Steepest descent minimization applied to a simple seismic inversion

Transcription

1 Steepest descent minimization applied to a simple seismic inversion Sylvain Barbot Friday, December 10, Problem Definition The problem one tries to address is the estimation of the location of a seismic wave source. One considers the seismic wave to propagate straight from the epicenter to the seismograph without reflection. Besides, one knows the wave propagation velocity v. And the relation 1 holds: (x xi )2 + (y y i )2 t i = v Equation 1 is called the direct model: (1) d = g(m) One has a set of time of wave front arrival measured from different seismographs scattered on the surface (the problem has been reduced to a two-dimensional space). One writes t i the time of arrival measured from the seismograph i d = C d = Where d is data column matrix and C d the corresponding covariance matrix. One finally writes the a priori information on model parameters m prior and its covariance matrix C m. These two last objects are necessary for the gradient formulation but as no real information is available on the seismic source, one defines arbitrarily the seismic source in the middle of the seismograph but with a high uncertainty (the source might be around a 10km circle): m prior = [ ] C m = We first draw the posterior probability density [ ] (2) (3) σ m (m) = exp ( S(m)) (4) 1

2 Figure 1: Probability Density σ m. Left: x and y vary from 0 to 20 km; Right: a zoom, x and y vary from 14 to 16 km. where 2 S(m) = (g(m) d obs ) T C 1 d This σ m as represented in figure 1. (g(m) d obs )+(m m prior ) T C 1 m (m m prior ) (5) 2

3 2 Implementation and Results 2.1 Steepest Descent Algorithm OOne then implements the steepest descent algorithm to minimize S(m). The steepest ascent vector γ is defined by relation 6: γ k = C m G T k C 1 d (g(m k) d obs ) + (m k m prior ).... G k = δf i δf i δx δy.. (6) And we use the following algorithm which converges towards the best estimation in the least square sense fitting the data parameters with respect to our direct model expression: m k+1 = m k εγ k (7) Finally, the a posteriori covariance matrix holds: C m = (G T k C 1 d G k + C 1 m ) 1 (8) One has implemented the steepest descent algorithm on IDL to find numerically the solution of this simple seismic inversion and to display the probability density shape. One has chosen ε arbitrarily with the trial and error method. Values close to 1 fail because too large to allow the convergence: we have chosen the value because we have verified in a first computation that the steepest ascent vector has components of the order of 10 4, and we want our jumps to be of the order of one km. Should the value have not ensures convergence, we would have taken half the value. After iterations, the computer answers: Least Squares Solution: Covariance Matrix:

4 2.2 Quasi-Newton Algorithm One has then implemented the quasi-newton algorithm. The difference holding in the calculation of the hessian matrix ξ k at each iteration: The algorithm becomes: ξ k = (G T k C 1 D G k + C 1 D ) 1 (9) γ k = G T k C 1 d (g(m k) d obs ) + C 1 M (m k m prior ) (10) m k+1 = m k ξ k γ k After 5 iterations, the programs displays: Least Squares Solution: Covariance Matrix:

5 3 Appendix FUNCTION seismic_proba_density d_obs = TRANSPOSE([3.12, 3.26, 2.98, 3.12, 2.84, 2.98]) data_cov = DIAG_MATRIX(REPLICATE(.1^2, 6)) m_prior = TRANSPOSE([4, 15.5]) prior_cov = DIAG_MATRIX(REPLICATE(10.^2, 2)) S = FLTARR(100, 100) FOR l = 0, DO BEGIN y = l / 5. FOR c = 0, DO BEGIN x = c / 5. S[c, l] = 1./2. * (TRANSPOSE(seismic_direct([x, y]) - d_obs) ## $ INVERT(data_cov) ## (seismic_direct([x, y]) - d_obs)) S[c, l]+= 1./2. * (TRANSPOSE([[x], [y]] - m_prior) ## $ INVERT(prior_cov) ## ([[x], [y]] - m_prior)) ENDFOR ENDFOR distribution = EXP(- S) END Table 1: IDL code calculating the probability density ρ M = exp ( S(m)) 5

6 PRO seismic_inversion iteration = & eps = 1e-5 d_obs = TRANSPOSE([3.12, 3.26, 2.98, 3.12, 2.84, 2.98]) data_cov = DIAG_MATRIX(REPLICATE(.1^2, 6)) m_prior = TRANSPOSE([10, 2]) prior_cov = [[1d, 0d], [0, 100d^2]] Mk = m_prior FOR k = 0, iteration - 1 DO BEGIN B = seismic_direct(mk) - d_obs LUDC, data_cov, index matrice = TRANSPOSE(LUSOL(data_cov, index, TRANSPOSE(B))) Gk = seismic_gradient(mk) vector = prior_cov ## TRANSPOSE(Gk) ## matrice + (Mk - m_prior) mk = mk - eps * vector ENDFOR post_cov = INVERT(TRANSPOSE(Gk) ## $ INVERT(data_cov) ## Gk + INVERT(data_cov)) PRINT, Least Squares Solution:, mk PRINT, Covariance Matrix:, post_cov Table 2: Procedure running the steepest descent algorithm core. 6

7 FUNCTION seismic_direct, m x = [3., 3., 4., 4., 5., 5.] y = [15., 16., 15., 16., 15., 16.] v = 5 RETURN, TRANSPOSE(SQRT((m[0] - x)^2 + (m[1] - y)^2) / v) END FUNCTION seismic_gradient, m x = [3., 3., 4., 4., 5., 5.] y = [15., 16., 15., 16., 15., 16.] v = 5 result = [[(m[0] - x) / (v * SQRT((m[0] - x)^2 + (m[1] - y)^2))],$ [(m[1] - y) / (v * SQRT((m[0] - x)^2 + (m[1] - y)^2))]] RETURN, TRANSPOSE(result) END Table 3: Ancillary functions calculating the direct model and the gradient matrix. 7

8 PRO seismic_inversion_quasi_newton iteration = 5 d_obs = TRANSPOSE([3.12, 3.26, 2.98, 3.12, 2.84, 2.98]) data_cov = DIAG_MATRIX(REPLICATE(.1^2, 6)) m_prior = TRANSPOSE([10, 2]) prior_cov = [[1d, 0d], [0, 100d^2]] Mk = m_prior FOR k = 0, iteration - 1 DO BEGIN B = seismic_direct(mk) - d_obs LUDC, data_cov, index matrice = TRANSPOSE(LUSOL(data_cov, index, TRANSPOSE(B))) Gk = seismic_gradient(mk) hessian = INVERT(TRANSPOSE(Gk) ## idata_cov ## Gk + iprior_cov) vector = TRANSPOSE(Gk) ## matrice + iprior_cov ## (Mk - m_prior) mk = mk - hessian ## vector ENDFOR PRINT, Least Squares Solution:, mk PRINT, Covariance Matrix:, hessian Table 4: Procedure running the Quasi-Newton algorithm 8