Education Ph.D. Candidate, Rice University, Houston, TX, USA, 08/ Present. M.S, Shanghai Jiao Tong University, Shanghai, China, 09/ /2009

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1 Yin Huang Education Ph.D. Candidate, Rice University, Houston, TX, USA, 08/ Present Dissertation Topic: Nonlinear Extended Waveform Inversion M.A. with Master Thesis: Transparency property of one dimensional acoustic wave equations Relevant courses: Numerical Differential Equations; Optimization; High Performance Computing; Geophysical Data Analysis M.S, Shanghai Jiao Tong University, Shanghai, China, 09/ /2009 Dissertation topic: Comparision of many numerical methods for saddle point system arising from the mixed finite element method of elliptic problems with nonsmooth coefficients Research Interests Seismic waveform inversion Forward and inverse problems for non-homogeneous medium High performace computing 1

2 Linearized Extended Waveform Inversion: Reduced Objective Function and Its Gradient Yin Huang Advisor: William Symes The Rice Inversion Project Annual Review Meeting April 18,

3 Introduction Seismic reflection inversion Over-determined: highly redundant in the observed data; (Gauthier et al., 1986; Santosa & Symes, 1989; Virieux & Operto, 2009) Extended model fitting Under-determined: model has more degree of freedom than data. Data misfit + extended modeling + differential semblance smooth objective function of model parameter. (Symes & Kern 1994, Symes, 1999, Stolk & Symes 2003) 3

4 Introduction Seismic reflection inversion Over-determined: highly redundant in the observed data; (Gauthier et al., 1986; Santosa & Symes, 1989; Virieux & Operto, 2009) Extended model fitting Under-determined: model has more degree of freedom than data. Data misfit + extended modeling + differential semblance smooth objective function of model parameter. (Symes & Kern 1994, Symes, 1999, Stolk & Symes 2003) 3

5 Introduction Seismic reflection inversion Over-determined: highly redundant in the observed data; (Gauthier et al., 1986; Santosa & Symes, 1989; Virieux & Operto, 2009) Extended model fitting Under-determined: model has more degree of freedom than data. Data misfit + extended modeling + differential semblance smooth objective function of model parameter. (Symes & Kern 1994, Symes, 1999, Stolk & Symes 2003) 3

6 Waveform Inversion and Extended Modeling Extended Waveform Inversion: Given data d D, find m M so that (Symes, 1986, 1991; Biondi and Almomin, 2012) F [ m] d. M = {m(x)} physical model space: velocity, density, bulk modulus,... M = {m(x, h)} extended model space, M M. D data space. F : M D forward map: acoustic, elastic... F : M D extended forward map. Extended model separation: m m l + δ m. Background model is physical. Reflectivity is extended. 4

7 Waveform Inversion and Extended Modeling Extended Waveform Inversion: Given data d D, find m M so that (Symes, 1986, 1991; Biondi and Almomin, 2012) F [ m] d. M = {m(x)} physical model space: velocity, density, bulk modulus,... M = {m(x, h)} extended model space, M M. D data space. F : M D forward map: acoustic, elastic... F : M D extended forward map. Extended model separation: m m l + δ m. Background model is physical. Reflectivity is extended. 4

8 Linearized Inversion Given data d D, find m l, δ m so that D F [m l ]δ m d F [m l ]. D F is the derivative, or Born approximation. (Symes and Carazzone, 1991; Chauris and Noble, 2001; Mulder and ten Kroode, 2002; Shen and Symes, 2008; Symes 2008.) Objective function: J[m l, δ m] = 1 2 D F [m l ]δ m (d F [m l ]) 2 + α2 Aδ m 2 2 A annihilator, Aδm = 0 for all δm M. 5

9 Linearized Inversion Given data d D, find m l, δ m so that D F [m l ]δ m d F [m l ]. D F is the derivative, or Born approximation. (Symes and Carazzone, 1991; Chauris and Noble, 2001; Mulder and ten Kroode, 2002; Shen and Symes, 2008; Symes 2008.) Objective function: J[m l, δ m] = 1 2 D F [m l ]δ m (d F [m l ]) 2 + α2 Aδ m 2 2 A annihilator, Aδm = 0 for all δm M. 5

