Wojciech Dębski

Information on elastic parameters obtained from the amplitudes of reflected waves

Seismic amplitude variation with offset data contain information on the elastic parameters of geological layers. As the general solution of the inverse problem consists of a probability over the space of all possible earth models, we look at the probabilities obtained using amplitude variation with offset (AVO) data for different choices of elastic parameters. A proper analysis of the information in the data requires a nontrivial definition of the probability defining the state of total ignorance on different elastic parameters (seismic velocities, Lame’s parameters, etc.). We conclude that mass density, seismic impedance, and Poisson’s ratio constitute the best resolved parameter set when inverting seismic amplitude variation with offset data.

New observations of Gulf of Gdansk seismic events

Abstract Seismicity of northern Poland and of the adjacent area of the Baltic Sea has been considered weak, if any. Historical reports mention just a couple of seismic events felt in the region, while observatory seismology has recorded just a single swarm of seismic events in the Gulf of Gdansk. The lack of information is caused not only by the weak seismicity of the area, but also by the lack of seismological observatories in the region. This situation is changing with the installment of a new seismological station Suwalki in northeastern Poland, and two temporary very broadband stations of the GEOFON network, one in northwestern Poland, and the other on Oeland Island, Sweden. Just months after the installations, we are able to detect and locate numerous seismic events in the Gulf of Gdansk. This does not necessarily mean that Gulf of Gdansk is a seismically active area, as there are indications that the events may be caused by human activity. One of the objectives of this paper was to obtain reliable location solutions with the best possible location error control. Since only a few stations are in operation in the nearby area and little is known about the velocity structure, the Bayesian inversion method has been used to locate event epicenters. This method allows to minimize location errors caused by insufficient knowledge of local crust velocity structure and non-Gaussian statistics of measurement/modeling errors. The method, being fully nonlinear, does not introduce instabilities typical for inverse approaches which uses linearization.

Probabilistic Inverse Theory

Abstract Geophysical investigations which commenced thousands of years ago in China from observations of the Earth shaking caused by large earthquakes (Lee et al., 2003) have gone a long way in their development from an initial, intuitive stage to a modern science employing the newest technological and theoretical achievements. In spite of this enormous development, geophysical research still faces the same basic limitation. The only available information about the Earth comes from measurement at its surface or from space. Only very limited information can be acquired by direct measurements. It is not surprising, therefore, that geophysicists have contributed significantly to the development of the inverse theory—the theory of inference about sought parameters from indirect measurements. For a long time this inference was understood as the task of estimating parameters used to describe the Earths structure or processes within it, like earthquake ruptures. The problem was traditionally solved by using optimization techniques following the least absolute value and least squares criteria formulated by Laplace and Gauss. Today the inverse theory faces a new challenge in its development. In many geophysical and related applications, obtaining the model “best fitting” a given set of data according to a selected optimization criterion is not sufficient any more. We need to know how plausible the obtained model is or, in other words, how large the uncertainties are in the final solutions. This task can hardly be addressed in the framework of the classical optimization approach. The probabilistic inverse theory incorporates a statistical point of view, according to which all available information, including observational data, theoretical predictions and a priori knowledge, can be represented by probability distributions. According to this reasoning, the solution of the inverse problem is not a single, optimum model, but rather the a posteriori probability distribution over the model space which describes the probability of a given model being the true one. This path of development of the inverse theory follows a pragmatic need for a reliable and efficient method of interpreting observational data. The aim of this chapter is to bring together two elements of the probabilistic inverse theory. The first one is a presentation of the theoretical background of the theory enhanced by basic elements of the Monte Carlo computational technique. The second part provides a review of the solid earth applications of the probabilistic inverse theory.

Bayesian Approach to Tomographic Imaging of Rock-mass Velocity Heterogeneities

Detailed imaging of the Earth subsurface structure has both scientific and practical aspects. From a scientific point of view knowledge of the Earth’s structure is necessary for understanding various processes. Practical aspects include such issues as localization and description of natural resources deposits. Although huge progress has been made in this field, there are still a lot of questions not answered yet. One of them is the question of a relation between observed seismicity and the earth’s structure. In this paper we address this issue and argue that the probabilistic (Bayesian) approach should be used. Since this inversion method introduces some additional complexity to the already difficult seismic tomography technique, we decided to describe the basic steps of Bayesian tomographic imaging from data preparation to analysis of imaging results. The methodological considerations are illustrated by examples of imaging for four mining regions within the Rudna (Poland) copper mine.

