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# nafta mathemat

Published on: Mar 3, 2016

#### Transcripts - nafta mathemat

• 1. Mathematical representation models and applications on seismic tomography M. Arvanitis and B. D. Al-Anazi REVIEW New mathematical techniques have contributed substantially to the improvement of the geophysical prospecting methods, like traveltime seismic tomography. Thanks to these new techniques, the time to solve an inverse problem has been reduced dramatically making seismic tomography applicable to a great number of problems even in three dimensions. New raytracing and wavefront techniques provide a more flexible parameterization. Advancement from the least squares technique to today’s back-projection method’s, for example, has improved tomographic methods. Key words: tomography, raytracing, wave front, grid points, travel time INTRODUCTION What does tomography mean as a word? Its origin is found in the Greek word “tomo”, which means slice. Therein lies the basic idea: if we take many 2-D slices, ac- cording to the central slice theorem28, we can reconstruct the whole 3-D image of an object. Thanks to the same theorem, we can easily construct 2-D sections from 1-D lines, which can be measured in experiments. Seismic travel time tomography can be defined as the reconstruction of the Earth’s velocity model, using the seismic waves travel time deviations from a reference ve- locity model, better known as starting or background model. A starting model is an initial guess, an estimate of a velocity model. Seismic tomography as we know it to- day originated in 1974 as “3-D inversion”. At first, seis- mologists were very skeptical about the new method and its results. The whole attitude changed dramatically in the mid-1980’s, when iterative methods were intro- duced, to facilitate the calculation of large and sparse matrixes that occurred from the seismological data.6,23 Believability in the method was linked to the first global tomographic results9,37,11,which correlated satisfactory with the geoid. As credibility of the method grew seismol- ogists renamed “3-D inversion” to what is known today as seismic tomography. We can classify seismic tomography in two main categories; travel time and amplitude tomography. In this paper we will focus only on travel time tomography. Regarding the nature of the seismic waves, travel time tomography can be divided into refraction, reflection and diffraction tomography. Referring to the source, whether it is a natural earthquake or a shot, we carve up tomogra- phy into passive and active tomography, respectively. Seismic tomography is an imaging technique that uses seismic waves generated by earthquakes and explosions to create computer-generated, three-dimensional images of Earth's interior. This is how seismologists infer the different layers in the Earth. How is this done? The time it takes for a seismic wave to arrive at a seismic station from an earthquake can be used to calculate the speed along the wave's ray path. By using first arrival times of P waves recorded by seismic stations all over the world, scientists are able to define slower or faster regions deep in the Earth The simplest case of seismic tomography is to estimate P-wave velocity. Several methods have been developed for this purpose, e.g., refraction traveltime tomography, fi- nite-frequency traveltime tomography, reflection traveltime tomography, waveform tomography.36 To obtain a higher-resolution image one has to abandon the infinite-frequency approximations of ray theory that are applicable to the time of the wave 'onset' and instead measure travel times (or amplitudes) over a time window of some length using cross-correlation. Finite-frequency tomography takes the effects of wave diffraction into ac- count, which makes the imaging of smaller objects or anomalies possible.24 The raypaths are replaced by volumetric sensitivity kernels, often named 'banana-doughnut' kernels in global tomography, because their shape may resemble a banana, whereas their cross-section looks like a doughnut, with, at least for direct P and S waves, zero sensitivity of the travel time on the geometrical ray path. In finite-frequency tomography, travel time and amplitude anomalies are frequency-dependent, which leads to an increase in resolution. To exploit the information in a seismogram to the full- est, one uses waveform tomography. In this case, the seismograms are the observed data. In seismic explora- tion, the forward model is usually governed by the acous- tic wave equation. This is an approximation to the elastic wave propagation.36 Elastic waveform tomography is much more difficult than acoustic waveform tomogra- phy. The acoustic wave equation is numerically solved by some numerical schemes such as finite-difference and fi- nite-element methods. In global tomography the inverse problem for elastic waves can be handled by adjoint methods. PARAMETERIZATION Lets consider two closely spaced points in the medium; the inverse of the local wavespeed associated with these NAFTA 60 (9) 495-498 (2009) 495
