Anna Avdeeva Dmitry Avdeev and Marion Jegen. 31 March Introduction to 3D MT inversion code x3di

Transcription

1 Anna Avdeeva Dmitry Avdeev and Marion Jegen 31 March 211

2 Outline 1 Essential Parts of 3D MT Inversion Code 2 Salt Dome Overhang Study with 3

3 Outline 1 Essential Parts of 3D MT Inversion Code 2 Salt Dome Overhang Study with 3

4 How 3D inverse problem is commonly solved Traditionally solution is sought as a stationary point of a penalty function ϕ(m, λ) = ϕ d (m) + λϕ s (m) m min (1) ϕ d (m) data misfit ϕ s (m) Tikhonov-type stabilizer λ regularization parameter m vector of model parameters (conductivities) ϕ d (m) = 1 d obs F (m) 2 (2) 2 F (m) is a forward problem mapping ϕ s (m) = 1 W(m m ref ) 2 (3) 2 m ref vector of model parameters, for some reference model, which usually include some a priori information.

5 Where are the differences between 3D inversion codes? 1 model parameters (σ, ρ, log ρ, log σ etc.) 2 forward problem solver (FD, FE or IE) 3 optimization method (GN, QN, LMQN, NLCG etc.) 4 form of the data misfit ϕ d 5 form of the stabilizer ϕ s

6 Where are the differences between 3D inversion codes? 1 model parameters (σ, ρ, log ρ, log σ etc.) 2 forward problem solver (FD, FE or IE) 3 optimization method (GN, QN, LMQN, NLCG etc.) 4 form of the data misfit ϕ d 5 form of the stabilizer ϕ s

7 Where are the differences between 3D inversion codes? 1 model parameters (σ, ρ, log ρ, log σ etc.) 2 forward problem solver (FD, FE or IE) - X3D 3 optimization method (GN, QN, LMQN, NLCG etc.) 4 form of the data misfit ϕ d 5 form of the stabilizer ϕ s

8 Where are the differences between 3D inversion codes? 1 model parameters (σ, ρ, log ρ, log σ etc.) 2 forward problem solver (FD, FE or IE) - X3D 3 optimization method (GN, QN, LMQN, NLCG etc.) 4 form of the data misfit ϕ d 5 form of the stabilizer ϕ s

9 method All iterative methods work at the same manner. At each iteration l they find: 1 a search direction vector p (l) 2 a step length α (l) using an inexact line search along p (l) 3 the next, improved model given by m (l+1) = m (l) + α (l) p (l) The difference is in how a specific method finds the search direction and in what price is paid for this. NLCG p (l) = g (l) + γ (l) p (l 1), where γ (l) = Newton, GN H (l) p (l) = g (l) QN {m (i), g (i) : i = 1,..., l} p (l) LMQN {m (i), g (i) : i = l n cp,..., l} p (l) (l) g(l) g g (l 1) g(l 1) H (l) ij = 2 ϕ m i m j (m (l) ), g (l) i = ϕ m i (m (l) )

10 method All iterative methods work at the same manner. At each iteration l they find: 1 a search direction vector p (l) 2 a step length α (l) using an inexact line search along p (l) 3 the next, improved model given by m (l+1) = m (l) + α (l) p (l) The difference is in how a specific method finds the search direction and in what price is paid for this. NLCG p (l) = g (l) + γ (l) p (l 1), where γ (l) = Newton, GN H (l) p (l) = g (l) QN {m (i), g (i) : i = 1,..., l} p (l) LMQN {m (i), g (i) : i = l n cp,..., l} p (l) (l) g(l) g g (l 1) g(l 1) H (l) ij = 2 ϕ m i m j (m (l) ), g (l) i = ϕ m i (m (l) )

11 Adjoint approach gradients requires 2 forward modellings at each frequency Important: Straightforward calculation of the gradients would require N + 1 forward modellings, i.e. in N/2 times more Example: number of model parameters = 3 Single forward modelling requires 4 min on PC Straightforward calculation of the single gradient requires 8 days Usually 2 iterations(gradients) is needed Total inversion time is 4 years.

