Hydrogeodesy : a Bayesian perspective

Abstract

While historically focused on local scales, modern hydrologic studies have increasingly adopted a global perspective, recognizing water as a finite resource and the interconnection between regions. This global perspective puts hydrology within the water cycle framework, offering a comprehensive view of water dynamics across regions and scales. Despite this framework’s conceptual clarity, accurately quantifying the global water cycle remains challenging due to the complexity of capturing localized and large-scale patterns, variations in topography, climate, and land use, as well as temporal variability. These complexities hinder comprehensive measurements, resulting in knowledge gaps around key water cycle components, including river discharge, surface water storage, soil moisture dynamics, and subsurface water storage and flow. Inspired by the existing knowledge gaps in the water cycle, an emerging field known as Hydrogeodesy comes to the forefront. Hydrogeodesy is the discipline that uses terrestrial and primarily spaceborne geodetic data, both geometric and gravimetric, to support global water cycle quantification. Utilizing technologies such as satellite altimetry, gravimetry, imaging, InSAR, GNSS, and GNSS-Reflectometry, hydrogeodesy offers direct or indirect measurements of key water cycle components, including terrestrial water storage, and river discharge, significantly advancing our understanding of water dynamics. Despite advancements in spaceborne geodetic sensors, hydrogeodesy faces challenges such as limitations in the spatiotemporal resolution of satellite measurements, measurement uncertainties, unobserved variables, inconsistencies in background models, and the difficulty of separating aggregated measurements. Possible solutions to these challenges involve combining different data types, including satellite, ground-based observations, and model outputs, to benefit from their complementary strengths. However, this presents its own challenges, as it requires reconciling datasets with varying resolutions, accuracies, and temporal scales. To address some of the challenges listed above, Bayesian approaches offer viable solutions by providing probabilistic interpretations and uncertainty quantification. Bayesian approaches offer a robust framework for updating prior knowledge with new data to yield a posterior distribution, enabling a probabilistic interpretation and explicit uncertainty estimation of parameters. This is especially valuable in hydrogeodesy, where parameters like river discharge, soil moisture, and groundwater storage are often estimated indirectly and carry substantial uncertainties. This habilitation thesis provides a foundational discussion on Bayesian modeling and statistics and demonstrates the versatility and power of Bayesian methods in enhancing our understanding of water cycle components by presenting three distinct Bayesian applications in hydrogeodesy. The first study applies a Bayesian approach, specifically the Kalman filter, to estimate river discharge using spaceborne geodetic measurements. In hydrogeodetic studies, the Kalman filter and dynamic systems are especially valuable, as they enable the integration of multiple data sources and the continuous updating of estimates with incoming measurements. This is particularly beneficial for river systems, which inherently function as a dynamic system. To assess this potential, a method is introduced that uses the cyclostationary properties of discharge as prior information, while observed altimetric discharge data provide the likelihood. Together, these yield a posterior providing an unbiased daily discharge estimate. The method is applied to the Niger River basin and its main tributaries and validated against in situ data from 18 gauges. Results show a high average Correlation Coefficient (CC) of 0.9 and an average relative Root Mean Squared Error (RMSE) and bias of 15%. This method effectively estimates daily river discharge across entire basins and shows promise for global application, especially in data-scarce regions. With satellite altimetry data from multiple virtual stations and historical discharge data, daily discharge estimates with an error under 20% could be attainable in many river basins worldwide. The growing availability of spaceborne geodetic data, such as that provided by SWOT, further enhances this potential by delivering comprehensive measurements of river height and width, along with global discharge estimates. In most real-world applications, including hydrogeodesy, the Gaussianity assumption required by the Kalman filter does not hold, limiting its applicability. Inspired by this challenge, and motivated by the need to overcome the limitations of the poor spatial resolution of the GRACE and GRACE-FO missions, the second study proposes a Bayesian method to downscale GRACE data, proposing a nonparametric method to infer the posterior distribution directly, without any assumption for the likelihood or posterior. The prior distribution is obtained based on GRACE data values using the monthly variation of GRACE data. To model the likelihood functions, copulas are employed to capture dependencies among multivariate distributions. Monthly empirical copulas are constructed and fitted to