The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

[1] EGM2008 is a spherical harmonic model of the Earth's gravitational potential, developed by a least squares combination of the ITG-GRACE03S gravitational model and its associated error covariance matrix, with the gravitational information obtained from a global set of area-mean free-air gravity anomalies defined on a 5 arc-minute equiangular grid This grid was formed by merging terrestrial, altimetry-derived, and airborne gravity data Over areas where only lower resolution gravity data were available, their spectral content was supplemented with gravitational information implied by the topography EGM2008 is complete to degree and order 2159, and contains additional coefficients up to degree 2190 and order 2159 Over areas covered with high quality gravity data, the discrepancies between EGM2008 geoid undulations and independent GPS/Leveling values are on the order of ±5 to ±10 cm EGM2008 vertical deflections over USA and Australia are within ±11 to ±13 arc-seconds of independent astrogeodetic values These results indicate that EGM2008 performs comparably with contemporary detailed regional geoid models EGM2008 performs equally well with other GRACE-based gravitational models in orbit computations Over EGM96, EGM2008 represents improvement by a factor of six in resolution, and by factors of three to six in accuracy, depending on gravitational quantity and geographic area EGM2008 represents a milestone and a new paradigm in global gravity field modeling, by demonstrating for the first time ever, that given accurate and detailed gravimetric data, asingle global model may satisfy the requirements of a very wide range of applications

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2011JB008916

The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)

We present Bedmap2, a new suite of gridded products describing surface elevation, ice-thickness and the seafloor and subglacial bed elevation of the Antarctic south of 60° S. We derived these products using data from a variety of sources, including many substantial surveys completed since the original Bedmap compilation (Bedmap1) in 2001. In particular, the Bedmap2 ice thickness grid is made from 25 million measurements, over two orders of magnitude more than were used in Bedmap1. In most parts of Antarctica the subglacial landscape is visible in much greater detail than was previously available and the improved data-coverage has in many areas revealed the full scale of mountain ranges, valleys, basins and troughs, only fragments of which were previously indicated in local surveys. The derived statistics for Bedmap2 show that the volume of ice contained in the Antarctic ice sheet (27 million km3) and its potential contribution to sea-level rise (58 m) are similar to those of Bedmap1, but the mean thickness of the ice sheet is 4.6% greater, the mean depth of the bed beneath the grounded ice sheet is 72 m lower and the area of ice sheet grounded on bed below sea level is increased by 10%. The Bedmap2 compilation highlights several areas beneath the ice sheet where the bed elevation is substantially lower than the deepest bed indicated by Bedmap1. These products, along with grids of data coverage and uncertainty, provide new opportunities for detailed modelling of the past and future evolution of the Antarctic ice sheets.

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Bedmap2: improved ice bed, surface and thickness datasets for Antarctica

Freshwater availability is changing worldwide. Here we quantify 34 trends in terrestrial water storage observed by the Gravity Recovery and Climate Experiment (GRACE) satellites during 2002–2016 and categorize their drivers as natural interannual variability, unsustainable groundwater consumption, climate change or combinations thereof. Several of these trends had been lacking thorough investigation and attribution, including massive changes in northwestern China and the Okavango Delta. Others are consistent with climate model predictions. This observation-based assessment of how the world’s water landscape is responding to human impacts and climate variations provides a blueprint for evaluating and predicting emerging threats to water and food security.

/pdf/emerging-trends-in-global-freshwater-availability-73pgdps0j7.pdf

Emerging trends in global freshwater availability.

We discuss several classes of improvements to gravity solutions from the Gravity Recovery and Climate Experiment (GRACE) mission. These include both improvements in background geophysical models and orbital parameterization leading to the unconstrained spherical harmonic solution JPL RL05, and an alternate JPL RL05M mass concentration (mascon) solution benefitting from those same improvements but derived in surface spherical cap mascons. The mascon basis functions allow for convenient application of a priori information derived from near-global geophysical models to prevent striping in the solutions. The resulting mass flux solutions are shown to suffer less from leakage errors than harmonic solutions, and do not necessitate empirical filters to remove north-south stripes, lowering the dependence on using scale factors (the global mean scale factor decreases by 0.17) to gain accurate mass estimates. Ocean bottom pressure (OBP) time series derived from the mascon solutions are shown to have greater correlation with in situ data than do spherical harmonic solutions (increase in correlation coefficient of 0.08 globally), particularly in low-latitude regions with small signal power (increase in correlation coefficient of 0.35 regionally), in addition to reducing the error RMS with respect to the in situ data (reduction of 0.37 cm globally, and as much as 1 cm regionally). Greenland and Antarctica mass balance estimates derived from the mascon solutions agree within formal uncertainties with previously published results. Computing basin averages for hydrology applications shows general agreement between harmonic and mascon solutions for large basins; however, mascon solutions typically have greater resolution for smaller spatial regions, in particular when studying secular signals.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2014JB011547

Improved methods for observing Earth's time variable mass distribution with GRACE using spherical cap mascons

Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10 cm in terms of geoid heights, and 3 mGal in terms of gravity anomalies, globally at a resolution of 100 km (degree/order 200).

/pdf/first-goce-gravity-field-models-derived-by-three-different-55q4c3j26r.pdf

First GOCE gravity field models derived by three different approaches

EIGEN-6C3 - The latest Combined Global Gravity Field Model including GOCE data up to degree and order 1949 of GFZ Potsdam and GRGS Toulouse

GFZ Potsdam and GRGS Toulouse have a long-time close cooperation in the field of global gravity field determination. Here we focus on (1) GOCE gravity field determination and (2) computation of high resolution combined gravity field models. Such data products play a fundamental role in geodesy and Earth sciences, ranging from practical purposes, like precise orbit determination, to scientific applications, like investigations of the density structure of the Earth’s interior. Here we present our combined gravity field model EIGEN-6C4 which is the fourth release of EIGEN-6C (EIGEN = European Improved Gravity model of the Earth by New techniques).

EIGEN-6C4 - The latest combined global gravity field model including GOCE data up to degree and order 1949 of GFZ Potsdam and GRGS Toulouse

CNES/GRGS 10-day gravity field models (release 2) and their evaluation

Precise measurements of the Earth's time-varying gravitational field from the NASA/Deutsches Zentrum fur Luft- und Raumfahrt Gravity Recovery and Climate Experiment (GRACE) mission allow unprecedented tracking of the transport of mass across and underneath the surface of the Earth and give insight into secular, seasonal, and subseasonal variations in the global water supply. Several groups produce these estimates, and while the various gravity fields are similar, differences in processing strategies and tuning parameters result in solutions with regionally specific variations and error patterns. This study examined the spatial, temporal, and spectral variations between the different gravity field products and developed an ensemble gravity field solution from the products of four such analysis centers. The solutions were found to lie within a certain analysis scatter regardless of the local relative water height variation, and the ensemble model is clearly seen to reduce the noise in the gravity field solutions within the available scatter of the solutions.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2013GL058632

Sean Bruinsma

Papers

First GOCE gravity field models derived by three different approaches

EIGEN-6C3 - The latest Combined Global Gravity Field Model including GOCE data up to degree and order 1949 of GFZ Potsdam and GRGS Toulouse

EIGEN-6C4 - The latest combined global gravity field model including GOCE data up to degree and order 1949 of GFZ Potsdam and GRGS Toulouse

CNES/GRGS 10-day gravity field models (release 2) and their evaluation

Ensemble prediction and intercomparison analysis of GRACE time‐variable gravity field models