Assimilation of spatially distributed water levels into a shallow-water flood model. Part I: Mathematical method and test case

International audienceRecent applications of remote sensing techniques produce rich spatially distributed observations for flood monitoring. In order to improve numerical flood prediction, we have developed a variational data assimilation method (4D-var) that combines remote sensing data (spatially distributed water levels extracted from spatial images) and a 2D shallow water model. In the present paper (part I), we demon- strate the efficiency of the method with a test case. First, we assimilated a single fully observed water level image to identify time-independent parameters (e.g. Manning coefficients and initial conditions) and time-dependent parameters (e.g. inflow). Second, we combined incomplete observations (a time ser- ies of water elevations at certain points and one partial image). This last configuration was very similar to the real case we analyze in a forthcoming paper (part II). In addition, a temporal strategy with time over- lapping is suggested to decrease the amount of memory required for long-duration simulation

Adjoint accuracy for the full-Stokes ice flow model: limits to the transmission of basal friction variability to the surface

International audienceThis work focuses on the numerical assessment of the accuracy of an adjoint-based gradient in the perspective of variational data assimilation and parameter identification in glaciology. Using noisy synthetic data, we quantify the ability to identify the friction coefficient for such methods with a non-linear friction law. The exact adjoint problem is solved, based on second order numerical schemes, and a comparison with the so called ''self-adjoint'' approximation, neglecting the viscosity dependency to the velocity (leading to an incorrect gradient), common in glaciology, is carried out. For data with a noise of $1\%$, a lower bound of identifiable wavelengths of $10$ ice thicknesses in the friction coefficient is established, when using the exact adjoint method, while the ''self-adjoint'' method is limited, even for lower noise, to a minimum of $20$ ice thicknesses wavelengths. The second order exact gradient method therefore provides robustness and reliability for the parameter identification process. In other respect, the derivation of the adjoint model using algorithmic differentiation leads to formulate a generalization of the ''self-adjoint'' approximation towards an incomplete adjoint method, adjustable in precision and computational burden

Multi-Regime Shallow Free-Surface Flow Models for Quasi-Newtonian Fluids

International audienceThe mathematical modeling of thin free-surface laminar flows for quasi-Newtonian fluids (power-law rheology) is addressed with a particular attention to geophysical flows (e.g. ice or lava flows). Asymptotic thin-layer flow models (one-equation and two-equation models) consistent with various viscous regimes, corresponding to different basal boundary conditions (from adherence to pure slip), are derived. The challenge being to derive models consistent from slip to no-slip basal boundary condition, though at the price of balancing small friction by small mean slope. Starting from reference flows (the steady-state uniform ones) corresponding to different shear regimes, the exact expressions of all fields (\bsigma, \bu, p) are calculated formally by a perturbation expansion method.The calculations are such that all field expressions remain valid for any laminar viscous regimes. The calculations are presented either in a mean slope coordinatesystem with local variations of the topography or in the Prandtl coordinate system, hence valid in presence of any non flat basal topography.Formal error estimates proving the consistency of the derivations are stated. An unified one-equation model (lubrication type in the depth variable $h$) is derived at order $1$. Next, few unified two-equation models in variable $(q,h)$ (shallow water type) are stated and discussed.The classical first order models from the literature are recovered if considering the corresponding particular cases (generally, flat bottom with a particular regime and/or specific basal boundary condition). Two one-dimensional numerical examples illustrate the robustness of these new multi-regime formulations (the change of flow regimes being due either to a sharp change of the mean-slope topography or to a sharp change of basal boundary condition)

DassFlow v1.0: a variational data assimilation software for river flows

Dassflow is a computational software for river hydraulics (floods), especially designed for variational data assimilation. The forward model is based on the bidimensional shallow-water equations, solved by a finite volume method (HLLC approximate Riemann solver). It is written in Fortran 95. The adjoint code is generated by the automatic differentiation tool Tapenade. Thus, Dassflow software includes the forward solver, its adjoint code, the full optimization framework (based on the M1QN3 minimization routine) and benchmarks. The generation of new data assimilation twin experiments is easy. The software is interfaced with few pre and post-processors (mesh generators, GIS tools and visualization tools), which allows to treat real data

