most effective satellite data in terms of obviously reducing operational prediction errors. However, there are still significant difficulties in the application of AMSU-A low-level channel data assimilation over land. One of them is the inaccurate surface skin temperature (SKT) of the background on land areas, which leads to significant uncertainty in the accuracy of simulating brightness temperature (BT) in these channels. Therefore, improving the accuracy of SKT in the background field is a direct way to improve the assimilation effect of these low-level channel data over land. In this study, both high-spatio-temporal-resolution automatic weather station (AWS) observation data from China in September 2021 and the AMSU-A observation data from NOAA-15/18/19 and MetOp-A were used. Based on the Advanced Research version of the Weather Research and Forecast model (WRF-ARW) and Gridpoint Statistical Interpolation (GSI) assimilation system, we first analyzed the differences in SKT between AWS observations and model simulations and then attempted to directly replace the simulated SKT with the observation data. On this basis, the differences in BT simulation effects over the land area of Southwest China before and after replacement were meticulously analyzed and compared. In addition, the impacts of SKT replacement in areas with different terrain elevations and in cloudy areas were also evaluated. The results indicate that the SKTs of background fields were generally lower than the surface observations, whereas the diurnal variation in SKT was not well simulated. After replacing the SKT of the background field with station observations, the BT differences between the observation and background (O–B, observation minus background) were remarkably reduced, especially for channels 3–5 and 15 of the AMSU-A. The volume of data passing the GSI quality control significantly increased, and the standard deviation of O–B decreased. Further analysis showed that the improvement effect was better in areas at an elevation above 1600 m. Moreover, introducing SKT observations leads to a significant and stable improvement over BT simulations in cloudy areas over land.

Evaluation of data assimilation strategies on improving the performance of crop modeling based on a novel evapotranspiration assimilation framework

Recently, data assimilation (DA) has garnered significant attention. Integration of DA approaches and crop models could diminish model uncertainties and improve the precision of model simulations. While previous research extensively focused on assimilating leaf area index (LAI) or soil moisture (SM), the feasibility and effectiveness of assimilating evapotranspiration (ET) have been rarely explored. In this study, we proposed a novel framework of ET assimilation. Then, together with commonly assimilated LAI and SM, we evaluated the performance of this new method in simulating the key indicators (i.e., LAI and ET in daily and interannual scales, and the crop yield) based on the long-term eddy covariance observations and well-calibrated crop model. DA strategies we utilized to evaluate consist of two approaches (i.e., Ensemble Kalman filter (EnKF) and EnKF with simultaneous state-parameter estimation (EnKF-SSPE)) and combinations of three assimilated observations (i.e., LAI, SM, and ET).Our results demonstrate that joint assimilation of LAI and ET with EnKF-SSPE performs best for wheat while joint assimilation of SM and ET with EnKF-SSPE is the best for maize. For a single observation, LAI and ET play a dominant role in DA for wheat and maize, respectively. This is because the interannual variability of wheat growth is primarily influenced by agricultural management (e.g., cultivar change) and can be represented by LAI. For maize which is mostly rainfed, water stress usually occurs. Therefore, ET, with its ability to reflect the water stress status, proves to be effective. EnKF-SSPE outperforms EnKF, exhibiting potential in revealing the parameter evolution during long-term crop modeling, especially when crop cultivars are regularly renewed. This study evaluates different observations and methods through DA based on a newly proposed sequential ET assimilation framework, which might be illuminating for future applications of DA.

Foundations for Universal Non‐Gaussian Data Assimilation

AbstractIn many applications of data assimilation, especially when the size of the problem is large, a substantial assumption is made: all variables are well‐described by Gaussian error statistics. This assumption has the advantage of making calculations considerably simpler, but it is often not valid, leading to biases in forecasts or, even worse, unphysical predictions. We propose a simple, but effective, way of replacing this assumption, by making use of transforming functions, while remaining consistent with Bayes' theorem. This method allows the errors to have any value of the skewness and kurtosis, and permits physical bounds for the variables. As such, the error distribution can conform better to the underlying statistics, reducing biases introduced by the Gaussian assumption. We apply this framework to a 3D variational data assimilation method, and find improved performance in a simple atmospheric toy model (Lorenz‐63), compared to an all‐Gaussian technique.

