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

More than 1.4 billion people depend on water from the Indus, Ganges, Brahmaputra, Yangtze, and Yellow rivers. Upstream snow and ice reserves of these basins, important in sustaining seasonal water availability, are likely to be affected substantially by climate change, but to what extent is yet unclear. Here, we show that meltwater is extremely important in the Indus basin and important for the Brahmaputra basin, but plays only a modest role for the Ganges, Yangtze, and Yellow rivers. A huge difference also exists between basins in the extent to which climate change is predicted to affect water availability and food security. The Brahmaputra and Indus basins are most susceptible to reductions of flow, threatening the food security of an estimated 60 million people.

http://science.sciencemag.org/content/sci/328/5984/1382.full.pdf

Climate change will affect the Asian water towers.

We combined an ensemble of satellite altimetry, interferometry, and gravimetry data sets using common geographical regions, time intervals, and models of surface mass balance and glacial isostatic adjustment to estimate the mass balance of Earth’s polar ice sheets. We find that there is good agreement between different satellite methods—especially in Greenland and West Antarctica—and that combining satellite data sets leads to greater certainty. Between 1992 and 2011, the ice sheets of Greenland, East Antarctica, West Antarctica, and the Antarctic Peninsula changed in mass by –142 ± 49, +14 ± 43, –65 ± 26, and –20 ± 14 gigatonnes year−1, respectively. Since 1992, the polar ice sheets have contributed, on average, 0.59 ± 0.20 millimeter year−1 to the rate of global sea-level rise.

/pdf/a-reconciled-estimate-of-ice-sheet-mass-balance-tca0inf12m.pdf

A reconciled estimate of ice-sheet mass balance

Supplementary Materials for A Reconciled Estimate of Ice-Sheet Mass Balance

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.

SUMMARY Most applications of the publicly released Gravity Recovery and Climate Experiment monthly gravity field models require the application of a spatial filter to help suppressing noise and other systematic errors present in the data The most common approach makes use of a simple Gaussian averaging process, which is often combined with a ‘destriping’ technique in which coefficient correlations within a given degree are removed As brute force methods, neither of these techniques takes into consideration the statistical information from the gravity solution itself and, while they perform well overall, they can often end up removing more signal than necessary Other optimal filters have been proposed in the literature; however, none have attempted to make full use of all information available from the monthly solutions By examining the underlying principles of filter design, a filter has been developed that incorporates the noise and full signal variance–covariance matrix to tailor the filter to the error characteristics of a particular monthly solution The filter is both anisotropic and nonsymmetric, meaning it can accommodate noise of an arbitrary shape, such as the characteristic stripes The filter minimizes the mean-square error and, in this sense, can be considered as the most optimal filter possible Through both simulated and real data scenarios, this improved filter will be shown to preserve the highest amount of gravity signal when compared to other standard techniques, while simultaneously minimizing leakage effects and producing smooth solutions in areas of low signal

/pdf/the-design-of-an-optimal-filter-for-monthly-grace-gravity-496ly17fz8.pdf

The design of an optimal filter for monthly GRACE gravity models

SUMMARY We discuss and develop further the methods of surface wave tomography in the frame of the geometric ray approximation. The general approach for determining the lateral phase or group velocity distribution, which is a standard 2-D tomography problem, involves linearization, representation of the unknown function as a series in some basis functions, and evaluation of the coefficients by the methods of linear algebra. If the wave paths cover the area under investigation non-uniformly, the basis functions should not be chosen a priori, but constructed proceeding from the pattern of paths. Different criteria for constructing the basis functions are compared, and a relation between them is considered. A more preferable approach is joint interpretation of phase and group velocity data for different periods, because it allows the information about phase velocity variations to be enlarged due to the use of the group velocity data. Both the phase and group traveltimes are represented as linear functionals of the unknown phase slowness corrections. A specific form of the data kernels allows the basis functions to be represented as a product of two functions, one depending on the horizontal coordinates, and the other on frequency.

/pdf/smoothness-criteria-in-surface-wave-tomography-37d96j1kc9.pdf

Smoothness criteria in surface wave tomography

The focus of the study is optimizing the technique for estimating geocenter motion and variations in J2 by combining data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission with output from an Ocean Bottom Pressure (OBP) model and a Glacial Isostatic Adjustment (GIA) model [Swenson et al., 2008; Sun et al., 2016]. First, we conduct an end-to-end numerical simulation study. We generate input time-variable gravity field observations by perturbing a synthetic Earth model with realistically simulated errors. We show that it is important to avoid large errors at short wavelengths and signal leakage from land to ocean, as well as to account for self-attraction and loading (SAL) effects. Second, the optimal implementation strategy is applied to real GRACE data. We show that the estimates of annual amplitude in geocenter motion are in line with estimates from other techniques, such as SLR and global GPS inversion. At the same time, annual amplitudes of C10 and C11 are increased by about 50% and 20%, respectively, compared to estimates based on [Swenson et al., 2008]. Estimates of J2 variations are by about 15% larger than SLR results in terms of annual amplitude. Linear trend estimates are dependent on the adopted GIA model, but still comparable to some SLR results.

Optimizing estimates of annual variations and trends in geocenter motion and J2 from a combination of GRACE data and geophysical models

Estimation of volume change rates of Greenland's ice sheet from ICESat data using overlapping footprints

An approach to handling colored observation noise in large least-squares (LS) problems is presented. The handling of colored noise is reduced to the problem of solving a Toeplitz system of linear equations. The colored noise is represented as an auto regressive moving-average (ARMA) process. Stability and invertability of the ARMA model allows the solution of the Toeplitz system to be reduced to two successive filtering operations using the inverse transfer function of the ARMA model. The numerical complexity of the algorithm scales proportionally to the order of the ARMA model times the number of observations. This makes the algorithm particularly suited for LS problems with millions of observations. It can be used in combination with direct and iterative algorithms for the solution of the normal equations. The performance of the algorithm is demonstrated for the computation of a model of the Earth’s gravity field from simulated satellite-derived gravity gradients up to spherical harmonic degree 300.

/pdf/how-to-handle-colored-observation-noise-in-large-least-353rqhi0ft.pdf

Pavel Ditmar

Papers

The design of an optimal filter for monthly GRACE gravity models

Smoothness criteria in surface wave tomography

Optimizing estimates of annual variations and trends in geocenter motion and J2 from a combination of GRACE data and geophysical models

Estimation of volume change rates of Greenland's ice sheet from ICESat data using overlapping footprints

How to handle colored observation noise in large least-squares problems