Efficient Preconditioning in 3D Marine Electromagnetic Geophysical - PowerPoint PPT Presentation

Efficient Preconditioning in 3D Marine Electromagnetic Geophysical Modeling N. Yavich, M. Malovichko, M. Zhdanov Applied Computational Geophysics Lab Moscow Institute of Physics and Technology September 2016

One of several Marine Electromagnetic (EM) Acquisition Setups Figure courtesy of pgs.com To plan a survey and interpret collected data, we have to solve low-frequency Maxwell’s equation repeatedly. 2

Electrical Conductivity Model Let’s consider a 3D heterogeneous conductivity Earth model composed of layered background with conductivity 𝜏 𝑐 (𝑨) and anomalous inclusions (bodies) 𝐸 with conductivity 𝜏 𝑏 + 𝜏 𝑐 . 𝜏 𝑐 𝐸 𝜏 𝑐 + 𝜏 𝑏 To have some simple measure of control on lateral contrast of the model, we assume there exist 𝛽 and 𝛾 such that, 𝛽 𝜏 𝑐 𝑨 ≤ 𝜏 𝑦, 𝑧, 𝑨 ≤ 𝛾 𝜏 𝑐 𝑨 0 < 𝛽 ≤ 1 ≤ 𝛾 < ∞, 𝜏 = 𝜏 𝑏 + 𝜏 𝑐 in 𝐸, . This inequality insures that the anomalous 𝜏 𝑐 in ℝ 3 \𝐸 inclusions are neither perfect conductors nor insulators. 3

Low- frequency Maxwell’s equations Within some finite volume 𝑊 , we formulate the secondary field low- frequency Maxwell’s equations: 𝑠𝑝𝑢 𝑠𝑝𝑢 𝐹 𝑏 − 𝑗 ω μ 0 𝜏 𝑐 𝐹 𝑏 = 𝑗 ω μ 0 𝜏 𝑏 (𝐹 𝑏 + 𝐹 𝑐 ) . 𝜏 𝑐 𝐹 𝑏 × 𝜉 = 0 . 𝐸 𝜏 𝑐 + 𝜏 𝑏 Here 𝐹 𝑏 is unknown, while layered Earth resposne 𝐹 𝑐 can be easily computed quasi- analytically. 𝑊 We will discuss efficient solution of the finite- difference (FD) discretization of the later 𝐹 = 𝐹 𝑏 + 𝐹 𝑐 equations. 4

FD System - 1 We use edge-based electric fields and edge-sampled conductivities on a non-uniform grid, 𝑂 𝑦 × 𝑂 𝑧 × 𝑂 𝑨 . The total number of unknowns is 𝑜 ≈ 3𝑂 𝑦 𝑂 𝑧 𝑂 𝑨 . 5

FD System - 2 We will use the following notations for FD operators and unknowns, 𝐹 𝑐 ≈ 𝑓 𝑐 , 𝐹 𝑏 ≈ 𝑓 𝑏 𝜏 𝑦, 𝑧, 𝑨 ≈ Σ , (diagonal matrices) 𝜏 𝑐 𝑨 ≈ Σ 𝑐 , 𝜏 𝑏 𝑦, 𝑧, 𝑨 ≈ Σ 𝑏 , 𝑠𝑝𝑢 𝑠𝑝𝑢 − 𝑗 ω μ 0 𝜏𝐽 ≈ 𝐵 , 𝑠𝑝𝑢 𝑠𝑝𝑢 − 𝑗 ω μ 0 𝜏 𝑐 𝐽 ≈ 𝐵 𝑐 FD secondary field formulation: 𝐵𝑓 𝑏 = 𝑗𝜕𝜈 0 Σ 𝑏 𝑓 𝑐 or 𝐵 𝑐 𝑓 𝑏 = 𝑗𝜕𝜈 0 Σ 𝑏 (𝑓 𝑏 + 𝑓 𝑐 ). This problems have typically 1 to 10 million unknowns. Their efficient solution is of major importance. 6

Major Preconditioning Approaches Sever major approaches are applicable to this problem. Some of them are • Geometric and algebraic multigrid, • ILU, ILUt, etc, • Domain decomposition methods, • Discrete separation of variables. We will base our presentation on discrete separation of variables, since it provides decent spectral properties of the preconditioned problem (will be proved later) and very memory economical. 7