Performance assessment of an iterative ensemble smoother with local analysis to assimilate big 4D seismic datasets applied to a complex pre-salt-like benchmark case

Abstract The use of 4D seismic (4DS) (or time-lapse seismic, TLS) in data assimilation (DA) makes the process more complex due to the higher amount of data to be assimilated, requiring more robust methods and better computational resources (processing capacity and memory). The development and application of permanent seismic monitoring technologies have increased in recent years, improving the overall 4D seismic quality in terms of signal resolution and repeatability. However, a massive amount of data is generated from the multiple monitors, making the incorporation of 4DS data in the DA process more complex. Therefore, robust DA methods capable of dealing with huge amount of data effectively and efficiently are essential. This paper aims to assess the performance of an iterative ensemble smoother method, named subspace ensemble randomized maximum likelihood with a local analysis, to assimilate a big dataset. The method was applied in a challenging pre-salt-like benchmark case with eight seismic surveys, one base, and seven monitors. The 4DS data are the impedance ratios (between two consecutive monitors) in 15 seismic horizons, totaling 105 maps to be assimilated. To our best knowledge, this is state of the art in terms of practical applications in DA. It was possible to assimilate all the data simultaneously: the 105 horizons for the 4DS data and the wells’ production and pressure data. The DA was successful in terms of results quality and method performance. We also ran a case assimilating only well data for comparison purposes.

Application of GOES-16 Atmospheric Temperature-Profile Data Assimilation in a Hurricane Forecast

This paper selects the case of the Atlantic hurricane “Michael” in 2018 to evaluate the accuracy of the GOES-16 atmospheric temperature profile during the hurricane and its effect on forecasting. Based on the weather research and forecasting (WRF) model, the assimilation of GOES-16 atmospheric temperature-profile products was achieved by using three-dimensional variational (3DVar) and the ensemble transform Kalman filter/three-dimensional variational (ETKF/3DVAR) hybrid system (Hybrid) systems. And the impact of geostationary satellite GOES-16 atmospheric temperature-profile data assimilation on a hurricane forecast is evaluated. The results show that, during the hurricane, the root mean square errors of the GOES-16 atmospheric temperature profile are all within 2 k at the height of 200–1000 hPa, and the quality of the data is generally good. Assimilating the GOES-16 atmospheric temperature-profile data can indeed effectively improve the analysis increment and improve the prediction results. The assimilation increment obtained by the hybrid system has obvious “flow-dependent” characteristics, which can reasonably improve the initial field of the model. Its temperature increment has an obvious spiral structure, which is in line with the characteristics of the hurricane, and the adjustment of the wind field and geopotential height field is also more beneficial to the development of the hurricane. It has a positive impact on the forecast of track, intensity, and precipitation, and the hybrid system is improved more obviously. In addition, from the RMSE of the analysis field and the forecast field relative to the observation data of different elements, the hybrid system is superior to the 3DVar system.

Constraining Flood Forecasting Uncertainties through Streamflow Data Assimilation in the Tropical Andes of Peru: Case of the Vilcanota River Basin

