Forward modelling, or, in other words, numerical solution of partial differential equations, is an essential part of a number of geophysical applications. Numerical solutions, in turn, are very often computed with use of the finite differences method. The method has many advantages such as simplicity, controlable error bounds, considerably modest memory usage. At the same time, the method posses a number of drawbacks such as compulsory preference for uniform grids, problematic with boundary conditions for higher order stencils, large geometrical error and weak approximating features. Therefore, we consider an competitive alternative to traditional grid based finite difference method, that is the class of the methods referred as grid-free or meshless kernel-based techniques. These methods, in general, do not require any predefined structures of computational nodes or grids, i.e. they can be applied in cases of irregular distribution of the nodes in computational domains with arbitrary shapes. These methods are able to deal effectively with various “non-classical” boundary conditions, and to adapt easily to multidimensional problems. The method will be illustrated by a few 2D examples.


Numerical solutions of partial differential equations (PDE) are in wide spread use in geophysical applications, such as, for example, full wave-form inversion, tomographic inversions, Green’s function approximation, etc. The solution procedure that is termed as forward modelling (FM), usually occupies a substantial time of the full-wave inversion procedure or some similar task, such as wave propagation modelling. It is clear that any improvement in performance and accuracy of the FM algorithm will directly influence the overall inversion quality and performance in terms of the total run time. Strictly speaking, in the above mentioned applications, one has to deal with a numerical solution of the wave equation derived for inhomogeneous media, or, in other cases, with the reduced wave equation, which is also known as the Helmholtz equation. Moreover, in Green’s function approximation using travel-times data, one has to also obtain the solution of the so-called transport equation for a wave field amplitude.

The typical way to numerically solve linear PDEs is to discretise them, i.e. approximate them by a linear system of equations. There are several ways of PDEs discretisation (Brenner and Scott, 1994; Peterson et al., 1998), which differ (i) on the problem formulation site, i.e. strong and weak problems, and (ii) on the approximation site, i.e. locally-defined polynomials and pointwise linear spaces. The simplest and most established approach is the finite difference (FD) approximations for the partial differential operators, which, in essence, are based on the low order Taylor series expansions, so that the differential operators are approximated by linear combinations of the discrete (pointwise) function values computed at the knots of a discretisation grid. The FD method is simple and straightforward to implement in a computer code. It can directly be applied to simple shape regions such as rectangles over uniform grids (meshes). However, the FD approach also posses a number of disadvantages which are listed below.


It is known that computation of wave propagation problems is limited by the wavelength that the discretisation grid can accurately represent.

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