10 Reduced Objective Function The value of the reduced objective function is the minimum value of J for fixed m l J[m l ] = min J[m l, δ m]. δ m Let Minimizer of J N[m l ] = D F [m l ] T D F [m l ] + α 2 A T A δ m[m l ] = N[m l ] 1 D F [m l ] T (d F [m l ]). 6

11 Gradient of J[m l ] Directional derivative D J[m l ]dm l = D ml J[m l, δ m]dm l + D δ m J[m l, δ m]d ml δ mdm l. Second term is 0, if δ m is solved exactly. Direct computation gives gradient J[m l ] = D 2 F [m l ] T [δ m, D F [m l ]δ m (d F [m l ])] +DF [m l ] T (D F [m l ]δ m (d F [m l ])). Second term is zero if apply an appropriate cutoff function to the data. D 2 F [ml ] T : M D such that for dml M, q M and φ D D 2 F [m l ][dm l, q], φ = dm l, D 2 F [m l ] T [q, φ]. 7

12 Gradient of J[m l ] Directional derivative D J[m l ]dm l = D ml J[m l, δ m]dm l + D δ m J[m l, δ m]d ml δ mdm l. Second term is 0, if δ m is solved exactly. Direct computation gives gradient J[m l ] = D 2 F [m l ] T [δ m, D F [m l ]δ m (d F [m l ])] +DF [m l ] T (D F [m l ]δ m (d F [m l ])). Second term is zero if apply an appropriate cutoff function to the data. D 2 F [m l ] T : M D such that for dm l M, q M and φ D D 2 F [m l ][dm l, q], φ = dm l, D 2 F [m l ] T [q, φ]. 7

13 Example: Computation of D 2 F [ml ] T For acoustic constant density wave equation: q M, φ D. Let ω solve the zero final value problem ( 1 2 ) t 2 ω = dx r φδ(x x r ). m 2 l D F [m l ] T φ = 2 m l dtω u. Let ω 0 solve the zero final value problem ( 1 2 ) ( ) q t 2 ω 0 = 2 ω. m l m 2 l D 2 F T [m l ][q, φ] = 2 dm l m l dh with ū the Born approximation wave field. dt (( ū + q ) ) u ω + ω 0 u. m l 8

14 Example: Computation of D 2 F [ml ] T For acoustic constant density wave equation: q M, φ D. Let ω solve the zero final value problem ( 1 2 ) t 2 ω = dx r φδ(x x r ). m 2 l D F [m l ] T φ = 2 m l dtω u. Let ω 0 solve the zero final value problem ( 1 2 ) ( ) q t 2 ω 0 = 2 ω. m l m 2 l D 2 F T [m l ][q, φ] = 2 dm l m l dh with ū the Born approximation wave field. dt (( ū + q ) ) u ω + ω 0 u. m l 8

15 Gradient needs a correction term Why need correction term? δ m J[m l, δ m] is not zero numerically. D ml δ m could be large. Neglect of the second term leads to large errors (Symes and Kern, 1994). Iteration method is used to solve for δ m, P k polynomial with degree k. δ m k [m l ] = P k (N[m l ])D F [m l ] T (d F [m l ]). D ml δ m k [m l ]dm l = D(P k (N[m l ]))dm l D F [m l ] T (d F [m l ]) +P k (N[m l ])D 2 F [ml ] T [dm l, d F [m l ]] P k (N[m l ])D F [m l ] T DF [m l ]dm l. 9

16 Gradient needs a correction term Why need correction term? δ m J[m l, δ m] is not zero numerically. D ml δ m could be large. Neglect of the second term leads to large errors (Symes and Kern, 1994). Iteration method is used to solve for δ m, P k polynomial with degree k. δ m k [m l ] = P k (N[m l ])D F [m l ] T (d F [m l ]). D ml δ m k [m l ]dm l = D(P k (N[m l ]))dm l D F [m l ] T (d F [m l ]) +P k (N[m l ])D 2 F [ml ] T [dm l, d F [m l ]] P k (N[m l ])D F [m l ] T DF [m l ]dm l. 9