analytical copulas, conditioned on specific quantile values, reflecting the dependency between GRACE and fine-scale data. A key advantage of this copula-supported Bayesian approach is its capacity to represent uncertainties in both data and models, even with variable input quality. The proposed downscaling approach is applied to the Amazon Basin, utilizing four different fine-scale datasets: WGHM, PCR-GLOBWB, SURFEX-TRIP, and the ensemble of flux data and soil moisture data from GLEAM and ASCAT. Validation is conducted against two independent datasets: space-based Surface Water Storage Change (SWSC) and GPS-observed Vertical Crustal Displacement Change (VCDR). In SWSC validation, downscaled results capture spatial variations in river storage with high CC and a relative RMSE of 26%. VCDR validation involves two analyses: comparing GPS-VCDR with TWSF-based VCDR using Green’s function convolution, where downscaled products yield RMSE values between 2.27 and 5.65 mm/month, outperforming input fine-scale data with 14 mm/month RMSE. In terms of CC, downscaled results achieve an average value of -0.81 versus -0.73 for the input. The proposed Bayesian framework effectively downscales GRACE data, with performance highly dependent on input data quality. The copula-supported Bayesian approach offers valuable uncertainty quantification even with inconsistent input data. This method aids in understanding water storage variations in small catchments, supporting local hydrological studies, and can be applied to other water cycle parameters as an alternative to traditional methods. Although a direct posterior is obtained for each grid cell in the downscaling study, spatial dependencies among neighboring grid cells are not considered. Graphical models are particularly well-suited for capturing such spatial dependencies. To address this limitation - and inspired by the challenge of noisy water level estimates from satellite altimetry over inland water bodies - the third study presents a Bayesian approach that formulates a probabilistic graphical model known as a Markov Random Field (MRF), with a Maximum A Posteriori estimation of the MRF (MRF-MAP) as the objective. There to improve inland altimetry, a retracking method is proposed. Unlike conventional retracking methods that target a single waveform point, a holistic approach by identifying retracking lines within 2D radargrams, treating the radargram as a segmented image. This segmentation divides the radargram into Front and Back segments, resembling a binary image segmentation task. The proposed MRF-MAP framework uses spatial dependencies as prior information, with the likelihood based on the temporal evolution of pixel labels across groundtrack cycles. Two temporal energy functions are applied: 1D, based solely on pixel intensity, and 2D, which includes both intensity and bin values, with the posterior probability maximized using the maxflow algorithm. The maxflow algorithm is then applied to obtain MAP solution, yielding a segmented radargram where the retracking line is defined as the boundary between segments. The proposed retracker method is applied to both pulse-limited and SAR altimetry datasets across nine U.S. lakes and reservoirs with varying altimetry characteristics. Validated against in situ data, the proposed method improves RMSE by approximately 0.25 m with the 1D temporal energy function and 0.51 m with the 2D function. The main advantage of the proposed method is its robustness against unexpected waveform variations, making it especially valuable for complex radargrams where conventional retrackers often deliver outliers. By integrating both spatial and temporal information, this method offers a more comprehensive understanding of the data and has broad applicability, such as improving the classification of SWOT pixel cloud points by incorporating spatiotemporal detail. Through these case studies, the thesis illustrates the advantages of Bayesian approaches in improving the accuracy and reliability of hydrological estimates - such as river discharge, terrestrial water storage, and water level measurements - derived from spaceborne geodetic sensors. By integrating theoretical insights with practical applications, the thesis demonstrates how Bayesian methods can effectively improve spatiotemporal resolution, obtain uncertainties, enhance data fusion, and accommodate the complexities inherent in hydrological systems. This combination of foundational knowledge and real-world examples shall establish a base for advancing the use of Bayesian approaches in hydrogeodetic research and beyond. Moreover, by highlighting the challenges in hydrogeodesy, this thesis provides a clear direction for future research and development in the field. It emphasizes critical areas requiring attention, such as improving the spatial and temporal resolution of hydrological estimates, addressing inherent uncertainties in geodetic observations, and developing more effective methods for assimilating diverse data sources. The thesis encourages the refinement of geodetic data processing techniques and the adoption of probabilistic frameworks, such as Bayesian modeling, in future work. Building upon the work presented here, future studies can ultimately achieve more accurate and reliable insights into the Earth's systems.