Variable damping in seismic tomography based on ray coverage

In the seismic tomography problem, the subsurface slowness distribution is estimated from the travel-times computed along generally curved rays. For a dense and uniform set of rays, the slowness distribution will be correctly reconstructed. However, for an uneven distribution of rays, the estimated slowness distribution may be influenced by the ray configuration. For the linear case, the seismic tomography problem can be written as A(x-xO)=(do bs _d pred) where dobs and d pred are the observed and predicted travel­ times, x and Xo are the true and prior slowness distributions, and A is the (mxn) matrix of partial derivatives aTi I auj. Prior estimates of model uncertainty can be used to stabilize the inverse problem (Tarantola, 1987). The stochastic linear inverse can then be written where Cd is the data covariance matrix and C is the prior model covariance associated with xo xo. Operationally, the diagonal elements of C;: provide damping of the solution and the off­ diagonal elements provide smoothing. The final covariance estimate is then Cx =(A • C d 1 A +C;:)-1. One might consider using the fmal covariance matrix Cx instead of C in the inverse formula above, however this will lead to an incorrect estimate (Claerbout, xo 1992). In addition to the prior uncertainties in the model, C c~ also include a discretization xo correction for variable block sizes (Nolet, 1987). For the case of constant damping and equal ray coverage, the larger blocks will be emphasized at the expense of the smaller blocks. In a very simple example, Nolet (1987) used the underdetermined formulation to invert for two variable length blocks from one travel-time. Using a constant damping this leads to unequal estimates of the slowness parameters in which S11S2 = 1 1 /1 2 , where Si is the slowness and Ii is the length for each of the two blocks. Incorporating a variable model covariance with diagonal values of af and ai results in the ratio of the slowness estimates of S11S2 = af111 ai1 2 • In order to equalize the slowness estimates from the bias resulting from the unequal block sizes, the diagonal elements of the model covariance matrix can be chosen to be proportional to the inverse of the block lengths. In a 3-D model, the diagonal elements would be proportional to the inverse of the block volumes (Nolet, 1987). For spline parameterizations, the block volumes can be estimated …