17 Chebyshev iteration Chebyshev is preferred. Coefficients of P k does not depend on N[m l ], once spectrum bound of N[m l ] is known in advance. Easy to analyze D ml δ m k [m l ]. Coefficients in conjugate gradient depends on N[m l ]. Bound on derivative is not obvious to calculate. For operator A, want P k (A) A 1 I AP k (A) 0 Fix k, find P k with leading coefficient 1, such that P k = arg min p k max 1 λp k(λ) λ with λ [λ min, λmax], i.e. spectrum bound of operator A. NOTE: Chebyshev polynomial is the unique optimal solution. (Varga 1962) 10

18 Chebyshev iteration Chebyshev is preferred. Coefficients of P k does not depend on N[m l ], once spectrum bound of N[m l ] is known in advance. Easy to analyze D ml δ m k [m l ]. Coefficients in conjugate gradient depends on N[m l ]. Bound on derivative is not obvious to calculate. For operator A, want P k (A) A 1 I AP k (A) 0 Fix k, find P k with leading coefficient 1, such that P k = arg min p k max 1 λp k(λ) λ with λ [λ min, λmax], i.e. spectrum bound of operator A. NOTE: Chebyshev polynomial is the unique optimal solution. (Varga 1962) 10

19 Gradient with a correction term P k (N[m l]) is of order k 2. D(P k (N[m l ])) D ml N[m l ]P k (N[m l]) is of order k 2. D(P k (N[m l ]))r k = O(k 2 r k ). Normal residual r k = N[m l ]δ m k D F [m l ] T (d F [m l ]) decrease exponentially in k. D δ m J[m l, δ m]d ml δ mdm l = D ml (P k (N[m l ]))dm l D F [m l ] T (d F [m l ]), r k + P k (N[m l ])D 2 F [ml ] T [dm l, d F [m l ]], r k dm l, D 2 F [ml ] T [P k (N[m l ])r k, d F [m l ]]. Gradient with correction term J[m l ] D 2 F [ml ] T [δ m, D F [m l ]δ m (d F [m l ])] +D 2 F [m l ] T [P k (N[m l ])r k, d F [m l ]]. 11

20 Gradient with a correction term P k (N[m l]) is of order k 2. D(P k (N[m l ])) D ml N[m l ]P k (N[m l]) is of order k 2. D(P k (N[m l ]))r k = O(k 2 r k ). Normal residual r k = N[m l ]δ m k D F [m l ] T (d F [m l ]) decrease exponentially in k. D δ m J[m l, δ m]d ml δ mdm l = D ml (P k (N[m l ]))dm l D F [m l ] T (d F [m l ]), r k + P k (N[m l ])D 2 F [ml ] T [dm l, d F [m l ]], r k dm l, D 2 F [ml ] T [P k (N[m l ])r k, d F [m l ]]. Gradient with correction term J[m l ] D 2 F [ml ] T [δ m, D F [m l ]δ m (d F [m l ])] +D 2 F [m l ] T [P k (N[m l ])r k, d F [m l ]]. 11

21 Restarting Chebyshev iteration Simple scheme: Compute no. of iterations k and Chebyshev coefficients Initialize the iteration, compute Rayleigh Quotient (RQ) to get spectrum bound λ Run Chebyshev iteration until Get bigger λ, restart Chebyshev Or run k steps and return. NOTE: Performance depends on condition number. Good preconditioner is needed. Implementation is available in RVL. Restarting curve for depth-oriented extended waveform inversion (Liu et al. 2013). 12

22 Summary Reduced objective function Smooth in long scale model D 2 F [ml ] T of extended acoustic constant density forward operator Gradient Correction term to the gradient Chebyshev iteration Better than CG for operator with moderate condition number and uniformly distributed spectrum Restarting scheme for operators without prior information of spectrum Available in RVL 13

23 Future work Short term Fill in detailed mathematical analysis Find good preconditioner for shot-coordinate extension Long term Extend to nonlinear waveform inversion (D. Sun s PhD thesis) 14

24 Acknowledgements Great thanks to Current and former TRIP team members. Sponsors of The Rice Inversion Project. Wonderful audience. 15