• 2. points is the slowness. There are three kinds of slowness models: homogeneous and heterogeneous cells (2-D) or blocks (3-D) of constant slowness values and rectangular grids with slowness values assigned to the grid points with different interpolation schemas to specify the values between the grid points. In general, the use of cells is the most common parameterization but it is facing difficul- ties, since the sharp boundaries between the cells cannot be resolved. As for the grid parameterization, a fine regu- lar and irregular grid parameterization exist. The former parameterization is the purely tomographic approach while the latter one is closer to forward modelling. Regu- lar grid has the advantage of simplicity but it can cause over-parameterization when high resolution is required. Recent studies are focused on irregular grid using Delaunay triangles or Veronoi polygons to avoid such problems.4,41 The travel time for a ray is: ( )T s V r s= ò 1/ d (1) where V(r) is the unknown velocity for the ray-path S. We want to determine V(r) from N travel time measure- ments. Let To be the travel time for the starting model: ( )T s V r s0 0 01= ò / d (2) Whether we are not sure about the estimate of the start- ing model we can use a tau-p method from picked arrival times to form a reliable starting velocity model.2 Using Fermat.s principle, we can ignore the true ray-path and use the ray-path of the starting model instead of it. The delay time is: ( ) ( ) ( )dT T T s V r s s V r s s V V s= - = - » - »ò òò0 0 0 0 01 1 1 1/ / / /d d d ( ) ( )( )» - -ò s V r V r s0 0 2 d d (3) where: ( ) ( ) ( )dV r V r V r= - 0 (3a) Equation (3) comprises a linear system of equations, which can be changed in such a way to become more suit- able for computer processing. We parameterize the me- dium with I interpolation functions hi, which is the basis of the subspace of the Hilbert space of all possible mod- els V(r): ( ) ( )d gV r h rk k k= å (4) where k spans the integers from 1 to I, and function gk is the weight of the function hk. Considering a cell- parameterization we have: hi (r)=1, if r in cell I and hi (r)=0, anywhere else (5) Equation (3) can now be defined as: ( ) ( )[ ]d g gT s V r h r s Ak k k ik k k= - =òå å0 0 2 1/ d (6) where: ( ) ( )[ ]A s V r h r sik k= -ò 0 0 21/ d (6a) Equation (6) can be formulated for each shot in a ma- trix form and in terms of slowness as: Ms t= (7) Where s is the slowness vector, t the time vector and M is the matrix of lij, where lij is the length of the i-th ray-path through j-th cell. SOLUTION Foremost, we have to calculate the matrix elements Aik, which implies the finding of the ray-path. Two methods are commonly used in seismic tomography to find the ray path: ray-tracing and wavefront methods. The two-point ray-tracing finds ray-paths along which seismic energy is propagating and calculates the travel time. For a layered media, rays are traced by solving the differential equa- tions with a Runge-Kutta predictor-corrector scheme. To define in a better way the ray geometry and the slowness, we mainly have two methods: shooting and bending, in- spired both from ray-tracing. The former is based on continuous iterations until the end of a ray to meet a limit condition or by interpolating between close rays using hermite cubic interpolation.7 The latter uses a parameterization for a ray-path by the support points Vi of a third order B-spline. The location of the ray is a func- tion of the four nearest points: ( )Q u b V b V b V b Vi i i i= + + +- - - - +2 2 1 1 0 1 1 (8) Where b1 depends on u, 0 £ u £ 1, and is known for the different values of u. A conjugate gradient method33 is needed to find which of the support points V1 minimizes the time given by (1). Èervený5 uses the quadratic slow- ness, 1/V2, instead of slowness since it offers the simplest analytical solution in inhomogeneous medium. Latter techniques pored over the drawbacks of ray-tracing in- cluding the works of Zelt and Ellis49 who invented a ray-tracing technique with a trapezoidal parameteri- zation providing a rapid traveltime calculation and of Sethian and Popovici34 who presented the fast marching technique that can model turning rays, but with unsatis- factory accuracy. The wavefront (surfaces of equal travel time) methods are alternative techniques to ray-tracing. This method determines minimum ray-paths and traveltimes by ex- panding a wavefront in the whole model. Recent ad- vances in the ray-tracing methods have mainly been focused on wavefront methods rather than ray-tracing, for two basic reasons: ray-tracing is valid only for smooth velocity structures and it is significantly slower than any wavefront method. Vidale43,42 modified the wavefront method and the eikonal equation solver introducing a fi- nite difference procedure, to propagate traveltimes through a uniformly sampled grid. The eikonal solver finds the wavefront that forms concentric shells about the source and conducts the ray-paths from their shape. A defect of Vidalia’s method27 is that it fails when velocity contrasts are of the order of u u2 1 2/ > (9) (Hole et al.14 modified Vidalia’s algorithm to a more rapid algorithm using variable grid spacing. SPR, for 496 NAFTA 60 (9) 495-498 (2009) M. ARVANITIS AND B. D. AL-ANAZI MATHEMATICAL REPRESENTATION MODELS AND APPLICATIONS...