1. AGIP MILANO2000 07 07 1 Seismic data inversion Enrico Pieroni Ernesto Bonomi Emma Calabresu () Geophysics Area CRS4

2. AGIP MILANO2000 07 07 2 Inverse problems are among the most challenging in computational and applied science and have been studied extensively. Although there is no precise definition inverse problems are concerned with the determination of inputs or sources from observed outputs or responses. This is in contrast to direct problems, in which outputs or responses are determined using knowledge of the inputs or sources. Inverse problems are among the most challenging in computational and applied science and have been studied extensively. Although there is no precise definition inverse problems are concerned with the determination of inputs or sources from observed outputs or responses. This is in contrast to direct problems, in which outputs or responses are determined using knowledge of the inputs or sources. The Art of Inverse Problem inferring model parameters from output data

3. AGIP MILANO2000 07 07 3 “Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion” Pratt, Shin, Hicks Geohpys.J.Int. (1998) 133, 341- 362 “High resolution velocity model estimation from refraction and reflection data” Forgues, Scala, Pratt SEG 1998 “Seismic Waveform inversion in the frequency domain” Pratt, Geophysics Jan 11, 1999 “Nonmonotone Spectral Projected Gradient Methods in Convex Set” 1999, Birgin, Martinez, Raydan Presentation outline inversion framework mathematical framework steepest descent optim. lagrangian approach optimization loop newton optimization conjugate direction opt. 1d optimization constrained optimization test cases “Multiscale seismic waveform inversion” Bunks, Saleck, Zaleski, Chavent Geohpysics (1995) 60, 1457-1473 “Nonlinear inversion of seismic reflection data in a laterally invariant medium” Pica, Diet, Tarantola, Geohpysics (1990) 55, 284-292 “Pre-stack inversion of a 1D medium” Kolb, Collino, Lailly IEEE (1986) 74, 498-508

4. AGIP MILANO2000 07 07 4 Parameters: N x N y N z unknowns to recover: the velocity field c(x,y,z) Observed data/measurements: recorded data at a reference depth STACK(x,y,t) = P(x,y,z=0,t). Simulated data: wave-field propagation imposed by the acoustic wave equation using some trial velocity field Inversion: find the velocity field that minimizes some measure of the misfit between observed and simulated data Inversion framework We solved the inverse problem with a single shot acquisition. The generalization to multiple shots is straightforward and can result in a better inversion.

5. AGIP MILANO2000 07 07 5 Mathematical framework measure STACK(x,y,t) at same reference level z=0, produced by a single source try a guess c (0) (x,y,z) for the velocity field solving the acoustic wave equation, simulate the pressure field over the entire spatial domain (with adequate B.C. and I.C.) evaluate the error or cost function and if necessary its derivatives (cumbersome) update iteratively the velocity field, with the intent to minimize the error function iterate this procedure up to a fixed “error threshold”

6. AGIP MILANO2000 07 07 6 Steepest descent optimization The velocity updating technique is usually based on local informations, e.g. the gradient: problem: avoid local minima Fixed point = minima

7. AGIP MILANO2000 07 07 7 Wave equation Lagrangian approach From P and  evaluate the gradient PROBLEM: time alignment! Constrained minimization problem adjoint field A sort of wave equation with source term = residual error Back in time!

8. AGIP MILANO2000 07 07 8 do it=0, nit-1 ! call FMod do step = 0, nt-1 ! call BMod call LoadMeasField call AdjMod call PartialGrad call PartialCostF ! end do call Optimizer ! end do Inner loop: align in time both direct and adjoint fields to perform in-core gradient evaluation Optimization loop Record data at z=0 & on the boundaries Use information on the boundaries to backpropagate field P Load observed data With real and simulated measurements build the source term and solve for the adjoint field FMod BModAdjMod Evaluate partial cost function and gradient Update velocity field

9. AGIP MILANO2000 07 07 9 Newton optimization The optimization procedure can use also information from the Hessian (second derivative matrix) but this is very expensive for both computational (# direct propagation = # parameters) and storage requirements ( [N x N y N z ] 2 ) ! E.G. Newton, Quasi-Newton or Gauss-Newton methods: Thus, aiming to a 3D reconstruction, we decided to only use the gradient.

10. AGIP MILANO2000 07 07 10 Optimization techniques storage convergence

11. AGIP MILANO2000 07 07 11 Conjugate direction optimization To achieve better convergence we studied different conjugate direction algorithms: [1] Fletcher-Reeves [2] Polak-Ribiere [3] Hestenes-Stiefel (but we have not observed sensible differences) [1] [2] [3]

12. AGIP MILANO2000 07 07 12 1D optimization At each iteration step, for each fixed direction d and velocity c, find a scalar  such that the resulting error function (depending now on a single real parameter) be minimum, E.G. by line search, bisection, generalized decreasing conditions

13. AGIP MILANO2000 07 07 13 Constrained optimization Because of the box constraints over the velocity, we are forced to adopt the projected conjugate gradient:

14. AGIP MILANO2000 07 07 14 nx = 116 nz = 66 nt = 270 dx = 3. dz = 3. dt = 0.00065 thick_x = 0 thick_z = 0 rec_thick_x = 1 rec_thick_z = 1 z_record = 4 Nopt = 20 Niter = 100 We will consider inversion of small 2D synthetic data-sets. For a better tuning of the algorithms we used velocity field with no lateral variations, but the code is genuine 2D. Test cases

15. AGIP MILANO2000 07 07 15 Target: piecewise constant function Initial guess: straight line Very good result, small changes after 140 iterations...

