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Abstract: We revisit the problem of recovering wave speeds and density across a curved interface from reflected wave amplitudes. Such amplitudes have been exploited for decades in (exploration) seismology in this context. However, the analysis in seismology has been based on linearization and mostly flat interfaces. Here, we present an analysis without linearization and allow curved interfaces, establish uniqueness and provide a reconstruction, while making the notion of amplitude precise through a procedure rooted in microlocal analysis. PubDate: 2022-05-13

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Abstract: After a short introduction on the historical context, the paper deals with the existence of the solution of Parker’s ideal body problem, namely the body of minimum constant density generating a given external potential. A crucial element of the proof is the use of a recently introduced topological space of closed sets, closed and compact with the distance defined as the Lebesgue measure of the symmetric difference of a couple of sets. Such a space is indeed smaller than that of all closed sets of a given B, but larger than that of star-shaped Lipschitz domains, where previous studies of the inverse gravimetric problem (with constant density) have been conducted. However, with the present knowledge, it is only in this class that a uniqueness theorem holds. PubDate: 2022-05-11

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Abstract: Abstract Calibration with respect to a bottom friction parameter is standard practice within numerical coastal ocean modelling. However, when this parameter is assumed to vary spatially, any calibration approach must address the issue of overfitting. In this work, we derive calibration problems in which the control parameters can be directly constrained by available observations, without overfitting. This is achieved by carefully selecting the ‘experiment design’, which in general encompasses both the observation strategy, and the choice of control parameters (i.e. the spatial variation of the friction field). In this work we focus on the latter, utilising existing observations available within our case study regions. We adapt a technique from the optimal experiment design (OED) literature, utilising model sensitivities computed via an adjoint-capable numerical shallow water model, Thetis. The OED method uses the model sensitivity to estimate the covariance of the estimated parameters corresponding to a given experiment design, without solving the corresponding parameter estimation problem. This facilitates the exploration of a large number of such experiment designs, to find the design producing the tightest parameter constraints. We take the Bristol Channel as a primary case study, using tide gauge data to estimate friction parameters corresponding to a piecewise-constant field. We first demonstrate that the OED framework produces reliable estimates of the parameter covariance, by comparison with results from a Bayesian inference algorithm. We subsequently demonstrate that solving an ‘optimal’ calibration problem leads to good model performance against both calibration and validation data, thus avoiding overfitting. PubDate: 2022-04-30

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Abstract: Abstract Highly conductive thin casings pose a great challenge in the numerical simulation of well-logging instruments. Witty asymptotic models may replace the presence of casings by impedance transmission conditions in those numerical simulations. The accuracy of such numerical schemes can be tested against benchmark solutions computed semi-analytically in simple geometrical configurations. This paper provides a general approach to construct those benchmark solutions for three different models: one reference model that indeed considers the presence of the casing; one asymptotic model that avoids computations in the casing domain; and one asymptotic model that reduces the presence of the casing to an interface. Our technique uses a Fourier representation of the solutions, where special care has been taken in the analytical integration of singularities to avoid numerical instabilities. PubDate: 2022-04-29

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Abstract: Abstract The present paper deals with Rayleigh wave propagation in a homogeneous isotropic nonlocal magneto-thermoelastic solid with hall current and rotation. The considered thermoelastic solid is subjected to multi-dual-phase lag heat transfer. The Secular equations of Rayleigh waves are derived mathematically at the stress-free and thermally insulated boundaries. The values of stress components, temperature change, phase velocity, attenuation coefficient, penetration depth and specific loss have been computed numerically and depicted graphically. The effects of hall current and nonlocal parameter have been depicted on the various wave quantities. Some particular cases have also been deduced from the present investigation. PubDate: 2022-04-01

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Abstract: Abstract A variational principle is established by the semi-inverse method and used to solve approximately a nonlinear problem by the Ritz method. In this process,it may be difficult to solve a large system of algebraic equations,the Groebner bases theory (Buchberger’s algorithm) is applied to solve this problem. The results show that the variational approach is much simpler and more efficient. PubDate: 2022-02-04 DOI: 10.1007/s13137-022-00194-6

