Spatial Error Correlation Research Articles

A Novel Localized Fast Multipole Method for Computations With Spatially Correlated Observation Error Statistics in Data Assimilation

AbstractSeveral observation types (e.g., geostationary satellite and Doppler radar observations) have recently been found to exhibit strong spatial error correlations. Including these error statistics in data assimilation for numerical weather prediction can improve analysis quality and forecast skill. Moreover, it allows for increases in the spatial density of observations assimilated, which is needed for the provision of information on appropriate scales for high‐resolution forecasting. However, introducing correlated error statistics may increase the computational complexity and parallel communication costs of matrix‐vector products involving observation precision matrices (inverse observation error covariance matrices). Without new approaches, we cannot take full advantage of new observation uncertainty estimates. We develop a new numerical approximation method based on a particular type of fast multipole method and a domain localization approach. The basic idea is to divide the observation domain into boxes and then separate calculations of matrix‐vector products according to the partition. These calculations can be done in parallel with very low communication overheads. The new method is easy to implement and parallelize, and it is applicable to a wide variety of observation precision matrices. We applied the new method to a simple variational data assimilation problem and found that the computational cost of the variational minimization was dramatically reduced while preserving analysis accuracy across a range of scales. The new method has the potential to be used as an efficient technique for practical applications where a large number of observations with mutual error correlations need to be assimilated quickly.

SPATIAL MODELING OF POVERTY IN BENGKULU PROVINCE WITH MIXED GEOGRAPHICALLY WEIGHTED REGRESSION

The percentage of poor people in Bengkulu Province is high from year to year. The poverty rate in Bengkulu Province also tends to fluctuate. If there is a decrease in the poverty rate, the decrease is relatively small. Poverty in the regions of Bengkulu Province also varies from district to district, subdistrict, and village to village, because poverty data is spatial data that varies regionally. The diversity of poverty data in Bengkulu Province is influenced by spatial effects, namely spatial dependency and spatial heterogeneity. Spatial dependency occurs due to spatial error correlation in cross section data, while spatial heterogeneity occurs due to random area effects, which is the difference between one region and another. Therefore, classical methods are not qualified enough to analyze the resulting diversity. This research will model the poverty of each district/city in Bengkulu Province using Mixed Geographically Weighted Regression (MGWR), because this method is quite complex in modeling data that contains spatial heterogeneity and variations in geospatial data. This modeling aims to identify and analyze poverty indicators in Bengkulu Province spatially, namely based on poverty data in each district/city in Bengkulu Province. The results showed that by using the MGWR method, the variables that locally influence the percentage of extreme poor people in each district/city in Bengkulu Province are Female Household Head Gender and not having a waterheater . Meanwhile, the variable that has a global effect on the percentage of the extreme poor in each district/city in Bengkulu Province is not having a flat screen television ().

Estimating the uncertainty of sea-ice area and sea-ice extent from satellite retrievals

Abstract. The net Arctic sea-ice area (SIA) can be estimated from the sea-ice concentration (SIC) by passive microwave measurements from satellites. To be a truly useful metric, for example of the sensitivity of the Arctic sea-ice cover to global warming, we need, however, reliable estimates of its uncertainty. Here we retrieve this uncertainty by taking into account the spatial and temporal error correlations of the underlying local sea-ice concentration products. As 1 example year, we find that in 2015 the average observational uncertainties of the SIA are 306 000 km2 for daily estimates, 275 000 km2 for weekly estimates, and 164 000 km2 for monthly estimates. The sea-ice extent (SIE) uncertainty for that year is slightly smaller, with 296 000 km2 for daily estimates, 261 000 km2 for weekly estimates, and 156 000 km2 for monthly estimates. These daily uncertainties correspond to about 7 % of the 2015 sea-ice minimum and are about half of the spread in estimated SIA and SIE from different passive microwave SIC products. This shows that random SIC errors play a role in SIA uncertainties comparable to inter-SIC-product biases. We further show that the September SIA, which is traditionally the month with the least amount of Arctic sea ice, declined by 105 000±9000 km2 a−1 for the period from 2002 to 2017. This is the first estimate of a SIA trend with an explicit representation of temporal error correlations.