Flood modeling and forecasting are crucial for managing and preparing for extreme flood events, such as those in the Tropical Andes. In this context, assimilating streamflow data is essential. Data Assimilation (DA) seeks to combine errors between forecasting models and discharge measurements through the updating of model states. This study aims to assess the applicability and performance of streamflow DA in a sub-daily forecasting system of the Peruvian Tropical Andes using the Ensemble Kalman Filter (EnKF) and Particle Filter (PF) algorithms. The study was conducted in a data-sparse Andean basin during the period February–March 2022. For this purpose, the lumped GR4H rainfall–runoff model was run forward with 100 ensemble members in four different DA experiments based on IMERG-E and GSMaP-NRT precipitation sources and assimilated real-time hourly discharges at the basin outlet. Ensemble modeling with EnKF and PF displayed that perturbation introduced by GSMaP-NRT’-driven experiments reduced the model uncertainties more than IMERG-E’ ones, and the reduction in high-flow subestimation was more notable for the GSMaP-NRT’+EnKF configuration. The ensemble forecasting framework from 1 to 24 h proposed here showed that the updating of model states using DA techniques improved the accuracy of streamflow prediction at least during the first 6–8 h on average, especially for the GSMaP-NRT’+EnKF scheme. Finally, this study benchmarks the application of streamflow DA in data-sparse basins in the Tropical Andes and will support the development of more accurate climate services in Peru.

The impact of global navigation satellite system (GNSS) zenith total delay data assimilation on the short-term precipitable water vapor and precipitation forecast over Italy using the Weather Research and Forecasting (WRF) model

Abstract. The impact of assimilating GNSS-ZTD (global navigation satellite system–zenith total delay) on the precipitable water vapor and precipitation forecast over Italy is studied for the month of October 2019, which was characterized by several moderate to intense precipitation events, especially over northwestern Italy. The WRF (Weather Research and Forecasting) model, version 4.1.3, is used with its 3D-Var data assimilation system to assimilate ZTD observations from 388 GNSS receivers distributed over the country. The dataset was built collecting data from all the major national and regional GNSS permanent networks, achieving dense coverage over the whole area. The water vapor forecast is verified for the forecast hours of 1–6 h after the last data assimilation time. Results show that WRF underestimates the atmospheric water vapor content for the period, and GNSS-ZTD data assimilation improves this underestimation. The precipitation forecast is verified in the phases of 0–3 and 3–6 h after the last data assimilation time using more than 3000 rain gauges spread over Italy. The application of GNSS-ZTD data assimilation to a case study improved the precipitation forecast by increasing the rainfall maximum and by better focusing the precipitation pattern over northeastern Italy, with the main drawback being the prediction of false alarms. Considering the study over the whole period, GNSS-ZTD data assimilation had a positive impact on rainfall forecast, with an improvement in the performance up to 6 h and with statistically significant results for moderate to intense rainfall thresholds (25–30 mm (3 h)−1).

Numerical calculation of the turbulent flow past a surface mounted cube with assimilation of PIV data

The application of Data Assimilation (DA) methods in Computational Fluid Dynamics (CFD) problems is a concept actively being explored to couple CFD with Experimental Fluid Dynamics data. Here, Particle Image Velocimetry (PIV) data are assimilated in an OpenFOAM based CFD solver to calculate the velocity and pressure fields of the turbulent flow past a surface mounted cube inside an atmospheric boundary layer for three planes belonging to the symmetry plane of the flow. At first, the SIMPLE algorithm is used to correct both pressure and velocity fields, with the PIV data used to formulate the initial and boundary conditions. The Reynolds stresses are calculated directly from the PIV data instead of using a turbulence model. Next, we use two implementations of the nudging method and two formulations of the Kalman Filter in order to assimilate the PIV data into the iterative SIMPLE procedure. A grid independence study is performed, and the performance of the different methods is assessed. The CFD predicted pressure field is in good agreement with pressure measurements on the cube surface. The results also show that the SIMPLE based correction step already leads to a significant reduction of both the mean and the variance of the continuity errors as well as the difference between the original PIV data and the resulting velocity fields. The application of the DA methods, particularly the KF, leads to minor further improvement of the results but does improve convergence of the CFD solver.

