High-dimensional experiments for the downward continuation using the LRFMP algorithm

Time-dependent gravity data from satellite missions like GRACE-FO reveal mass redistribution in the system Earth at various time scales: long-term climate change signals, inter-annual phenomena like El Nino, seasonal mass transports and transients, e. g. due to earthquakes. For this contemporary issue, a classical inverse problem has to be considered: the gravitational potential has to be modelled on the Earth's surface from measurements in space. This is also known as the downward continuation problem. Thus, it is important to further develop current mathematical methods for such inverse problems. For this, the (Learning) Inverse Problem Matching Pursuits ((L)IPMPs) have been developed within the last decade. Their unique feature is the combination of local as well as global trial functions in the approximative solution of an inverse problem such as the downward continuation of the gravitational potential. In this way, they harmonize the ideas of a traditional spherical harmonic ansatz and the radial basis function approach. Previous publications on these methods showed proofs of concept. Here, we consider the methods for high-dimensional experiments settings with more than 500 000 grid points which yields a resolution of 20 km at best on a realistic satellite geometry. We also explain the changes in the methods that had to be done to work with such a large amount of data. The corresponding code (updated for big data use) is available at https://doi.org/10.5281/zenodo.8223771 under the licence CC BY-NC-SA 3.0 Germany

Downscaling GRACE total water storage change using partial least squares regression

The Gravity Recovery And Climate Experiment (GRACE) satellite mission recorded temporal variations in the Earth’s gravity field, which are then converted to Total Water Storage Change (TWSC) fields representing an anomaly in the water mass stored in all three physical states, on and below the surface of the Earth. GRACE provided a first global observational record of water mass redistribution at spatial scales greater than 63000 km2. This limits their usability in regional hydrological applications. In this study, we implement a statistical downscaling approach that assimilates 0.5° × 0.5° water storage fields from the WaterGAP hydrology model (WGHM), precipitation fields from 3 models, evapotranspiration and runoff from 2 models, with GRACE data to obtain TWSC at a 0.5° × 0.5° grid. The downscaled product exploits dominant common statistical modes between all the hydrological datasets to improve the spatial resolution of GRACE. We also provide open access to scripts that researchers can use to produce downscaled TWSC fields with input observations and models of their own choice

The role of two-point functions in geodesy and their classification

In geodesy, two-point functions appear as covariance functions, convolution kernels like the Green functions, transfer functions of the gravity field functionals and filter kernels. Knowledge of their structure both in the spatial and the spectral domains opens vistas not only for understanding their behaviour, but also enabling their design. Here, we develop the two-point functions in terms of spherical harmonic functions and discuss their structure. We identify homogeneity and isotropy as the two key structural properties of the two-point functions that provide a solid basis for their classification

Basin-scale runoff prediction: An Ensemble Kalman Filter framework based on global hydrometeorological data sets

In order to cope with the steady decline of the number of in situ gauges worldwide, there is a growing need for alternative methods to estimate runoff. We present an Ensemble Kalman Filter based approach that allows us to conclude on runoff for poorly or irregularly gauged basins. The approach focuses on the application of publicly available global hydrometeorological data sets for precipitation (GPCC, GPCP, CRU, UDEL), evapotranspiration (MODIS, FLUXNET, GLEAM, ERA interim, GLDAS), and water storage changes (GRACE, WGHM, GLDAS, MERRA LAND). Furthermore, runoff data from the GRDC and satellite altimetry derived estimates are used. We follow a least squares prediction that exploits the joint temporal and spatial auto- and cross-covariance structures of precipitation, evapotranspiration, water storage changes and runoff. We further consider time-dependent uncertainty estimates derived from all data sets. Our in-depth analysis comprises of 29 large river basins of different climate regions, with which runoff is predicted for a subset of 16 basins. Six configurations are analyzed: the Ensemble Kalman Filter (Smoother) and the hard (soft) Constrained Ensemble Kalman Filter (Smoother). Comparing the predictions to observed monthly runoff shows correlations larger than 0.5, percentage biases lower than ± 20%, and NSE-values larger than 0.5. A modified NSE-metric, stressing the difference to the mean annual cycle, shows an improvement of runoff predictions for 14 of the 16 basins. The proposed method is able to provide runoff estimates for nearly 100 poorly gauged basins covering an area of more than 11,500,000 km2 with a freshwater discharge, in volume, of more than 125,000 m3/s

What is the spatial resolution of GRACE satellite products for hydrology?