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Abstract: Abstract Vertical infiltration of water plays an important role in the recharged of contaminated water and enhanced moisture content in the unsaturated porous media. The mathematical model used for such type of phenomenon is Burger's equation. Unsaturated porous media are analyzed by solving Burger's equation using the variational iterative modeling and homotopy perturbation method. When considering all moisture contents, it appears that the cumulative coefficient is unchanged. It is also shown that the soil's moisture content decreases with depth (y) and time (t). The results indicate that this method is very efficient and can be useful to solve large-scale problems that arise in civil engineering, geology, material science, and fossil fuel problems. PubDate: 2022-01-06 DOI: 10.1007/s13137-021-00193-z

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Abstract: Abstract Hydro-morphodynamic modelling is an important tool that can be used in the protection of coastal zones. The models can be required to resolve spatial scales ranging from sub-metre to hundreds of kilometres and are computationally expensive. In this work, we apply mesh movement methods to a depth-averaged hydro-morphodynamic model for the first time, in order to tackle both these issues. Mesh movement methods are particularly well-suited to coastal problems as they allow the mesh to move in response to evolving flow and morphology structures. This new capability is demonstrated using test cases that exhibit complex evolving bathymetries and have moving wet-dry interfaces. In order to be able to simulate sediment transport in wet-dry domains, a new conservative discretisation approach has been developed as part of this work, as well as a sediment slide mechanism. For all test cases, we demonstrate how mesh movement methods can be used to reduce discretisation error and computational cost. We also show that the optimum parameter choices in the mesh movement monitor functions are fairly predictable based upon the physical characteristics of the test case, facilitating the use of mesh movement methods on further problems. PubDate: 2021-12-09 DOI: 10.1007/s13137-021-00191-1

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Abstract: Abstract Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution of such problems can be non trivial, both from a mathematical and a geometrical standpoint. Indeed 3D-1D coupling requires to operate in non standard function spaces, and, also, simulation geometries can be complex for the presence of multiple intersecting domains. Recently, a PDE-constrained optimization based formulation has been proposed for such problems, proving a well posed mathematical formulation and allowing for the use of non conforming meshes for the discrete problem. Here an unconstrained optimization formulation of the problem is derived and an efficient gradient based solver is proposed for such formulation, along with a suitable preconditioner to speed up the iterative solver. Some numerical tests on quite complex configurations are discussed to show the viability of the method . PubDate: 2021-12-08 DOI: 10.1007/s13137-021-00192-0

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Abstract: Abstract In the current work, we propose the fractal Whitham–Broer–Kaup equation which can well describe the propagation of shallow water travelling along unsmooth boundary (such as the fractal seabed). By the Semi-inverse method, we establish its fractal variational principle, which is proved to have a strong minimum condition by He–Weierstrass theorem. Then the fractal variational method is used to seek its solitary wave solution. The impact of the fractal order on the behaviors of the solitary wave is presented through the 3-D plots and 2-D curves. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences. PubDate: 2021-10-22 DOI: 10.1007/s13137-021-00189-9

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Abstract: Abstract A mathematical model is considered for Rayleigh–Bénard convection of mantle where the viscosity depends strongly on both temperature and pressure defined in an Arrhenius form. The model is solved numerically for extremely large viscosity variations across a unit aspect ratio cell using a modified cut-off viscosity law, and steady solutions are obtained. The aim is to investigate the convection pattern with internal heating at a very high viscosity variation in the presence of high Rayleigh number. The study also investigates the relation between temperature dependent parameter and pressure dependent parameter in a basally heated convection cell. The numerical simulation is performed using the finite element method based PDE solver and the results are presented through figures, tables and graphs. PubDate: 2021-10-22 DOI: 10.1007/s13137-021-00190-2