Impact of Assimilation of the Tropical Cyclone Strong Winds Observed by Synthetic Aperture Radar on Analyses and Forecasts

Abstract A few high-wind observations have been obtained from satellites over the ocean around tropical cyclones (TCs), but the impact of data assimilation of such observations over the sea on forecasting has not been clear. The spaceborne synthetic aperture radar (SAR) provides high-resolution and wide-area ocean surface wind speed data around the center of a TC. In this study, the impact of data assimilation of the ocean surface wind speed of SAR (OWSAR) on regional model forecasts was investigated. The assimilated data were estimated from SAR on board Sentinel-1 and RADARSAT-2. The bias of OWSAR depends on wind speed, the observation error variance depends on wind speed and incidence angle, and the spatial observation error correlation depends on the incidence angle. The observed OWSAR is screened using the variational quality control method with the Huber norm. In the case of Typhoon Hagibis (2019), OWSAR assimilation modified the TC low-level inflow, which also modified the TC upper-level outflow. The propagation of this OWSAR assimilation effect from the surface to the upper troposphere was given by a four-dimensional variational method that searches for the optimal solution within strong constraints on the time evolution of the forecast model. Statistical validation confirmed that errors in the TC intensity forecast decreased over lead times of 15 h, but this was not statistically significant. The validation using wind profiler observations showed that OWSAR assimilation significantly improved the accuracy of wind speed predictions from the middle to the upper level of the troposphere. Significance Statement The purpose of this study was to demonstrate the impact of the assimilation of ocean surface wind speed by synthetic aperture radar (SAR) on regional model predictions. In the case of tropical cyclones, ocean surface wind speed assimilation modified inflows in the lower layer and outflows in the upper layer. The results indicate that the SAR assimilation improves the accuracy of wind speed forecasts in the middle to upper troposphere.

Including the Horizontal Observation Error Correlation in the Ensemble Kalman Filter: Idealized Experiments with NICAM-LETKF

Abstract Densely observed remote sensing data such as radars and satellites generally contain significant spatial error correlations. In data assimilation, the observation error covariance matrix is usually assumed to be diagonal, and the dense data are thinned or spatially averaged to compensate for neglecting the spatial observation error correlation. However, in theory, including the spatial observation error correlation in data assimilation can make better use of the dense data. This study performs perfect model observing system simulation experiments (OSSEs) using the nonhydrostatic icosahedral atmospheric model (NICAM) and the local ensemble transform Kalman filter (LETKF) to assess the impact of assimilating horizontally dense and error-correlated observations. The condition number of the observation error covariance matrix, defined as the ratio of the largest to smallest eigenvalues, is important for the numerical stability of the LETKF computation. A large condition number makes it difficult to compute the ensemble transform matrix correctly. Reducing the condition number by reconditioning is found effective for stable computation. The results show that including the horizontal observation error correlation with reconditioning makes the analysis more accurate but requires 6 times more computations than the case with the diagonal observation error covariance matrix. Significance Statement It is important to effectively utilize observations in data assimilation for more accurate weather prediction. Spatially dense observations are known to have an error correlation that is ignored in the data assimilation. This study explores assimilating dense observations by explicitly including observation error correlations with an idealized experiment. The results shows that the analysis is improved by including the observation error correlations. Also, the condition number of the observation error covariance matrix is essential for stable computations.

A criterion and incremental design construction for simultaneous kriging predictions

In this paper, we further investigate the problem of selecting a set of design points for universal kriging, which is a widely used technique for spatial data analysis. Our goal is to select the design points in order to make simultaneous predictions of the random variable of interest at a finite number of unsampled locations with maximum precision. Specifically, we consider as response a correlated random field given by a linear model with an unknown parameter vector and a spatial error correlation structure. We propose a new design criterion that aims at simultaneously minimizing the variation of the prediction errors at various points. We also present various efficient techniques for incrementally building designs for that criterion scaling well for high dimensions. Thus the method is particularly suitable for big data applications in areas of spatial data analysis such as mining, hydrogeology, natural resource monitoring, and environmental sciences or equivalently for any computer simulation experiments. We have demonstrated the effectiveness of the proposed designs through two illustrative examples: one by simulation and another based on real data from Upper Austria.

