Innovations in Wave Processes Modelling and Decision Making

This chapter presents a brief description of chapters devoted to the innovations in wave processes modelling and decision making (grid-characteristic method and applications). Grid-characteristic method is a direct finite-difference numerical method for obtaining full-wave solution of hyperbolic systems of equations. One can use different types of grids such as the regular, triangular, tetrahedral, and nested grids. This method is often used for modelling of the acoustic and isotropic/anisotropic elastic waves in heterogeneous media. Also, the original analytical algorithms for interpolation on the unstructured triangular and tetrahedral meshes are developed. The study of wave processes might be used in different applied areas, e.g. geophysics, non-destructive testing of different objects and materials, ultrasonic testing, seismic stability investigation, and ultrasonic operations modelling. Original investigation of composite materials delamination and non-destructive testing is done. The geological faults zones study is performed. Also, the migration techniques of acoustic and elastic fields are developed.

The chapter develops the analytical formulae for high-order interpolation on the unstructured triangular grids, such as the polynomial interpolation, piecewise linear interpolation, and hybrid interpolation. The interpolation might be used during the creation of new unstructured triangular or regular gird instead of previous ones as an element of numerical method for finding 2D solutions on the unstructured triangular grids and visualization of some 2D field, as well as during the images’ creation or converting. Also the hierarchical nested unstructured triangular grids are discussed in this chapter. This type of grids can be used as an element of numerical method on the unstructured triangular grids for the visualization, creation, and transformation of 2D images. The more the numerical mathematicians work, the faster the software executes and the lesser hardware resources are needed for obtaining the same solution. Analytical formulae reduce the recourses needed, for example the software operation time and amount of dynamic computer memory. In this chapter, one can find the analytical expressions ready for use. The deduced analytical expressions and formed tables help to achieve the huge numerical modelling results in a case of the hardware resources’ deficiency.

Analytical expressions for polynomial interpolations of high-orders are discussed in the chapter. Also in this Chapter several approaches for hybrid interpolation are discussed. Using algorithms for the piecewise linear interpolation one can obtain the hierarchical nested unstructured tetrahedral grids. The recurrent formulae for recalculation of the local reference points’ indices to the global reference points’ indices are discussed in the chapter as well. An example of hybridization called hybrid parabolic—linear interpolation is considered. Another example of hybridization based on a limiter is considered as well. An example of approach based on both hierarchical nested unstructured grids and hybridization called parabolic interpolation on the reference points for interpolation of fourth order is offered in the chapter.

This chapter develops analytical expressions for piecewise linear interpolation on unstructured tetrahedral meshes. These expressions are crucial because most of the arithmetic operations are performed analytically and these operations no longer need to be repeated during using the software. So, these expressions allow to perform the supercomputer calculations with less amount of resources as dynamic computer memory. The interpolation on unstructured grids plays a key role for numerical solving of last amount of problems in seismic exploration of oil and gas, non-destructive testing of different up-to-date complex materials, investigation of seismic stability of different complex objects like nuclear power plants, and especially for summation of injuries of human bodies for medicine.

In this chapter, a family of grid-characteristic methods for numerical simulation is considered. These methods are developed and used to solve a wide range of applied problems: traumatology, ultrasound studies of the human body, ultrasonic operations, seismic exploration of oil and gas, seismic resistance of residential and industrial facilities, non-destructive testing of railways and innovative materials including composites, development territories with complex natural conditions, shock effects on complex-shaped structures, and global seismic of various planets of the solar system. The methods allow to simulate the wave processes in heterogeneous media of complex topology and dynamic process of destruction of these media. Also these methods help to investigate clearly small heterogeneous features that represent breaks in the integration domain. Grid-characteristic methods are used to solve the hyperbolic systems of equations describing the wave processes. In this chapter, the elastic waves in isotropic and anisotropic cases and acoustic waves are considered. The method is well paralleled and actively implemented in software using the high-performance computing systems.

Delamination caused by low-velocity strike is considered as one of the most dangerous failure types. The destruction of contact between the plies or composite components significantly lowers the residual strength of the material but cannot be determined by visual inspection. These failures can mostly be determined by ultrasound testing, however, it requires a long time and cannot be carried out on site, which increases the maintenance cost. Both delamination emergence and ultrasound diagnostic results are determined by wave processes in viscoelastic media. The grid-characteristic method used in this chapter shows good results verified on various experimental data. The results of numerical modelling of delamination and its diagnostics are given in this chapter.

In this chapter, the application of the grid-characteristic method to solving seismic prospecting problems is considered. The characteristic seismogeological models, including Marmousi and SEG/EAGE Salt Model, are considered, wave patterns and seismograms are presented. The cases of 2D and 3D modelling, curvilinear boundaries between geological layers, fractured layers taking into account the topology of the Earth’s surface, construction of seismograms for both 2D and 3D seismic survey cases, and vertical seismic profiling are considered. The investigation of the performance of a software complex developed on the basis of a grid-characteristic method for modelling hydrocarbon deposits of various computational complexity was performed. Also, faults zones of a different nature, both about the length of the faults and the type of geological environment inside these faults, was studied. A detailed analysis of spatial dynamic wave patterns is carried out, and predictive conclusions are made about the nature of the seismograms obtained, which were actually confirmed in the respective seismograms. It will be shown in this Chapter that typical analytical tests cannot guarantee that software gives an opportunity for the geologist to develop right conclusions. This problem can be solved only by understanding the physical basis of the phenomena under consideration and the peculiarities of the operation of the difference methods used in the software, simultaneously. This suggests the method called Wave Logica, fragments of which are also given in the Chapter.

This chapter presents a new method of migration of the elastic wavefield. It is based on the Born approximation of the forward modelling operator for the elastic, which is obtained as an extension of the Born approximation for the acoustic field to the case of the elastic wavefield propagation. We present a detail mathematical derivation of the migration operators for the acoustic and elastic cases. The numerical experiments based on these operators are conducted for a set of synthetic multilayered models with curvilinear boundaries between the layers. We also present a comparative study of the migration images produced by the migration of the acoustic and elastic wavefields and examine the sources of false boundaries appeared in some of these images.

Seismic method is one of the main methods of geophysical exploration. Interpretation of seismic data requires their transformation into the images of the subsurface geological formation. These images can be produced by migration of the observed seismic data from the surface of observation in the lower half space. The traditional migration algorithms are based on the solution of the wave equation for the seismic field in the reverse time. In this chapter, we develop an algorithm of migration based on the elastic field equations, which describes more accurately the seismic field propagation than the simple acoustic wave equation. Our method uses the Kirchhoff and Rayleigh integral formulas for elastic wavefields. Considering that the elastic model approximates the propagation of the seismic fields in the geological media better than the acoustic model, the developed approach can significantly improve the quality of interpretation of seismic data in complex geological formations.