The heterogeneous Helmholtz problem with spherical symmetry: Green’s operator and stability estimates

We study wave propagation phenomena modelled in the frequency domain by the Helmholtz equation in heterogeneous media with focus on media with discontinuous, highly oscillating wave speed. We restrict to problems with spherical symmetry and will derive explicit representations of the Green’s operator and stability estimates which are explicit in the frequency and the wave speed.

Efficient gravity inversion of basement relief using a versatile modeling algorithm

We have developed a novel approach to compute, in an efficient and versatile way, the gravity anomaly produced by an arbitrary, discrete 3D distribution of density contrast. The method allows adjustable precision and is particularly suited for the interpretation of sedimentary basins. Because the gravity field decays with the square of the distance, we use a discrete Green’s operator that may be made much smaller than the whole study area. For irregularly positioned observations, this discrete Green’s operator must be computed just at the first iteration, and because each of its horizontal layers presents a center of symmetry, only one-eighth of its total elements need to be calculated. Furthermore, for gridded data on a plane, this operator develops translation symmetry for the whole region of interest and has to be computed just once for a single arbitrary observation position. The gravity anomaly is obtained as the product of this small operator by any arbitrary density contrast distribution in a convolution-like operation. We use the proposed modeling to estimate the basement relief of a [Formula: see text] basin with density contrast varying along [Formula: see text] only using a smaller Green’s operator at all iterations. The median of the absolute differences between relief estimates, using the small and a large operator (the latter covering the whole basin) has been approximately 9 m for a 3.6 km deep basin. We also successfully inverted the anomaly of a simulated basin with density contrast varying laterally and vertically, and a real anomaly produced by a steeply dipping basement. The proposed modeling is very fast. For 10,000 observations gridded on a plane, the inversion using the proposed approach for irregularly spaced data is two orders of magnitude faster than using an analytically derived fitting, and this ratio increases enormously with the number of observations.

Some Weighted Norm Estimates for the Composition of the Homotopy and Green’s Operator

We establish theAr(D)-weighted integral inequality for the composition of the HomotopyTand Green’s operatorGon a bounded convex domain and also motivated it to the global domain by the Whitney cover. At the same time, we also obtain some(p,q)-type norm inequalities. Finally, as applications of above results, we obtain the upper bound for theLpnorms ofT(G(u))or(T(G(u)))Bin terms ofLqnorms ofuordu.