16. AGIP MILANO2000 07 07 16 Log ! The cost function decreases of about 4 orders of magnitude. The steepest slope is obtained in the first ~20 iterations. A second sudden jump comes as the velocity gets the second ridge!

17. AGIP MILANO2000 07 07 17 Target: piecewise constant function Initial guess: straight line After ~10 iterations we get the first ridge...

18. AGIP MILANO2000 07 07 18 We see the steepest slope in the first ~10 iterations, a ‘plateau’ seems to follow!

19. AGIP MILANO2000 07 07 19 We take one of the last iterated field (#11) and freeze the gradient of the first (20) layers In ~20 iterations we reach both the first and the second ridges!

20. AGIP MILANO2000 07 07 20 After ~5 iterations the main ridge is detected!

21. AGIP MILANO2000 07 07 21 Target: piecewise constant function Initial guess: straight line but it does not matches the ‘trend’ Iterated velocity field Things goes wrong if the low frequency trend is not included in the initial guess...

22. AGIP MILANO2000 07 07 22

23. AGIP MILANO2000 07 07 23 Freezing the first 20 layers, the 1st discontinuity gets worse but we better recover the 2nd one... Here we start from #2 of previous iterations...

24. AGIP MILANO2000 07 07 24

25. AGIP MILANO2000 07 07 25 Target: parabola Initial guess: straight line Good! After ~170 iterations things does not change too much!

26. AGIP MILANO2000 07 07 26 Log ! 3 orders of magnitude decreasing! Steepest slope in the very first (~5) steps

27. AGIP MILANO2000 07 07 27 Target: parabola We start from the previous velocity field (#60) and freeze the gradient at the first layers (#20) In ~10 iterations we get a really good result!

28. AGIP MILANO2000 07 07 28 In the first ~8 iterations we have the steepest slope...

29. AGIP MILANO2000 07 07 29 Target: parabola + sin Initial guess: straight line Iterated velocity field Nice! The greatest part is done in the first ~100 iterations!

30. AGIP MILANO2000 07 07 30 Cost Function Log ! As observed, the big is done in the first ~100 iterations!

31. AGIP MILANO2000 07 07 31 Target: parabola + sin We start from a previous iteration (#20) and freeze the gradient at the first (20) layers Not good as before: we only get the medium trend!

32. AGIP MILANO2000 07 07 32

33. AGIP MILANO2000 07 07 33 The main problem is the presence of a large number of local minima. To get rid of them is possible to linearize the direct model (eg Born approx.), to have a convex cost function or adopt some multi-scale approach: large to small spatial scale, or low to high time frequencies but: loose refracted/multiply reflected waves, ecc. ecc. but: the ultra-low frequency (the velocity field trend) components don’t produce reflected waves, thus must be already present in the initial guess. Some preliminary conclusions

34. AGIP MILANO2000 07 07 34 Advantages of the time frequency domain high data compression rate (~10) uncoupled problems in  embarassing parallelism large to small spatial scale approach, inverting separately small and large frequencies quickest and scalable approach The advantage of the time domain is the intuitive comprehension of the involved fields and results Time versus Frequency Domain

35. AGIP MILANO2000 07 07 35 Extra time!

36. AGIP MILANO2000 07 07 36 Spectral conjugate gradient Spectral Conjugate Gradient Method the advantage is that in this way the conjugate direction (-  g) contains some explicit information on the Hessian matrix.

37. AGIP MILANO2000 07 07 37 In geophysic application, the number of parameters is very large, this motivates the choice of a conjugate gradient minimization algorithm Without uphill movements (a<0) in the line search procedure, none optimization method will prevent the trapping inside the local minima Modified Nonlinear Conjugate Gradient

38. AGIP MILANO2000 07 07 38 In our approach a can be either positive, describing a movement in the descent direction of p k, or negative. For a negative, the line search is similar Line search (  >0)

39. AGIP MILANO2000 07 07 39 very noisy function, presenting oscillations up to small scales (many local minima) after 7 steps both Wolfe conditions are satisfied Allowing a<0, the algorithm can visit and leave most local minima Analytical 1D example

40. AGIP MILANO2000 07 07 40 the function is a sum of a simple convex quadratic and low-amplitude high frequency perturbation (N=2) after 8 steps both Wolfe conditions are satisfied Allowing a<0, the algorithm can visit and leave most local minima Analytical 2D example

41. AGIP MILANO2000 07 07 41 same function as before, with N=32 standard gradient based minimization methods are not satisfactory with such a noisy function on nontrivial analytical examples, our approach converges quickly towards the global minimum Analytical 32D example

42. AGIP MILANO2000 07 07 42 The number of parameters plays a crucial role in the choice of the algorithm to minimize the cost function j(p) in the parameter space storage Without uphill movements in the line search procedure, none optimization method will prevent the trapping inside the local minima The landscape of the cost function presents many local minima convergence Number of parameters

43. AGIP MILANO2000 07 07 43 The number p of parameters impacts on the choice of the optimization strategy: for very small p the gradient can be computed numerically for small p, use the gradient and the Hessian to compute the search directions -exact Hessian (Newton) - approximation of the Hessian as the iteration progresses (Quasi-Newton). for large p, use only the gradient to compute the search directions -nonlinear conjugate gradient for very large p, use stochastic methods -simulated annealing The number p of parameters impacts on the choice of the optimization strategy: for very small p the gradient can be computed numerically for small p, use the gradient and the Hessian to compute the search directions -exact Hessian (Newton) - approximation of the Hessian as the iteration progresses (Quasi-Newton). for large p, use only the gradient to compute the search directions -nonlinear conjugate gradient for very large p, use stochastic methods -simulated annealing Number of parameters