The New Algorithm for Fast Probabilistic Hypocenter Locations

The spatial location of sources of seismic waves is one of the first tasks when transient waves from natural (uncontrolled) sources are analysed in many branches of physics, including seismology, oceanology, to name a few. It is well recognised that there is no single universal location algorithm which performs equally well in all situations. Source activity and its spatial variability in time, the geometry of recording network, the complexity and heterogeneity of wave velocity distribution are all factors influencing the performance of location algorithms. In this paper we propose a new location algorithm which exploits the reciprocity and time-inverse invariance property of the wave equation. Basing on these symmetries and using a modern finite-difference-type eikonal solver, we have developed a new very fast algorithm performing the full probabilistic (Bayesian) source location. We illustrate an efficiency of the algorithm performing an advanced error analysis for 1647 seismic events from the Rudna copper mine operating in southwestern Poland.

Time Scales: Towards Extending the Finite Difference Technique for Non-homogeneous Grids

Tremendous progress in seismology over last years is greatly due to availability of high quality seismic waveforms. Their availability prompts the new mathematical and numerical algorithms for their more detailed analysis. This analysis usually takes a form of the inverse problems—an estimation of physical parameters from seismic waveforms called the full waveform inversion (FWI). No matter which inversion algorithm is used, the FWI technique requires precise modeling of synthetic seismograms for a given lithological model. This is by no means a trivial task from the algorithmic point of view, as it requires solving (usually numerically) the wave equation describing propagation of seismic waves in complex 3D media, taking into account such effects as spatial heterogeneities of media properties, anisotropy, and energy attenuation, to name a few. Although many numerical algorithms have been developed to handle this task, there is still a need for further development as there is no single universal approach equally good for all tasks in hand. In this chapter, the possibility of using the Time Scale Calculus formalism to advance the synthetic seismograms calculation is discussed. This modern approach developed the late 1990s with the aim of unifying analytical and numerical calculations provides the very promising basement for developing new computational methods for seismological, or more general geophysical applications. In this chapter we review the basic elements of the Time Scale Calculus keeping in mind its application in seismology but also we extend the initial concept of Hilger’s derivative towards the backward-type and central-type derivatives using the unified approach and compare their properties for various time scales. Using these results we define the second order differential operators (laplacians) and provide explicit formulas for different time scales. Finally, the formalism of time scales is used for solving 1D linear, acoustic wave equation for a velocity model with large velocity discontinuities. Based on this simple example we demonstrate that even in such a simple case using an extension of the classical finite difference schemata towards irregular grid leads to a significant improvement of computational efficiency.

Robust and Accurate Seismic/Acoustic Ray Tracer

Recent development of high resolution seismic tomography and an increasing necessity of high precision seismic (acoustic) source locations calls for robust and very precise numerical methods of an estimation of seismic (acoustic) wave travel times and propagation ray paths. In this contribution I present an algorithm based on a parametrisation of ray paths by series of the Chebyshev polynomials. This pseudo-spectral method combined with the accurate Gauss-Lobbato integration procedure allows to reach a very high relative accuracy of travel time calculation of order of 1e-6 or better. Employing of the Genetic Algorithm based optimizer to seek for the shortest travel time path assures an extreme robustness of the algorithm which allows the method to be used in complicated 3D geological structures.

Dynamic Stress Drop for Selected Seismic Events at Rudna Copper Mine, Poland

In this paper, we report on an analysis of rupture processes of mining-induced events using the Empirical Green Function approach. The basic goal of this analysis is to estimate the dynamic stress drop—the quantity describing frictional forces at a slipping fault plane during the unstable part of an earthquake. The presented results cover 40 selected events with magnitude

Particle-Based DEM Model for Simulating Brittle Cracks Evolution in Rock-like Materials During the Tensional Fracturing Process

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