FD GF Preconditioner – 1 Matrix 𝐵 𝑐 can be efficiently factorized, −1 𝑣 is at most 𝑃(𝑂 𝑦 𝑂 𝑧 𝑂 𝑨 𝑂 𝑦 + 𝑂 𝑧 ) so that complexity to compute 𝐵 𝑐 operations and auxiliary memory 𝑃 𝑜 . Consequently, we may used as a preconditioner, −1 𝐵 𝑓 𝑏 = 𝑗𝜕𝜈 0 𝐵 𝑐 −1 Σ 𝑏 𝑓 𝑐 or 𝑓 𝑏 = 𝑗𝜕𝜈 0 𝐵 𝑐 −1 Σ 𝑏 (𝑓 𝑏 + 𝑓 𝑐 ) 𝐵 𝑐 This is pretty much an equivalent of the IE formulation −1 is the FD Green’s function (GF) of the layered media. since 𝐵 𝑐 How good will be this preconditioner? 8

FD GF Preconditioner – 2 We studied the respective eigenvalue problem, −1 𝐵 𝑤 = 𝜇 𝑤 , 𝐵 𝑐 to understand properties of this preconditioner. −1 𝐵 ≈ |𝜇 max | |𝜇 min | ≤ 𝛾 𝑑𝑝𝑜𝑒 𝐵 𝑐 𝛽 9

Contraction Operator – 1 Let us try to improve the later result. Our formulation of the Maxwell’s equations imply the energy equality, ∗ ⋅ 𝐾 𝑏 𝑒𝑊 = 0. ‍ 𝜏 𝑐 𝐹 𝑏 2 𝑒𝑊 + 𝑆𝑓 ‍ 𝐹 𝑏 𝑊 𝑊 It also holds at the discrete level, 1 2 𝑓 𝑏 ∥ 2 +𝑆𝑓(𝑓 𝑏 ∗ , 𝑘 𝑏 ) = 0. ∥ Σ 𝑐 The equality can be used to transform the FD system. Introduce, 𝐿 1 = 1 𝐿 2 = 1 −1/2 , −1/2 , 2 Σ + Σ 𝑐 Σ 𝑐 2 Σ − Σ 𝑐 Σ 𝑐 𝑓 𝑏 = 𝐿 1 𝑓 𝑏 . 10

Contraction Operator – 2 Then 𝑓 𝑏 will satisfy, 𝐽 − 𝐷 𝑓 𝑏 = 𝑔, where, 1 1 2 + 𝐽 𝐿 2 𝐿 1 2 𝐵 𝑐 −1 Σ 𝑐 −1 , 𝐷 = 2𝑗 ω μ 0 Σ 𝑐 1 −1 Σ 𝑏 𝑓 𝑐 . 2 𝐵 𝑐 𝑔 = 𝑗 ω μ 0 Σ 𝑐 Interestingly, 𝐷 < 1. Thus we will refer this transformation as the contraction operator (CO) preconditioner. 11

Contraction Operator – 3 𝐽 − 𝐷 𝑣 = 𝜇 𝑣. 𝐷 ≤ 1 − 2 min (𝛽, 1 𝛾) =: 𝛿. 1 It can be proved, 𝑑𝑝𝑜𝑒 𝐽 − 𝐷 ≤ max 1 𝛽 , 𝛾 . Comparison of the two condition numbers leads us to a conclusion. When the bodies are only resistive or conductive, the covergence of iterative solvers will be similar is similar. In case of resistive and conductive bodies, CO will provide faster convergence. 12

Marine resistivity model of a hydrocarbon deposit 0.3 2 1 5 4 200 9 7 12 16 100 Ohm·m indicated 13

Sampled model/ towed source and receiver array Source Receiver array Colors indicate Ohm·m 14

Performance comparison 172 x 96 x 83 computational grid, 4’034’327 unknowns BiCGStab, 𝜁 =1e-8 Performance of the solver at one of the source positions: FD GF Contraction Preconditioner operator Iterations/ time, s Iterations/ time, s 78 / 445 31 / 180 We observed a speed up of 2.5 times!

Towed Streamer Data Sensitivity – 1 We modeled responses at 32 setup locations for models with and without deposit. vs. 16

Towed Streamer Data Sensitivity – 2 Below is the ratio the responses. We good data sensitivity: 48% amplitude anomaly, 32 ° phase anomaly. half-offset common mid-point 17

Summary • We designed, analyzed, and tested two preconditioners for 3D electromagnetic low-frequency modeling. • Our analysis and tests showed that convergence of iterative solvers applied to CO preconditioned system is faster or same than that applied to GF preconditioned system. • We also demonstrated applicability of the approaches to marine geophysical EM modeling. 18