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No / In evaluating the resistance of sediment particles to entrainment by the action of the flow in a river, the grain geometry is usually characterized using representative sizes. This approach has been dictated, initially by lack of physical insight, but more recently by the lack of analytical tools able to describe the 3-D nature of surface grain organization on water-worked sediment beds. Laboratory experiments are presented where mixed grain size beds were mobilized under a range of hydraulic and sediment input conditions. Detailed bed topography was measured at various stages. Statistical tools have been adopted which describe the degree of surface organization on water-worked sediment bed surfaces. The degree of particle organization and the bed stability can be evaluated in relative terms using the properties of the probability density distribution of the bed surface elevations and in absolute terms using a properly defined 2-D structure function. The methods described can be applied directly to natural water-worked surfaces given the availability of appropriate bed surface elevation data sets.

Bed topography identification in open channel and glacier flows is of paramount importance for the study of the respective flows. In the former, the knowledge of the channel bed topography is required for modelling the hydrodynamics of open channel flows, fluvial hydraulics, flood propagation, and river flow monitoring. Indeed, flow models based on the Shallow Water Approximation require prior information on the channel bed topography to accurately capture the flow features. While in the latter, usable bedrock topographic information is very important for glacier flow modellers to accurately predict the flow characteristics. Experimental techniques to infer the bed topography are usually used but are mostly time consuming, costly, and sometimes not possible due to geographical restrictions. However, the measurement of free surface elevation is relatively easy. Alternative to experimental techniques, it is therefore important to develop fast, easy-to-implement, and cost-effective numerical methods. The inverse of the classical hydrodynamic problem corresponds to the determination of hydraulic parameters from measurable quantities. The forward problem uses model parameters to determine measurable quantities. New one-shot and direct pseudo-analytical and numerical approaches for reconstructing the channel bed topography from known free surface elevation data is developed for one-dimensional shallow water flows. It is shown in this work that instead of treating this inverse problem in the traditional partial differential equation (PDE)-constrained optimization framework, the governing equations of the direct problem can be conveniently rearranged to obtain an explicit PDE for the inverse problem. This leads to a direct solution of the inverse problem which is successfully tested on a range of benchmark problems and experimental data for noisy and noiseless free surface data. It was found that this solution approach creates very little amplification of noise. A numerical technique which uses the measured free surface velocity to infer the channel bed topography is also developed. The one-dimensional shallow water equations along with an empirical relationship between the free surface and the depth averaged velocities are used for the inverse problem analysis. It is shown that after a series of algebraic manipulation and integration, the equation governing the inverse problem simplifies to a simple integral equation. The proposed method is tested on a range of analytical and experimental benchmark test cases and the results confirm that, it is possible to reconstruct the channel bed topography from a known free surface velocity distribution of one-dimensional open channel flows. Following the analysis of the case of one-dimensional shallow water flows, a numerical technique for reconstructing the channel bed topography from known free surface elevation data for steep open channel flows is developed using a modified set of equations for which the zero-inertia shallow water approximation holds. In this context, the shallow water equations are modified by neglecting inertia terms while retaining the effects of the bed slope and friction terms. The governing equations are recast into a single first-order partial differential equation which describes the inverse problem. Interestingly, the analysis shows that the inverse problem does not require the knowledge of the bed roughness. The forward problem is solved using MacCormack’s explicit numerical scheme by considering unsteady modified shallow water equations. However, the inverse problem is solved using the method of characteristics. The results of the inverse and the forward problem are successfully tested against each other. In the framework of full two-dimensional shallow water equations, an easy-to-implement and fast to solve direct numerical technique is developed to solve the inverse problem of shallow open channel flows. The main underlying idea is analogous to the idea implemented for the case of one-dimensional reconstruction. The technique described is a “one-shot technique” in the sense that the solution of the partial differential equation provides the solution to the inverse problem directly. The idea is tested on a set of artificial data obtained by first solving the forward problem. Glaciers are very important as an indicator of future climate change or to trace past climate. They respond quickly compared to the Antarctica and Greenland ice sheets which make them ideal to predict climate changes. Glacier bedrock topography is an important parameter in glacier flow modelling to accurately capture its flow dynamics. Thus, a mathematical technique to infer this parameter from measured free surface data is invaluable. Analogous to the approaches implemented for open channel flows, easy-to-implement direct numerical and analytical algorithms are developed to infer the bedrock topography from the knowledge of the free surface elevation in one space dimension. The numerical and analytical methods are both based on the Shallow Ice Approximation and require the time series of the ablation/accumulation rate distribution. Moreover, the analytical method requires the knowledge of a non-zero glacier thickness at an arbitrary location. Numerical benchmark test cases are used to verify the suitability and applicability of the algorithms.