DassFow-Shallow, Variational Data Assimilation for Shallow-Water Models: Numerical Schemes, User and Developer Guides

DassFlow is a computational software for free-surface flows includingvariational data assimilation (4D-VAR), sensitivity analysis, calibration features (adjoint method). The code version "shallow" solves shallow-water like models (Saint-Venant's type).The other version (ALE, not detailed in the present document) includes free-surface Stokes like models (low Reynolds, power-law rheology, ALE surface dynamics). All source files are written in Fortran 2003 / MPI. For more details and references, please consult DassFlow website.In the present manuscript, we describe: the equations, the compilation/execution instructions, the input / output files (user guide), the finite volume schemes, few validation test cases included in the archive, and the code structure (developer guide)

Existence and uniqueness of solution to an adaptive elasticity model

International audienceIn this work we study the existence and unicity of solutions to an {\sl Adaptive Elasticity Model} applied to bone remodeling. Specifically, we consider the model derived by Cowin and Hegedus, directly from continuum mechanics theory. We use a fixed point argument in order to prove the existence of solutions and a straightforward adaptation of the Cowin and Nachlinger analysis in order to prove uniqueness

Robust finite volume schemes for 2D shallow water models. Application to flood plain dynamics

This study proposes original combinations of higher order Godunov type finite volume schemes and time discretization schemes for the 2d shallow water equations, leading to fully second-order accuracy with well-balanced property. Also accuracy, positiveness and stability properties in presence of dynamic wet/dry fronts is demonstrated. The test cases are the classical ones plus extra new ones representing the geophysical flow features and difficulties

Inverse computational algorithms for flood plain dynamic modelling

Flood plain dynamic modelling remains a challenge because of the complex multi-scale data, data uncertainties and the uncertain heterogeneous flow measurements. Mathematical models based on the 2d shallow water equations are generally suitable but wetting-drying processes can be driven by small scale data features. The present study aims at deriving an accurate and robust direct solver for dynamic wet-dry fronts and a variational inverse method leading to sensitivity analyses and data assimilation processes. The numerical schemes and algorithms are assessed on academic benchmarks representing well some flood dynamic features and a real test case (Lèze river, southwestern of France). Original sensitivity maps with respect to the (friction , topography) fields are performed and discussed. Furthermore, the identification of inflow discharges (time series) or roughness coefficients defined by land covers (spatially distributed parameters) demonstrate the relevance of the approach and the algorithm efficiency. Inverse computational methods may contribute to breakthrough in flood plain modelling

DassFlow v1.0: a variational data assimilation software for river flows

Dassflow is a computational software for river hydraulics (floods), especially designed for variational data assimilation. The forward model is based on the bidimensional shallow-water equations, solved by a finite volume method (HLLC approximate Riemann solver). It is written in Fortran 95. The adjoint code is generated by the automatic differentiation tool Tapenade. Thus, Dassflow software includes the forward solver, its adjoint code, the full optimization framework (based on the M1QN3 minimization routine) and benchmarks. The generation of new data assimilation twin experiments is easy. The software is interfaced with few pre and post-processors (mesh generators, GIS tools and visualization tools), which allows to treat real data

Variational data assimilation for 2D fluvial hydraulics simulations

International audienceA numerical method for model parameters identification is presented for a river model based on a finite volume discretization of the bidimensional shallow water equations. We use variational data assimilation to combine optimally physical information from the model and observation data of the physical system in order to identify the value of model inputs that correspond to a numerical simulation which is consistent with reality. Two numerical examples demonstrate the efficiency of the method for the identification of the inlet discharge and the bed elevation. An application to real data on the Pearl River for the identification of boundary conditions is presented