Improving UAV‐SfM photogrammetry for modelling high‐relief terrain: Image collection strategies and ground control quantity

AbstractImage collection strategies and ground control points (GCPs) are of particular importance for uncrewed aerial vehicle combined with Structure‐from‐Motion (UAV–SfM) photogrammetry, and the generalization of their effects has proved elusive. This study designed various photogrammetric scenarios to investigate the effects of image collection strategies, ground control quantity, and their interaction on digital elevation model (DEM) errors and their spatial structure in high‐relief terrain. The results of 1.77 × 105 UAV–SfM scenarios provide insights for improving UAV–SfM practices. A high image capture angle (20–40°) enhances camera calibration quality decreasing the magnitude and spatial correlation of errors. High camera inclination reduces the sensitivity of mean and standard deviation of error to flying height but not the spatial correlation of error. Including additional data (e.g. supplemented convergent images; images captured at multiple flying heights) has only a minor effect if imagery is highly inclined. GCPs provide more effective constraints than image collection strategies. The mean error and standard error decline quickly with a small number of GCPs and then become stable in all scenarios, but the spatial correlation of error can be further improved with increasing GCPs. However, the effects of GCP quantity do interact with image collection strategies. High camera inclination reduces requirements for GCPs, whilst strategies combining different flying heights and image orientations have little effect on necessary GCP quantity. The distribution of GCPs still affects the errors, but the effect of GCP distribution becomes less important with an increase in the number of GCPs. Finally, we show that UAV–SfM photogrammetric quality assessment should routinely assess the spatial dependence of error using a statistic like Moran's I.

An Efficient BDS-3 Long-Range Undifferenced Network RTK Positioning Algorithm

In 2020, the BeiDou-3 global navigation satellite system (BDS-3) was officially completed and put into service. Currently, network real-time kinematic (RTK) technology is considered the main means through which to improve the positioning accuracy of the BeiDou navigation satellite system (BDS). This paper proposes a long-range undifferenced network RTK (URTK) algorithm, based on multi-frequency observation data of the BDS. First, the multi-frequency phase integer ambiguity resolution (AR) model considering atmospheric error parameters is designed, and the multi-frequency phase integer ambiguity of the long-range BDS reference station is determined. Then, the undifferenced integer ambiguity of each reference station is obtained, using linear variation based on the accurately determined phase integer ambiguity between reference stations, and the undifferenced observation error of each reference station is calculated. Considering the weakening spatial correlation of the observation errors between long-range stations, undifferenced classification error corrections of a reference station network are separated, according to different error characteristics. Finally, the inverse distance weighting method is employed to calculate the classification undifferenced error correction of the rover station. The rover station corrects the observation error through applying the undifferenced error correction to achieve high-precision positioning. The measured data of a long-range continuous operation reference station (CORS) network are selected for an experiment. The results show that the proposed algorithm can quickly and accurately realize the resolution of the BDS integer ambiguity of a reference station network and establish an undifferenced area error correction model in order to achieve accurate classification of undifferenced error correction values for a rover station. In China, the BDS-3 is superior to the global positioning system (GPS) in terms of the satellite number, position dilution of precision (PDOP) value, AR success rate, stability, and convergence time. The results show that the AR success rate, stability, and convergence time increase with the operational frequency, and the BDS-3 can achieve centimeter-level positioning of single-system rover stations without relying on the GPS.

The changing food environment and neighborhood prevalence of type 2 diabetes

In this ecological study, we used longitudinal data to assess if changes in neighborhood food environments were associated with type 2 diabetes mellitus (T2DM) prevalence, controlling for a host of neighborhood characteristics and spatial error correlation. We found that the population-adjusted prevalence of fast-food and pizza restaurants, grocery stores, and full-service restaurants along with changes in their numbers from 1990 to 2010 were associated with 2015 T2DM prevalence. The results suggested that neighborhoods where fast-food restaurants have increased and neighborhoods where full-service restaurants have decreased over time may be especially important targets for educational campaigns or other public health-related T2DM interventions.

