Sudhish Kumar Bakku sudhi mit edu Modeling Elastic Wave Propagation in layered medium with Surface Topography Term Report for CSE 2 18 086 2008 0 Abstract The objective of the Term Project is to study the Finite Difference Methods to better approximate and model the effect of topography on Wave Propagation First wave propagation is modeled using Standard Staggered grid in a 2D homogenous medium in a 1000m 500m space bounded by rigid surfaces on three sides and a free surface on the fourth side Then PML boundaries conditions are introduced to suppress the waves from rigid surfaces Vacuum method and Image method for Surface topography are discussed Both are implemented for horizontal plane surface and compared Vacuum method is chosen for modeling Surface Topography A dipping layer is introduced into the model now To take a step further multi grid variable grid is developed and implemented for plane surface case 1 Introduction Understanding wave propagation is particularly important for processing the seismic signal recorded during Seismic Survey for Oil Exploration With conventional oil reserves depleting we are now forced to explore oil in extreme terrains as well Due to the uneven topography most of the energy sent into the ground turns out on seismograms as scatter noise It is therefore natural to understand the wave propagation in such topography for us to remove the noise due to scattering We do are interested in exploring the possibilities of using these surface scatters as secondary sources to better image the reservoir and enhance resolution Conventionally finite differences are used to model the wave propagation But approximating the topography is a problem due to the complexity in approximating the topography with a Cartesian grid and the resolution required It is also noticed that the corners of the grids would act as diffractors and disturb the solution These aspects are observed in the project I have applied the model to the Seismic Survey done in a field trip to compare how well we could do to simulate the wave propagation The Seismic Survey is done in the California Desert at Vidal River Side Mountains The geological structure to be imaged was the fault below Sudhish Kumar Bakku sudhi mit edu A schematic diagram of the feature is shown below We have deployed Geophones along the surface at every 30m for 1 kilometer at equal intervals and shot Betsy gun at certain source locations at every 30m along the seismic line to record the seismogram Analyzing the seismograms by refraction seismology we have arrived at the above model I would like to verify if our model would reproduce the observed seismograms I would like to present the physics behind wave propagation and then proceed to the modeling and results 2 Physics of Elastic Wave Propagation Consider a Homogenous Isotropic and 3 D infinite medium Let the material properties be given by E Young s Modulus K Bulk Modulus G Shear Modulus Poisson s ratio But all the above four parameters are inter related and can be expressed in terms of only two parameters Usually Lame s constants are used to denote the medium parameters These are related to the above parameters as E 3 2 K 2 3 G 2 Consider the infinitesimal cube presented below xx yy zz are the Normal Stresses acting on the faces of the element and xy yx zx xz yz zy are the shear stresses acting on the faces of the cube The strains are given by Sudhish Kumar Bakku sudhi mit edu v u x u w z x zx where are yz y z xy zz u v w w v v u x y w z yy y xx displacements along ve x y z directions Figure showing the directions of stress components Stress and strain are related as xx 2 xx yy 2 yy zz 2 zz xy xy yz yz zx zx Applying equation of motion to the above cube we arrive at 2 u xx xy xz 2 v yx yy yz 2 w zx zy zz 2 2 2 x y z x y z x y z t t t The above differential equations describe the wave motion For simplicity of modeling we consider a 2D model So our equations reduce to v u x y xx 2 Sudhish Kumar Bakku sudhi mit edu v u y x yy 2 v u xy x y 2u xx xy t 2 x y and 2 v yx yy t 2 x y We can rewrite the above equations as below known as Velocity Stress Formulation Since we would be recording velocities in field and boundary conditions are associated with stresses it is a better formulation V y Vx x y xx 2 1 2 xy y x x y 3 Vx xx xy t x y 4 5 V y y yy 2 V x x V V Vy t yx x yy y indicates derivative w r t time We would use finite difference scheme to solve the above system of differential equations 3 Standard Staggered Grid The two popular methods of modeling the elastic wave propagation are Standard Staggered Grid J Virieux et Al 1986 and Rotated Staggered Grid Saenger et al 1997 Standard Staggered grid is the traditional one Though rotated staggered is good at handling material discontinuity and easier to code and implement I have chosen Standard Staggered Grid because the surface condition Normal Stress is zero can be implemented well by image method And I see smart imaging of stresses as the best way for Topography Whereas rotated staggered method is not so comfortable in implementing the stress imaging methods due to rotation in the co ordinate axes Standard Staggered Grid is as below Consider a cell as shown Axial Stresses and Elastic modules xx yy lie at the centre of the cell shear stress and density xy at the corners and the velocities i Sudhish Kumar Bakku sudhi mit edu on the edges The velocities in x and y direction are staggered by half a grid distance in both directions Stresses lie in between the velocities and as a result they are also staggered Also the velocities and stresses are not coexistent in time They are staggered by half time step xy X xx yy y Vx Vy Implementing this grid to our differential equations our finite differences equations look like m 0 5 m 0 5 m 0 5 V V xx i j yy i j xy i j m 1 m 1 x i j y i j V m V m V m x i j 0 5 V m x i j 0 5 m 0 5 xx i j 2 t x V m y i 0 5 j V m y i 0 5 j m 0 5 yy i j 2 t y V m y i j 0 5 V m y i j 0 5 m 0 5 xy i j t x V m 1 y i j m 0 5 yy i 0 5 j m 0 5 yy i 0 5 j t y Note j is along x axis and I …