The increased temperatures in the Arctic accelerate the loss of land based ice stored in glaciers. The Greenland Ice Sheet is the largest ice mass in the Northern Hemisphere and holds ~10% of all the freshwater on Earth, equivalent to ~7 metres of global sea level rise. A few decades ago, the mass balance of the Greenland Ice Sheet was poorly known and assumed to have little impact on global sea level rise. The development of regional climate models and remote sensing of the ice sheet during the past decade have revealed a significant mass loss. To monitor how the Greenland Ice Sheet will affect sea levels in the future requires understanding the physical processes that govern its mass balance and movement. In the southeastern and central western regions, mass loss is dominated by the dynamic behaviour of ice streams calving into the ocean. Changes in surface mass balance dominate mass loss from the Greenland Ice Sheet in the central northern, southwestern and northeastern regions. Little is known about what the hydrological system looks like beneath the ice sheet; how well the hydrological system is developed decides the water’s impact on ice movement. In this thesis, I have focused on radar sounding measurements to map the subglacial topography in detail for a land-terminating section of the western Greenland Ice Sheet. This knowledge is a critical prerequisite for any subglacial hydrological modelling. Using the high-resolution ice thickness and bed topography data, I have made the following specific studies: First, I have analysed the geological setting and glaciological history of the region by comparing proglacial and subglacial spectral roughness. Second, I have analysed the subglacial water drainage routing and revealed a potential for subglacial water piracy between adjacent subglacial water catchments with changes in the subglacial water pressure regime. Finally, I have looked in more detail into englacial features that are commonly observed in radar sounding data from western Greenland. In all, the thesis highlights the need not only for accurate high-resolution subglacial digital elevation models, but also for regionally optimised interpolation when conducting detailed hydrological studies of the Greenland Ice Sheet. / De ökade temperaturerna i Arktis påskyndar förlusten av landbaserad is lagrad i glaciärer och permafrost. Grönlands inlandsis är den största ismassan på norra halvklotet och lagrar ca 10% av allt sötvatten på jorden, vilket motsvarar ca 7 meter global havsnivåhöjning. För ett par decennier sedan var inlandsisens massbalans dåligt känd och antogs ha liten inverkan på dagens havsnivåhöjning. Utvecklingen av regionala klimatmodeller och satellitbaserad fjärranalys av inlandsisen har under de senaste decenniet påvisat en betydande massförlust. För att förutse vilken inverkan inlandsisen har på framtida havsnivåhöjningar krävs en förståelse för de fysikaliska processerna som styr dess massbalans och isrörelse. I de sydöstra och centrala västra delarna av inlandsisen domineras massförlusten av dynamiska processer i isströmmar som kalvar ut i havet. Massförlusten i de centrala norra, sydvästra och nordöstra delarna domineras av isytans massbalans. Ytterst lite är känt om hur det hydrologiska systemet ser ut under inlandsisen; hur väl det hydrologiska systemet är utvecklat avgör vattnets påverkan på isrörelsen. I denna doktorsavhandling har jag använt markbaserade radarmätningar för att kartlägga den subglaciala topografin för en del av den västra landbaserade inlandsisen. Denna kunskap är en viktig förutsättning för att kunna modellera den subglaciala hydrologin. Med hjälp av rumsligt högupplöst data över istjockleken och bottentopografin har jag gjort följande specifika studier: Först har jag analyserat de geologiska och glaciologiska förhållandena i regionen genom att jämföra proglacial och subglacial spektralanalys av terrängens ytojämnheter. Sedan har jag analyserat den subglaciala vattenavrinningen och påvisat en potential för att avrinningsområdena kan ändras beroende på vattentryckförhållandena på botten. Slutligen har jag tittat mer i detalj på englaciala radarstrukturer som ofta observerats i radardata från västra Grönland. Sammanfattningsvis belyser avhandlingen behovet av inte bara noggranna rumsligt högupplösta subglaciala digitala höjdmodeller, utan även regionalt optimerad interpolering när detaljerade hydrologiska studier ska utföras på Grönlands inlandsis.