Trans-dimensional gravity and magnetic joint inversion for 3-D earth models

SUMMARY Studying 3-D subsurface structure based on spatial data is an important application for geophysical inversions. However, major limitations exist for conventional regularized inversion when applied to potential-field data. For example, global regularization parameters can mask model features that may be important for interpretation. In addition, 3-D inversions are typically based on data acquired in 2-D at the Earth’s surface. Such data may contain significant spatial error correlations in 2-D due to the choice of spatial sampling, acquisition geometry, ambient noise and model assumptions. These correlations can cause trade-offs with spatial resolution and should be accounted for. However, correlations are often ignored, particularly 2-D correlations in spatial data, such as potential field data recorded on the Earth’s surface. Non-linear Bayesian methods can address these shortcomings and we present a new hierarchical model for 2-D correlated errors. Nonetheless, limitations also exist. For example, non-linear Bayesian estimation requires numerical integration with a considerable computational burden to collect a posterior ensemble of models. For 3-D applications, this cost can be prohibitive. This paper presents a non-linear Bayesian inversion with trans-dimensional (trans-D) partitioning of space by a hierarchy of Voronoi nodes and planes (VP), and trans-D estimation of the data noise covariance matrix. The addition of planes permits the introduction of prior information which reduces non-uniqueness. The covariance matrix estimation uses a trans-D autoregressive (AR) noise model to quantify correlated noise on 2-D potential-field data. We address computational cost by wavelet compression in the forward problem and by basing susceptibility on an empirical relationship with density contrast. The method is applied to simulated data and field data from off-shore Abu Dhabi. With simulated data, we demonstrate that subsurface structures are well-resolved with the trans-D model that applies hierarchical VP partitioning. In addition, the model locally adapts based on data information without requiring regularization. The method is also successful in reducing 2-D error correlation via trans-D AR models in 2-D. From field data, the inversion efficiently resolves basement topography and two distinct salt diapirs with a parsimonious and data-driven parametrization. Results show a considerable reduction in 2-D spatial correlations of field data using the proposed trans-D AR model.

Multi-objective risk analysis for flood control operation of a complex reservoir system under multiple time-space correlated uncertainties

Many uncertainties are involved in flood control operation of a multi-reservoir system and result in risks for the system. Risk analysis for reservoir flood control operation is essential for decision making. Traditional flood control risk analysis only takes flood forecast error as the main risk source, ignoring the influence of dynamic control of flood-limited water level on reservoir initial water level for flood regulation. In this study, the spatial correlation of reservoir initial water level errors and the spatiotemporal correlation of flood forecast errors are identified via copula function. A risk analysis model considering both upstream and downstream is established for multi-objective flood control operation of a complex reservoir system, coupling multi-dimensional uncertainties of reservoir initial water levels, flood forecast errors, reservoir capacity curve errors, reservoir discharge curve errors and river flood routing errors. The impact of multiple risk sources with spatiotemporal correlations on reservoir flood control operation is then evaluated and the competitive tradeoff between flood control risks in upstream and downstream analyzed. The model is applied to a mixed four-reservoir system in Pi River Basin in China. Results indicated that 1) there is a strong correlation between the reservoir initial water level errors in space and also between the flood forecast errors in space and time, which can be effectively described by copula function, and ignoring the correlations will underestimate the flood control risk; 2) the two objective values of upstream and downstream show a clear competitive relationship, and the solution that prefers one objective has low risk marginal benefit, thus choosing a compromise solution can balance risks; 3) coupling the initial water level uncertainty reduces the flood control risk caused by the flood forecast errors to a certain extent, but also lead to the increase of extreme risk loss.

Uncertainty Analysis of Digital Elevation Models by Spatial Inference From Stable Terrain

The monitoring of Earth’s and planetary surface elevations at larger and finer scales is rapidly progressing through the increasing availability and resolution of digital elevation models (DEMs). Surface elevation observations are being used across an expanding range of fields to study topographical attributes and their changes over time, notably in glaciology, hydrology, volcanology, seismology, forestry and geomorphology. However, DEMs frequently contain large-scale instrument noise and varying vertical precision that lead to complex patterns of errors. Here, we present a validated statistical workflow to estimate, model, and propagate uncertainties in DEMs. We review the state-of-the-art of DEM accuracy and precision analyses, and define a conceptual framework to consistently address those. We show how to characterize DEM precision by quantifying the heteroscedasticity of elevation measurements, i.e. varying vertical precision with terrain- or sensor-dependent variables, and the spatial correlation of errors that can occur across multiple spatial scales. With the increasing availability of high-precision observations, our workflow based on independent elevation data acquired on stable terrain can be applied almost anywhere on Earth. We illustrate how to propagate uncertainties for both pixel-scale and spatial elevation derivatives, using terrain slope and glacier volume changes as examples. We find that uncertainties in DEMs are largely underestimated in the literature, and advocate that new metrics of DEM precision are essential to ensure the reliability of future Earth and planetary surface elevation assessments.