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Abstract: Abstract This work presents a new extension to B-Splines that enables them to model functions on directed tree graphs such as non-braided river networks. The main challenge of the application of B-splines to graphs is their definition in the neighbourhood of nodes with more than two incident edges. Achieving that the B-splines are continuous at these points is non-trivial. For both, simplification reasons and in view of our application, we limit the graphs to directed tree graphs. To fulfil the requirement of continuity, the knots defining the B-Splines need to be located symmetrically along the edges with the same direction. With such defined B-Splines, we approximate the topography of the Mekong River system from scattered height data along the river. To this end, we first test and validate successfully the method with synthetic water level data, with and without added annual signal. The quality of the resulting heights is assessed besides others by means of root mean square errors (RMSE) and mean absolute differences (MAD). The RMSE values are 0.26 m and 1.05 m without and with added annual variation respectively and the MAD values are even lower with 0.11 m and 0.60 m. For the second test, we use real water level observations measured by satellite altimetry. Again, we successfully estimate the river topography, but also discuss the short comings and problems with unevenly distributed data. The unevenly distributed data leads to some very large outliers close to the upstream ends of the rivers tributaries and in regions with rapidly changing topography such as the Mekong Falls. Without the outlier removal the standard deviation of the resulting heights can be as large as 50 m with a mean value of 15.73 m. After the outlier removal the mean standard deviation drops to 8.34 m. PubDate: 2021-09-22 DOI: 10.1007/s13137-021-00188-w

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Abstract: Abstract This paper tackles the issue of the computational load encountered in seismic imaging by Bayesian traveltime inversion. In Bayesian inference, the exploration of the posterior distribution of the velocity model parameters requires extensive sampling. The computational cost of this sampling step can be prohibitive when the first arrival traveltime prediction involves the resolution of an expensive number of forward models based on the eikonal equation. We propose to rely on polynomial chaos surrogates of the traveltimes between sources and receivers to alleviate the computational burden of solving the eikonal equation during the sampling stage. In an offline stage, the approach builds a functional approximation of the traveltimes from a set of solutions of the eikonal equation corresponding to a few values of the velocity model parameters selected in their prior range. These surrogates then substitute the eikonal-based predictions in the posterior evaluation, enabling very efficient extensive sampling of Bayesian posterior, for instance, by a Markov Chain Monte Carlo algorithm. We demonstrate the potential of the approach using numerical experiments on the inference of two-dimensional domains with layered velocity models and different acquisition geometries (microseismic and seismic refraction contexts). The results show that, in our experiments, the number of eikonal model evaluations required to construct accurate surrogates of the traveltimes is low. Further, an accurate and complete characterization of the posterior distribution of the velocity model is possible, thanks to the generation of large sample sets at a low cost. Finally, we discuss the extension of the current approach to more realistic velocity models and operational situations. PubDate: 2021-09-21 DOI: 10.1007/s13137-021-00184-0

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Abstract: Abstract Using one and two dimensional steady and advective nonlinear diffusion equation as examples, both, statistical and deterministic inversion approaches are explored. Inversions for material parameters, including but not limited to pure geometrical reconstructions or pointwise parameter updates, are presented. Following a gentle theoretical introduction to the inverse problems multiple algorithms, ranging from basic sampling to Hamiltonian Monte Carlo Markov Chains, are explored in terms of their implementation, area of application and performance. Additionally, a discussion of two sensitivity analysis methods using derivative information to gain deeper insights into the dynamics of the forward problem is given. PubDate: 2021-09-14 DOI: 10.1007/s13137-021-00186-y

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Abstract: Abstract Porous media have been a significant subject of research for a long time due to their applicability in various sciences. This paper investigates the heat transfer phenomenon in the porous media. A convergent solution is obtained for a two-dimensional time-fractional equation arising in a porous soil heat transfer. He's polynomial and He's variational iteration method are used to accomplish the required goals. The fractional derivative used in the article is described by He’s definition. He's fractional complex transform is used to convert the fractional differential equation into its traditional partner differential equation, which can be solved iteratively. Graphical representations of the results are provided to demonstrate the efficacy of the methods used. PubDate: 2021-09-08 DOI: 10.1007/s13137-021-00187-x

