Extension of spherical nonparametric estimators to nonisotropic kernels: An oceanographic application

In a recently published paper. spherical nonparametric estimators were applied to feature-track ensembles to determine a range of statistics for the atmospheric features considered. This approach obviates the types of bias normally introduced with traditional estimators. New spherical isotropic kernels with local support were introduced. Ln this paper the extension to spherical nonisotropic kernels with local support is introduced, together with a means of obtaining the shape and smoothing parameters in an objective way. The usefulness of spherical nonparametric estimators based on nonisotropic kernels is demonstrated with an application to an oceanographic feature-track ensemble.

The PML for rough surface scattering

In this paper we investigate the use of the perfectly matched layer (PML) to truncate a time harmonic rough surface scattering problem in the direction away from the scatterer. We prove existence and uniqueness of the solution of the truncated problem as well as an error estimate depending on the thickness and composition of the layer. This global error estimate predicts a linear rate of convergence (under some conditions on the relative size of the real and imaginary parts of the PML function) rather than the usual exponential rate. We then consider scattering by a half-space and show that the solution of the PML truncated problem converges globally at most quadratically (up to logarithmic factors), providing support for our general theory. However we also prove exponential convergence on compact subsets. We continue by proposing an iterative correction method for the PML truncated problem and, using our estimate for the PML approximation, prove convergence of this method. Finally we provide some numerical results in 2D.(C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

Hydraulic modeling of runoff over a rough surface under partial inundation

This study presents a numerical method to derive the Darcy- Weisbach friction coefficient for overland flow under partial inundation of surface roughness. To better account for the variable influence of roughness with varying levels of emergence, we model the flow over a network which evolves as the free surface rises. This network is constructed using a height numerical map, based on surface roughness data, and a discrete geometry skeletonization algorithm. By applying a hydraulic model to the flows through this network, local heads, velocities, and Froude and Reynolds numbers over the surface can be estimated. These quantities enable us to analyze the flow and ultimately to derive a bulk friction factor for flow over the entire surface which takes into account local variations in flow quantities. Results demonstrate that although the flow is laminar, head losses are chiefly inertial because of local flow disturbances. The results also emphasize that for conditions of partial inundation, flow resistance varies nonmonotonically but does generally increase with progressive roughness inundation.

A well-posed integral equation formulation for three-dimensional rough surface scattering

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space L-2 (Gamma) when the scattering surface G does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, kappa, for kappa > 0, if the coupling parameter h is chosen proportional to the wave number. In the case when G is a plane, we show that the choice eta = kappa/2 is nearly optimal in terms of minimizing the condition number.

Acoustic scattering by mildly rough unbounded surfaces in three dimensions

For a nonlocally perturbed half- space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single-and double-layer potential but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half- space Green's function. Due to the unboundedness of the surface, the integral operators are noncompact. In contrast to the two-dimensional case, the integral operators are also strongly singular, due to the slow decay at infinity of the fundamental solution of the three-dimensional Helmholtz equation. In the case when the surface is sufficiently smooth ( Lyapunov) we show that the integral operators are nevertheless bounded as operators on L-2(Gamma) and on L-2(Gamma G) boolean AND BC(Gamma) and that the operators depend continuously in norm on the wave number and on G. We further show that for mild roughness, i.e., a surface G which does not differ too much from a plane, the boundary integral equation is uniquely solvable in the space L-2(Gamma) boolean AND BC(Gamma) and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.