Spatial correlation in ground motion prediction errors in Central and Eastern North America

Risk analysis and risk-informed design of spatially distributed infrastructure systems for earthquake hazards require an understanding of and ability to model the spatial correlation of ground motion prediction errors. We assess this spatial correlation in Central and Eastern North America (CENA) by calculating ground motion residuals and semivariograms from earthquake recordings in the Next Generation Attenuation (NGA)-East database. Although data limitations prohibit the development of a reliable model to capture this correlation, we have made notable findings relevant to future risk analyses. The spatial correlation of ground motion prediction errors in CENA is larger than those previously published for shallow crustal regions, which agrees with the lower attenuation observed in CENA. Differences in correlation behavior is also observed between tectonic and induced event recordings. This is, in part, due to the characteristics of the system of stations in which they were recorded. In addition, the choice of ground motion model (GMM) used to calculate the predicted ground motions was found to have an impact on the resulting correlation of errors and we recommend that future CENA spatial correlation models be tailored to the specific infrastructure system and location that will be analyzed.

From consumer to enterprise grade: How the choice of four UAS impacts point cloud quality

AbstractUncrewed aerial systems (UAS), combined with structure‐from‐motion photogrammetry, has already proven to be very powerful for a wide range of geoscience applications and different types of UAS are used for scientific and commercial purposes. However, the impact of the UAS used on the accuracy of the point clouds derived is not fully understood, especially for the quantitative analysis of geomorphic changes in complex terrain. Therefore, in this study, we aim to quantify the magnitude of systematic and random error in digital elevation models derived from four commonly used UAS (XR6/Sony α6000, Inspire 2/X4s, Phantom 4 Pro+, Mavic Pro) following different flight patterns. The vertical error of each elevation model is evaluated through comparison with 156 GNSS reference points and the normal distribution and spatial correlation of errors are analysed. Differences in mean errors (−0.4 to −1.8 cm) for the XR6, Inspire 2 and Phantom 4 Pro are significant but not relevant for most geomorphological applications. The Mavic Pro shows lower accuracies with mean errors up to 4.3 cm, thus showing a higher influence of random errors. QQ plots revealed a deviation of errors from a normal distribution in almost all data. All UAS data except Mavic Pro exhibit a pure nugget semivariogram, suggesting spatially uncorrelated errors. Compared to the other UAS, the Mavic Pro data show trends (i.e. differences increase with distance across the survey—doming) and the range of semivariances is 10 times greater. The lower accuracy of Mavic Pro can be attributed to the lower GSD at the same flight altitude and most likely, the rolling shutter sensor has an effect on the accuracy of the camera calibration. Overall, our study shows that accuracies depend highly on the chosen data sampling strategy and that the survey design used here is not suitable for calibrating all types of UAS camera equally.

The interaction between activity choice and duration: Application of Copula-based and Nested-logit models

There is a relationship between choosing an activity and duration of that activity, especially for non-mandatory ones. Some previous studies have analyzed the decisions about an activity type and duration independently, though some others have used joint models. This paper contributes to the body of knowledge through using Nested-logit and Copula-based models for assessing the existence of interdependency or a hierarchy between non-mandatory activity choice and the relative duration. In the Nested-logit model, it is assumed that error terms of these decisions are interrelated, though one is influenced by another. In contrast, the Copula-based model can accommodate spatial error correlation across observational units without imposing a restrictive distribution assumption on the dependency structures between the error components. The data from Qazvin, a city in Iran, are used for estimating both Nested-logit and Copula-based models and the best variables explaining both choices for each model have been selected. The final models were compared in terms of log-likelihood at convergence and adjusted likelihood ratio index. The results indicated that there are some common influential observed and unobserved factors between these decisions. Also, Copula-based joint model with ρ_0^2 equals to 0.134 outperforms Nested-logit models and provides a better explanatory power.