12 3D data misfit ϕ d (σ) = 1 2 N S N T ] β ij tr [ĀT ij (σ)a ij (σ) i=1 j=1 σ = (σ 1,..., σ N ) T the vector of the electrical conductivities of the cells, N number of cells N S number of MT sites r i = (x i, y i, z = ) N T number of frequencies ω j A ij = Z ij D ij 2 2 matrices complex-valued predicted Z(r i, ω j ) impedance Z ij D ij complex-valued observed D(r i, ω j ) impedance β ij some positive weights ] tr [ĀT A = Ā 11 A 11 + Ā 12 A 12 + Ā 21 A 21 + Ā 22 A 22 (4)

13 Gradients: Adjoint approach ϕ d σ k N T = Re Maxwell s equation 2 ( j=1 p=1 V k E (p) j Adjoint Maxwell s equation u (p) j u (p) x E (p) x + u (p) y E (p) y 1ω j µσ(r)e (p) j 1ω j µσ(r)u (p) j = 1ω j µ + u (p) z E (p) z ) dv (5) = 1ω j µj (p) j (6) ( j (p) j ) + h (p) j j (p) j and h (p) j - horizontal electric and magnetic dipoles at the MT sites p = 1, 2 - polarization (7) The forward modelling is performed with X3D by (Avdeev et al., 22).

14 Grid and MT sites N= cells dx = dy = 4 km N S = 8 MT sites N T =3 periods: 1, 3, 1 s noise - 1% initial guess model: 5 Ωm halfspace

15 Logarithmic parametrization vs conductivities: inversion results

16 Logarithmic parametrization vs conductivities: convergence curves Misfit Misfit without additional regularization m= σ m=log 1 σ Target misfit n fg with additional regularization m= σ m=log 1 σ Target misfit log 1 (λ) log 1 (λ) n fg

17 Tikhonov-type stabilizers: Laplace ϕ s = dxdy αβγ Gradient ϕ s = dxdy [ ( m ) 2 + x αβγ [ 2 ] 2 m x m y m z 2 dz γ (8) αβγ ( ) 2 m + y ( ) ] 2 m dz γ (9) z αβγ

18 Gradient vs Laplace: inversion results

19 Gradient vs Laplace: inversion results Laplace Gradient True model

20 Gradient vs Laplace: convergence curves Misfit Laplace regul. Gradient regul. Target misfit n fg log 1 (λ)

21 Outline 1 Essential Parts of 3D MT Inversion Code 2 Salt Dome Overhang Study with 3

22 3D Model of a salt wall

23 1995 MT sites N= cells dx = dy =.25 km N S =1995 MT sites N T =5 frequencies: Hz noise - 5% initial guess model: 11 Ωm halfspace

24 Inversion result for model 2 with overhang MT sites. Horizontal slices km km km km 1 y (km) Inversion result z :.2.5 km Model 2: with overhang z :.2.5 km km km km km y (km) x (km) x (km) x (km) x (km) 1 1 Resistivity (Ωm) x (km) 1 2

25 Inversion result: model 1 vs model MT sites. Vertical slices

26 15 MT sites along three profiles N= cells dx = dy =.25 km N S =15 MT sites N T =5 frequencies: Hz noise - 5% initial guess model: 11 Ωm halfspace

27 Inversion result for model 2 with overhang. 15 MT sites. Horizontal slices Inversion result km km km km 1 A y (km) z :.2.5 km B 2 C Model 2: with overhang z :.2.5 km km km km km 1 A y (km) B 2 C x (km) x (km) x (km) x (km) 1 1 Resistivity (Ωm) x (km) 1 2

28 Inversion result: model 1 vs model MT sites. Vertical slices

29 Outline 1 Essential Parts of 3D MT Inversion Code 2 Salt Dome Overhang Study with 3

30 Logarithmic parametrization is beneficial in terms of inversion result and computational time based on Gradient suppresses spatial resistivity gradients The code produces encouraging results for salt dome overhang detectability

31 Acknowledgements We would like to thank Wintershall Holding AG, who funded the inversion study and 2 ocean bottom MT instruments

32 Thanks for your attention Avdeev, D. B. and Avdeeva, A. D. (29). Three-dimensional magnetotelluric inversion using a limited-memory QN optimization. Geophysics, 74(3), F45 F57. Avdeev, D. B., Kuvshinov, A. V., Pankratov, O. V., and Newman, G. A. (22). Three-dimensional induction logging problems, Part I: An integral equation solution and model comparisons. Geophysics, 67, Avdeeva, A. D. and Avdeev, D. B. (26). A limited-memory quasi-newton inversion for 1D magnetotellurics. Geophysics, 71, G191 G196.