The mass change information from the Gravity Recovery And Climate Experiment (grace) satellite mission is available in terms of noisy spherical harmonic coefficients truncated at a maximum degree (band-limited). Therefore, filtering is an inevitable step in post-processing of grace fields to extract meaningful information about mass redistribution in the Earth-system. It is well known from previous studies that a number can be allotted to the spatial resolution of a band-limited spherical harmonic spectrum and also to a filtered field. Furthermore, it is now a common practice to correct the filtered grace data for signal damage due to filtering (or convolution in the spatial domain). These correction methods resemble deconvolution, and, therefore, the spatial resolution of the corrected grace data have to be reconsidered. Therefore, the effective spatial resolution at which we can obtain mass changes from grace products is an area of debate. In this contribution, we assess the spatial resolution both theoretically and practically. We confirm that, theoretically, the smallest resolvable catchment is directly related to the band-limit of the spherical harmonic spectrum of the grace data. However, due to the approximate nature of the correction schemes and the noise present in grace data, practically, the complete band-limited signal cannot be retrieved. In this context, we perform a closed-loop simulation comparing four popular correction schemes over 255 catchments to demarcate the minimum size of the catchment whose signal can be efficiently recovered by the correction schemes. We show that the amount of closure error is inversely related to the size of the catchment area. We use this trade-off between the error and the catchment size for defining the potential spatial resolution of the grace product obtained from a correction method. The magnitude of the error and hence the spatial resolution are both dependent on the correction scheme. Currently, a catchment of the size ≈63,000 km 2 can be resolved at an error level of 2 cm in terms of equivalent water height

Re-assessing global water storage trends from GRACE time series

Monitoring changes in freshwater availability is critical for human society and sustainable economic development. To identify regions experiencing secular change in their water resources, many studies compute linear trends in the total water storage (TWS) anomaly derived from the Gravity Recovery and Climate Experiment (GRACE) mission data. Such analyses suggest that several major water systems are under stress (Rodell et al 2009 Nature 460 999–1002; Long et al 2013 Geophys. Res. Lett. 40 3395–401; Richey et al 2015 Water Resour. Res. 51 5217–38; Voss et al 2013 Water Resour. Res. 49 904–14; Famiglietti 2014 Nat. Clim. Change. 4 945–8; Rodell et al 2018 Nature 557 651–9). TWS varies in space and time due to low frequency natural variability, anthropogenic intervention, and climate-change (Hamlington et al 2017 Sci. Rep. 7 995; Nerem et al 2018 Proc. Natl Acad. Sci.). Therefore, linear trends from a short time series can only be interpreted in a meaningful way after accounting for natural spatiotemporal variability in TWS (Paolo et al 2015 Science 348 327–31; Edward 2012 Geophys. Res. Lett. 39 L01702). In this study, we first show that GRACE TWS trends from a short time series cannot determine conclusively if an observed change is unprecedented or severe. To address this limitation, we develop a novel metric, trend to variability ratio (TVR), that assesses the severity of TWS trends observed by GRACE from 2003 to 2015 relative to the multi-decadal climate-driven variability. We demonstrate that the TVR combined with the trend provides a more informative and complete assessment of water storage change. We show that similar trends imply markedly different severity of TWS change, depending on location. Currently more than 3.2 billion people are living in regions facing severe water storage depletion w.r.t. past decades. Furthermore, nearly 36% of hydrological catchments losing water in the last decade have suffered from unprecedented loss. Inferences from this study can better inform water resource management

A probabilistic approach to characterizing drought using satellite gravimetry

In the recent past, the Gravity Recovery and Climate Experiment (GRACE) satellite mission and its successor GRACE Follow‐On (GRACE‐FO), have become invaluable tools for characterizing drought through measurements of Total Water Storage Anomaly (TWSA). However, the existing approaches have often overlooked the uncertainties in TWSA that stem from GRACE orbit configuration, background models, and intrinsic data errors. Here we introduce a fresh view on this problem which incorporates the uncertainties in the data: the Probabilistic Storage‐based Drought Index (PSDI). Our method leverages Monte Carlo simulations to yield realistic realizations for the stochastic process of the TWSA time series. These realizations depict a range of plausible drought scenarios that later on are used to characterize drought. This approach provides probability for each drought category instead of selecting a single final category at each epoch. We have compared PSDI with the deterministic approach (Storage‐based Drought Index, SDI) over major global basins. Our results show that the deterministic approach often leans toward an overestimation of storage‐based drought severity. Furthermore, we scrutinize the performance of PSDI across diverse hydrologic events, spanning continents from the United States to Europe, the Middle East, Southern Africa, South America, and Australia. In each case, PSDI emerges as a reliable indicator for characterizing drought conditions, providing a more comprehensive perspective than conventional deterministic indices. In contrast to the common deterministic view, our probabilistic approach provides a more realistic characterization of the TWS drought, making it more suited for adaptive strategies and realistic risk management.DF