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

Establishing a quantitative description of global riverine fluxes is one of the main goals of contemporary hydrology and geomorphology. Here we study changes in global riverine water discharge and suspended sediment flux over a 50-year period, 1960–2010, applying a new version of the WBMsed (WBMsed v.2.0) global hydrological water balance model. A new floodplain component is introduced to better represent water and sediment dynamics during periods of overbank discharge. Validated against data from 16 globally distributed stations, WBMsed v.2.0 simulation results show considerable improvement over the original model. Normalized departure from an annual mean is used to quantify spatial and temporal dynamics in both water discharge and sediment flux. Considerable intra-basin variability in both water and sediment discharge is observed for the first time in different regions of the world. Continental-scale analysis shows considerable variability in water and sediment discharge fluctuations both in time and between continents. A correlation analysis between predicted continental suspended sediment and water discharge shows strong correspondence in Australia and Africa (R2 of 0.93 and 0.87 respectively), moderate correlation in North and South America (R2 of 0.64 and 0.73 respectively) and weak correlation in Asia and Europe (R2 of 0.35 and 0.24 respectively). We propose that yearly changes in intra-basin precipitation dynamics explain most of these differences in continental water discharge and suspended sediment correlation. The mechanism proposed and demonstrated here (for the Ganges, Danube and Amazon Rivers) is that regions with high relief and soft lithology will amplify the effect of higher than average precipitation by producing an increase in sediment yield that greatly exceeds increase in water discharge.

Global suspended sediment and water discharge dynamics between 1960 and 2010: Continental trends and intra-basin sensitivity

Remote estimation of river discharge from river width variations is an intriguing method for gauging rivers without conventional measurements. Entirely cloud-free imagery of an entire river reach is often rare, but partial coverage is more frequent. Discharge is estimated from spatially discontinuous imagery via construction of multiple width–discharge rating curves within a 62-km reach of the Tanana River, Alaska. The resulting discharge error is as low as 6.7% root mean squared error. Imagery covering <20% of the study reach can be used. Copyright © 2014 John Wiley & Sons, Ltd.

Using Width-Based Rating Curves from Spatially Discontinuous Satellite Imagery to Monitor River Discharge

Space-borne instruments can measure river water surface elevation, slope and width. Remote sensing of river discharge in ungauged basins is far more challenging, however. This work investigates the estimation of river discharge from simulated observations of the forthcoming Surface Water and Ocean Topography (SWOT) satellite mission using a variant of the classical variational data assimilation method “4D-Var”. The variational assimilation scheme simultaneously estimates discharge, river bathymetry and bed roughness in the context of a 1.5D full Saint Venant hydraulic model. Algorithms and procedures are developed to apply the method to fully ungauged basins. The method was tested on the Po and Sacramento Rivers. The SWOT hydrology simulator was used to produce synthetic SWOT observations at each overpass time by simulating the interaction of SWOT radar measurements with the river water surface and nearby land surface topography at a scale of approximately 1 m, thus accounting for layover, thermal noise and other effects. SWOT data products were synthesized by vectorizing the simulated radar returns, leading to height and width estimates at 200 m increments along the river centerlines. The ingestion of simulated SWOT data generally led to local improvements on prior bathymetry and roughness estimates which allowed the prediction of river discharge at the overpass times with relative root-mean-squared errors of 12.1% and 11.2% for the Po and Sacramento rivers respectively. Nevertheless, equifinality issues that arise from the simultaneous estimation of bed elevation and roughness may prevent their use for different applications, other than discharge estimation through the presented framework.

Discharge estimation in ungauged basins through variational data assimilation : The potential of the SWOT mission.