DASH: a MATLAB toolbox for paleoclimate data assimilation

Abstract. Paleoclimate data assimilation (DA) is a tool for reconstructing past climates that directly integrates proxy records with climate model output. Despite the potential for DA to expand the scope of quantitative paleoclimatology, these methods remain difficult to implement in practice due to the multi-faceted requirements and data handling necessary for DA reconstructions, the diversity of DA methods, and the need for computationally efficient algorithms. Here, we present DASH, a MATLAB toolbox designed to facilitate paleoclimate DA analyses. DASH provides command line and scripting tools that implement common tasks in DA workflows. The toolbox is highly modular and is not built around any specific analysis, and thus DASH supports paleoclimate DA for a wide variety of time periods, spatial regions, proxy networks, and algorithms. DASH includes tools for integrating and cataloguing data stored in disparate formats, building state vector ensembles, and running proxy (system) forward models. The toolbox also provides optimized algorithms for implementing ensemble Kalman filters, particle filters, and optimal sensor analyses with variable and modular parameters. This paper reviews the key components of the DASH toolbox and presents examples illustrating DASH's use for paleoclimate DA applications.

High-order moment closure models with random batch method for efficient computation of multiscale turbulent systems.

We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatiotemporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimensions by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz '96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches. The models provide effective tools for a wide range of real-world applications in prediction, uncertainty quantification, and data assimilation.

Benefits and Cautions in Data Assimilation Strategies: An Example of Modeling Groundwater Recharge.

Assimilating recent observations improves model outcomes for real-time assessments of groundwater processes. This is demonstrated in estimating time-varying recharge to a shallow fractured-rock aquifer in response to precipitation. Results from estimating the time-varying water-table altitude (h) and recharge, and their error covariances, are compared for forecasting, filtering, and fixed-lag smoothing (FLS), which are implemented using the Kalman Filter as applied to a data-driven, mechanistic model of recharge. Forecasting uses past observations to predict future states and is the current paradigm in most groundwater modeling investigations; filtering assimilates observations up to the current time to estimate current states; and FLS estimates states following a time lag over which additional observations are collected. Results for forecasting yield a large error covariance relative to the magnitude of the expected recharge. With assimilating recent observations of h, filtering and FLS produce estimates of recharge that better represent time-varying observations of h and reduce uncertainty in comparison to forecasting. Although model outcomes from applying data assimilation through filtering or FLS reduce model uncertainty, they are not necessarily mass conservative, whereas forecasting outcomes are mass conservative. Mass conservative outcomes from forecasting are not necessarily more accurate, because process errors are inherent in any model. Improvements in estimating real-time groundwater conditions that better represent observations need to be weighed for the model application against outcomes with inherent process deficiencies. Results from data assimilation strategies discussed in this investigation are anticipated to be relevant to other groundwater processes models where system states are sensitive to system inputs.

Data assimilation application in prediction of flowrate for a sustainable groundwater resource: Falaj Al-Khatmain, Oman

Data assimilation application in prediction of flowrate for a sustainable groundwater resource: Falaj Al-Khatmain, Oman

How Sampling Errors in Covariance Estimates Cause Bias in the Kalman Gain and Impact Ensemble Data Assimilation

Abstract Localization is the key component to the successful application of ensemble data assimilation (DA) to high-dimensional problems in the geosciences. We study the impact of sampling error and its amelioration through localization using both analytical development and numerical experiments. Specifically, we show how sampling error in covariance estimates accumulates and spreads throughout the entire domain during the computation of the Kalman gain. This results in a bias, which is the dominant issue in unlocalized ensemble DA, and, surprisingly, we find that it depends directly on the number of independent observations but only indirectly on the state dimension. Our derivations and experiments further make it clear that an important aspect of localization is a significant reduction of bias in the Kalman gain, which in turn leads to an increased accuracy of ensemble DA. We illustrate our findings on a variety of simplified linear and nonlinear test problems, including a cycling ensemble Kalman filter applied to the Lorenz-96 model. Significance Statement The dampening of long-range correlations has been the key to the success of ensemble data assimilation in global numerical weather prediction. In this paper, we show how noise in covariance estimates propagates through the state estimation process and corrupts state estimates. We show that this noise results in a bias and that this bias depends on the number of observations and not, as might be expected, on the state dimension. We go on to show how dampening long-range covariances through a process referred to as “localization” helps to mitigate the detrimental effects of this sampling noise.