EFFECT OF SPATIAL CORRELATION ON THE PERFORMANCES OF MODERNIZED GPS AND GALILEO IN RELATIVE POSITIONING

In the context of processing GNSS (Global Navigation Satellite System) data, it is known that the estimation of the ionospheric delays decreases the strength of the observation model and makes significant the time required to fix the ambiguities namely in case of long baselines. However, considering the double-differenced (DD) ionospheric delays as stochastic quantities, the redundancy in this case increases and leads to the reduction of time of fixing the ambiguities. The approach developed in the present paper makes two considerations: 1) the DD ionospheric delays are assumed as stochastic quantities and, 2) the spatial correlation of errors is accounted for based on a simple model of correlation. A simulation is made and aims to study the effect of these two mentioned considerations on the performances of the three multifrequency GNSSs; modernized GPS, Galileo and BDS which are not yet in full capability. For each GNSS, dual-frequency combinations of frequencies as well as triple-frequency combination are investigated in the simulation. The performances studied include: the time to fix the ambiguities with high success rate and the precision of coordinates in static relative positioning with varying baseline length. A method is developed to derive what we call the spatial correlation model which approximately gives the covariance between the individual errors belonging to two stations. Furthermore, the stochastic models that follow from accounting and neglecting the spatial correlation are developed. The LAMBDA (Least-squares Ambiguity Decorrelation Adjustment) method is implemented for ambiguity decorrelation. The results show that the time to fix the ambiguities caused by accounting the spatial correlation is less than the time of fix without the spatial correlation. Also, a slight superiority of Galileo in terms of performances is seen compared to the other GNSS. For all the dualfrequency combinations investigated, when processing a baseline length of 500 km with accounted spatial correlation, the time needed to successfully fix the ambiguities lies between 5 and 9 min, whereas it becomes only between 2.5 and 3 min for all the triple-frequency combinations, this is with a sampling time of 5 s. In addition, for all different combinations, the coordinates precision is less than 8 mm even for 500 km. We think that these high performances result from: 1) the precise codes of future GNSS signals, 2) the high redundancy in the observations equation and, 3) taking into account the spatial correlation in the definition of the stochastic model.

Influence of the GEO satellite orbit error fluctuation correction on the BDS WADS zone correction

Decimeter-level service is provided by the BeiDou satellite navigation system wide area differential service (BDS WADS) for users who collect carrier phase measurements. However, the fluctuations in Geostationary Earth Orbit (GEO) satellite orbit errors reduce the spatial correlation of orbit errors. These fluctuations not only decrease the accuracy and stability of zone correction service provided by BDS WADS, but also shorten its effective range. In this paper, we proposed an algorithm to weaken the influence of GEO satellite orbit error fluctuations and verified the method using data from eight sparsely distributed zones. The results show that orbit errors can be stabilized using orbit fluctuation corrections, and the positioning precision and stability of the BDS WADS can be improved simultaneously. Under normal circumstances, the horizontal and vertical positioning accuracy of users within 1000 km from the center of the zone can reach 0.19 m and 0.34 m. Furthermore, the effective range is increased. The positioning performance within 1800 km could reach 0.24 m and 0.38 m for the horizontal and vertical components, respectively.

A practical assimilation approach to extract smaller‐scale information from observations with spatially correlated errors: An idealized study

AbstractIt is still common to neglect the spatial error correlations of assimilated observations in numerical weather prediction systems because no practical approach is available to account for them when the number of observations with correlated error is large or when these observations are non‐uniformly distributed. Instead, it is common practice to inflate observation error variances to avoid overfitting large scales and spatially thin observations to reduce error correlations between remaining observations, although both methods generally sacrifice small‐scale information. Inspired by previous work on assimilating the difference between adjacent observations (so‐called spatial difference observations), this study aims at combining direct observations with spatial difference observations in the assimilation to extract both large‐ and smaller‐scale information from observations with spatially correlated errors, while still neglecting these error correlations. Experiments performed in a simplified 1D context over a periodic domain show that the combined approach is numerically equivalent to directly assimilating observations available at every grid point using non‐diagonal observation error covariances based on a first‐order autoregressive correlation function. In a case where observation error correlations have a different structure (e.g. Gaussian), the true observation error correlations are not perfectly taken into account by the combined approach, but it still efficiently extracts information to correct the scales that have the most errors. Combining direct observations with spatial difference observations proves complementary and experimental results show lower analysis errors for both large and intermediate scales. More specifically, while neglecting spatial observation error correlations, the combined approach provides results with lower analysis error than the direct approach, especially when spatial error correlations are large and when direct observations are spatially thinned.