More than one-third of Earth’s landmass is drained by rivers that seasonally freeze over. Ice transforms the hydrologic1,2, ecologic3,4, climatic5 and socio-economic6–8 functions of river corridors. Although river ice extent has been shown to be declining in many regions of the world1, the seasonality, historical change and predicted future changes in river ice extent and duration have not yet been quantified globally. Previous studies of river ice, which suggested that declines in extent and duration could be attributed to warming temperatures9,10, were based on data from sparse locations. Furthermore, existing projections of future ice extent are based solely on the location of the 0-°C isotherm11. Here, using satellite observations, we show that the global extent of river ice is declining, and we project a mean decrease in seasonal ice duration of 6.10 ± 0.08 days per 1-°C increase in global mean surface air temperature. We tracked the extent of river ice using over 400,000 clear-sky Landsat images spanning 1984–2018 and observed a mean decline of 2.5 percentage points globally in the past three decades. To project future changes in river ice extent, we developed an observationally calibrated and validated model, based on temperature and season, which reduced the mean bias by 87 per cent compared with the 0-degree-Celsius isotherm approach. We applied this model to future climate projections for 2080–2100: compared with 2009–2029, the average river ice duration declines by 16.7 days under Representative Concentration Pathway (RCP) 8.5, whereas under RCP 4.5 it declines on average by 7.3 days. Our results show that, globally, river ice is measurably declining and will continue to decline linearly with projected increases in surface air temperature towards the end of this century. An analysis based on Landsat imagery shows that the extent of river ice has declined extensively over past decades and that this trend will continue under future global warming.

The past and future of global river ice

Constraining Remote River Discharge Estimation Using Reach-Scale Geomorphology

Constraining the Assimilation of SWOT Observations With Hydraulic Geometry Relations

Exploring the Factors Controlling the Error Characteristics of the Surface Water and Ocean Topography Mission Discharge Estimates

At‐many‐stations hydraulic geometry (AMHG), while useful for estimating river discharge from satellite data, remains empirical and has yet to be reconciled with the at‐a‐station hydraulic geometry (AHG) from which it was originally derived. Here we present evidence, using United States Geological Survey field measurements of channel hydraulics for 155 rivers, that AMHG can be hydraulically and geomorphically reconciled with AHG. Our results indicate that AMHG is rightly understood as an expression of a river‐wide model of hydraulics driven by changes in slope imposed upon AHG physics. The explanatory power of AHG and this river‐wide model combine to determine whether AMHG exists: if both AHG and the river‐wide model adequately describe hydraulics, then we show that AMHG is a necessary mathematical consequence of these two phenomena. We also orient these findings in the context of river discharge estimation and other applications. Plain Language Summary Hydraulic geometry (HG) is an empirical phenomenon that predicts river width, depth, and velocity given river discharge and is fundamental to our ability to predict floods, river habitats, and water availability for human and ecosystem use. Recently, a new form of HG was discovered (at‐many‐stations HG, or AMHG), and it has been successfully deployed in a range of applications and proven to exist in a wide variety of rivers. However, the novel phenomenon of AMHG remains empirical with puzzling and seemingly contradictory links to other, traditional forms of HG, which are well understood and have been studied since the 1950s. AMHG is also not manifested in all rivers, adding to confusion as to its origin. Here, we show for the first time that we can reconcile AMHG with all other traditional variants of HG and gain a complete understanding of how andwhen AMHG occurs in rivers. This puts the most puzzling aspects of AMHG—why is it observed in some rivers but not others, and what causes the phenomenon—to rest. We have fundamentally changed the conception of AMHG and suggest this work as a basis for all future AMHG research.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2019GL084529

C. B. Brinkerhoff

Papers

Constraining Remote River Discharge Estimation Using Reach-Scale Geomorphology

Constraining the Assimilation of SWOT Observations With Hydraulic Geometry Relations

Exploring the Factors Controlling the Error Characteristics of the Surface Water and Ocean Topography Mission Discharge Estimates

Reconciling at-a-Station and at-Many-Stations Hydraulic Geometry Through River-Wide Geomorphology

Lake Morphometry and River Network Controls on Evasion of Terrestrially Sourced Headwater CO 2