Sismik Prospeksiyon ( ingilizce ) Introduction to Seismology - 8 Introduction to Seismology March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005 ? Go back to the observations again and look at the deviation from the model. Objective: we need to find a model that minimize ?t Errors caused by the model and the source location are usually combined together, and the noise is assumed to be white, with a Gaussian distribution. If we know the 1D model, we can apply Snell’s law to estimate the geometry: T obs = ? 1 d l c x 3D () Ray Path Note that 1 () = k ?Tx () cx If c change, the ray path changes. We end up with a nonlinear problem. Thus, we should try to linearize the inversion, using the Fermat’s Principle. If we change a little bit the ray path around the optimum, we’ll end up with a small change in travel time. We have two kinds of “deviations” from the reference ray: 1. First contribution: effect of changes in velocity ?c. 2. Second contribution: effect of change in the ray path. Fermat’s principle says that we can ignore it. 6 Introduction to Seismology March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005 Linearization of the travel time Travel time residual: Observation = ?t 3D =T obs -T ref = ? 1 dl - ? 1 dl 0 x true c x () reference c 0 () 3D 3D structure path Fermat s ' P r inceple 1 1 ? dl 0 - ? dl 0 c x c x reference () reference 0 () 3D 3D path path = ? ? 2 c dl 0 = ? ( s () -s 0 () x ) 0 x dl c reference 0 reference 3D 3D path path = ?sx d l ? ( () ) 0 refer ence 3D path In linearizing the problem, we get rid of the unknown ray. We can do our calculation in a reference earth model. ? The travel time tomography is an iterative process: ? Create 1D model ? Ray tracing and get new rays in the model ? Update ray geometry ? Get the reference ray related to the 3D t = T obs - T ref (3D) (The reference model does not have to be a 1D model.) Linearization of the hypocenter mislocation with t 0 the origin time, ( ,,) the location of the earthquake. xyz Then we try to solve for ?s , ?t , ? x , ? y , ?z . 7 Introduction to Seismology March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005 Inverse Problem First, we need to discretize the problem, i.e. parameterization: For example, plane wave summation: ? xyz i k x ?t ) = ??? ? ( ,,, ? )e ( ·- dkd? Then, we inject µ into the equation: 8 Introduction to Seismology March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005 where G i : Green functions, solution of a point source. We need to do a convolution with a point perturbation in order to get the observations. M : Number of model parameters. N : Number of observations. In general, M ? N . As a consequence, the matrix A is not square. Multiplying the equation by A T , we can get the solution: Back to our specific inverse problem: One way is to take k h as a series of cells/blocks, with a value for x inside the cell k and zero otherwise. We have ? i t s d l =? ? = 1 M cell k k s = ? ? () ik d l where i : event-station pair. () ik dl : path length. Rewrite the equation in the matrix form: 9 Introduction to Seismology March 10/12, 2008 Adapted from notes 2/22, 3/28, 2005 ? ?l 11 … ?l 1M ?? ?s 1 ? ? ?t 1 ? ? ?? ? ? ? Am = ik ·= ? ?? ? ? ? ? ?l … ?l ?? ?s ? ? ?t ? ? N1 NM?? M ? ? N ? where each ray gives a row in the matrix. We have an average wavespeed along the ray. In order to construct a model vector, we need to get data from different rays crossing each other. A is a sparse matrix. If we look at one ray: T A will have only ~100 elements non-zero. The good thing about sparse matrix is that AAis approximately diagonal. The problem is that there are many singularities, which make the inversion unstable (in that case, we need to add a damping factor or regularize the problem). One possibility is to not use cells of the same size. Consequently, it reduces the number of cells; the inverse matrix is less singular. Nevertheless, the computation time increases. Another way is take h k as spherical harmonics (in global seismology). 10