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Abstract: Abstract This work presents a hybrid modeling approach to data-driven learning and representation of unknown physical processes and closure parameterizations. These hybrid models are suitable for situations where the mechanistic description of dynamics of some variables is unknown, but reasonably accurate observational data can be obtained for the evolution of the state of the system. In this work, we propose machine learning to account for missing physics and then data assimilation to correct the prediction. In particular, we devise an effective methodology based on a recurrent neural network to model the unknown dynamics. A long short-term memory (LSTM) based correction term is added to the predictive model in order to take into account hidden physics. Since LSTM introduces a black-box approach for the unknown part of the model, we investigate whether the proposed hybrid neural-physical model can be further corrected through a sequential data assimilation step. We apply this framework to the weakly nonlinear Lorenz model that displays quasiperiodic oscillations, the highly nonlinear chaotic Lorenz model, and two-scale Lorenz model. The hybrid neural-physics model yields accurate results for the weakly nonlinear Lorenz model with the predicted state close to the true Lorenz model trajectory. For the highly nonlinear Lorenz model and the two-scale Lorenz model, the hybrid neural-physics model deviates from the true state due to the accumulation of prediction error from one time step to the next time step. The ensemble Kalman filter approach takes into account the prediction error and updates the diverged prediction using available observations in order to provide a more accurate state estimate for the highly nonlinear chaotic Lorenz model and two-scale Lorenz system. The successful synergistic integration of neural network and data assimilation for low-dimensional system shows the potential benefits of the proposed hybrid-neural physics model for complex dynamical systems. PubDate: 2021-09-04 DOI: 10.1007/s13137-021-00185-z

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Abstract: Abstract In this survey paper, we present a multiscale post-processing method in exploration. Based on a physically relevant mollifier technique involving the elasto-oscillatory Cauchy–Navier equation, we mathematically describe the extractable information within 3D geological models obtained by migration as is commonly used for geophysical exploration purposes. More explicitly, the developed multiscale approach extracts and visualizes structural features inherently available in signature bands of certain geological formations such as aquifers, salt domes etc. by specifying suitable wavelet bands. PubDate: 2021-07-30 DOI: 10.1007/s13137-021-00179-x

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Abstract: Abstract We propose a new procedure for obtaining thermal properties (thermal conductivity, diffusivity, and capacity) of marine sediments from the heat-pulse Lister-type probe method. Established methods suffered from ill-posed inversion when aiming for the determination of sediment thermal diffusivities. Our new approach is based on the first-order Taylor expansion of the temperature decay function as before, but including important modifications/extensions: (1) we introduce the thermal contact resistance into the formulation of the temperature decay function, (2) we improve numerical properties of the inversion procedure itself by introducing a regularization method including constraints, and (3) we use an adaptive choice of time frames for thermal decay inversion. Here, we demonstrate that integrating the thermal contact resistance into the procedure improves inversion results for the thermal diffusivity considerably. Moreover, we show that including regularization and constraints based on a-priori knowledge of the admissible range of the corresponding parameter values leads to stable inversion results. PubDate: 2021-07-29 DOI: 10.1007/s13137-021-00183-1

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Abstract: Abstract The aim of this article focuses on deriving a fractal modification of the nonlinear oscillator in a porous foundation vibration. The variational principle and frequency formula of the fractal nonlinear equation are found by simple methods that were proposed by Ji-Huan He. In order to make the error smaller, we modify the approximate analytical solution expression and prove its effectiveness through examples. PubDate: 2021-07-24 DOI: 10.1007/s13137-021-00181-3

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Abstract: Abstract Kadomtsev–Petviashvili (KP) equation is an important fluid equation. In the framework of Lax scheme, this paper first derives a new and more general variable-coefficient KP (vcKP) equation. Then the inverse scattering transform (IST) and the F-expansion method are respectively used to construct exact wave solutions of a special case of the derived vcKP equation and its extended case in which the constraint relationships between the coefficient functions are further weakened. The obtained wave solutions include rational solutions, which are constructed not only by the IST but also by the F-expansion method, and Jacobi elliptic function solutions. In the limit cases of the modulus parameter, the obtained Jacobi elliptic function solutions degenerate into hyperbolic function solutions and trigonometric function solutions as well as rational solutions. To gain more insights into some of the obtained solutions, dynamical evolution with novle features like the Z-shaped flat like-lump soliton and the inclined double periodic wave of whose are shown directly by pictures. The significance of this paper is to extend IST to the high-dimensional models with variable coefficients by taking the vcKP equation as an example, and obtain some new solutions and their novel solution structres of the vcKP equation by using the IST and the F-expansion method. PubDate: 2021-07-14 DOI: 10.1007/s13137-021-00182-2