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SIRT Method

The diagonal term of equation 2.9 is given as:
 
[LTL]jj = $\displaystyle \sum_{i}l_{ij}^2$ (2.10)

and is larger than the off-diagonal components. Therefore, the inverse to ${\bf L^TL}$can be approximated by a diagonal matrix, with the ith diagonal term equal to the reciprocal of equation 2.10. Therefore the update in slowness is given by
dsi = $\displaystyle \frac{ ({\bf L^T dt})_i }{[{\bf L^TL}]_{ii} }$ (2.11)

and is a SIRT-like (simultaneous iterative reconstruction tomography) method.
           MATLAB SIRT TOMOGRAPHY CODE

% t0     - input- observed traveltime data vector from ns shot gathers
%                 with ng traces per shot gather.
% s0     - input- starting slowness vector for a nx by nz slowness model
%                 with cell width dx.
%thresh  - input- estimated average traveltime error
%filter  - input- small (e.g., 4x4) low-pass averaging filter
% tc     -output- calculated traveltime data vector
% diag   -output- diagonal matrix of diagonal values of L'L

s1 = s0;thresh=1;
[L,tc,diag] = raytraveltime(s1,nx,nz,ns,ng,dx);
%
if thresh > 0;   
  residual = tc -t0;
  s2 = s1 - alpha*L'*residual./diag
  [L,tc,diag] = raytraveltime(s1,nx,nz,ns,ng,dx);
%
  average_t_error = sqrt((tc-t0)'*(tc-t0))/(ns*ng)
  if average_t_error < thresh; thresh = -1;
    else; s2 = conv2(s2,filter);end;
end
Let us now use the SIRT-like algorithm to invert traveltime data. The first example will demonstrate the use of refraction tomography, and the second example will invert transmission traveltimes from crosswell data.



Gerard Schuster
1998-07-29