Mitigating temporal aliasing effects of high-frequency geophysical fluid dynamics in satellite gravimetry

SUMMARY Temporal aliasing errors induced by high-frequency tidal and non-tidal mass variability in the Earth system are among the three most important error sources that limit the accuracy of present-day surface mass estimates from satellite gravimetry. By means of end-to-end simulations, we demonstrate that the Kalman Smoother approach developed by Kurtenbach et al. effectively captures non-tidal submonthly variability, and thereby reduces temporal aliasing errors way beyond the level of simply subtracting the standard dealiasing model AOD1B. Validation against in situ ocean bottom pressure observations confirms that the Kalman Smoother solutions published together with the ITSG-Grace2016 monthly gravity fields contain high-frequency signal over the oceans not predicted by AOD1B. The daily gravity fields therefore reduce aliasing artefacts in the monthly gravity fields, and at the same time provide observational evidence on submonthly bottom pressure variability presently not reflected in state-of-the-art numerical ocean circulation models. It is thus recommended to include a Kalman Smoother approach into any standard GRACE processing scheme. For a hypothetical double-pair configuration currently under consideration as a future mass change mission, we find that the benefit of the Kalman Smoother is much smaller thanks to the increased number of observations taken at different inclinations, which lead to generally reduced aliasing errors and much more isotropic spatial error correlations. We also reassess the idea of pre-eliminating low-resolution daily gravity fields and find large distortions in the monthly mean gravity solution at spatial wavelengths around the cut-off-degree of the daily fields. We thus recommend further study for any satellite gravity mission concept that critically relies on such pre-elimination schemes for reaching its science objectives.

Diagnosing spatial error structures in CO<sub>2</sub> mole fractions and XCO<sub>2</sub> column mole fractions from atmospheric transport

Abstract. Atmospheric inversions inform us about the magnitude and variations of greenhouse gas (GHG) sources and sinks from global to local scales. Deployment of observing systems such as spaceborne sensors and ground-based instruments distributed around the globe has started to offer an unprecedented amount of information to estimate surface exchanges of GHG at finer spatial and temporal scales. However, all inversion methods still rely on imperfect atmospheric transport models whose error structures directly affect the inverse estimates of GHG fluxes. The impact of spatial error structures on the retrieved fluxes increase concurrently with the density of the available measurements. In this study, we diagnose the spatial structures due to transport model errors affecting modeled in situ carbon dioxide (CO2) mole fractions and total-column dry air mole fractions of CO2 (XCO2). We implement a cost-effective filtering technique recently developed in the meteorological data assimilation community to describe spatial error structures using a small-size ensemble. This technique can enable ensemble-based error analysis for multiyear inversions of sources and sinks. The removal of noisy structures due to sampling errors in our small-size ensembles is evaluated by comparison with larger-size ensembles. A second filtering approach for error covariances is proposed (Wiener filter), producing similar results over the 1-month simulation period compared to a Schur filter. Based on a comparison to a reference 25-member calibrated ensemble, we demonstrate that error variances and spatial error correlation structures are recoverable from small-size ensembles of about 8 to 10 members, improving the representation of transport errors in mesoscale inversions of CO2 fluxes. Moreover, error variances of in situ near-surface and free-tropospheric CO2 mole fractions differ significantly from total-column XCO2 error variances. We conclude that error variances for remote-sensing observations need to be quantified independently of in situ CO2 mole fractions due to the complexity of spatial error structures at different altitudes. However, we show the potential use of meteorological error structures such as the mean horizontal wind speed, directly available from ensemble prediction systems, to approximate spatial error correlations of in situ CO2 mole fractions, with similarities in seasonal variations and characteristic error length scales.