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

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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.

Electrical Conductivity Model

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Let’s consider a 3D heterogeneous conductivity Earth model composed of layered background with conductivity 𝜏𝑐(𝑨) and anomalous inclusions (bodies) 𝐸 with conductivity 𝜏𝑏 + 𝜏𝑐. To have some simple measure of control on lateral contrast of the model, we assume there exist 𝛽 and 𝛾 such that, 𝛽 𝜏𝑐 𝑨 ≤ 𝜏 𝑦, 𝑧, 𝑨 ≤ 𝛾 𝜏𝑐 𝑨 0 < 𝛽 ≤ 1 ≤ 𝛾 < ∞, This inequality insures that the anomalous inclusions are neither perfect conductors nor insulators. 𝜏𝑐 𝜏𝑐 + 𝜏𝑏 𝜏 = 𝜏𝑏 + 𝜏𝑐 in 𝐸, 𝜏𝑐 in ℝ3\𝐸 .

𝐸

Low-frequency Maxwell’s equations

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Within some finite volume 𝑊, we formulate the secondary field low-frequency Maxwell’s equations: 𝑠𝑝𝑢 𝑠𝑝𝑢 𝐹𝑏 − 𝑗ωμ0𝜏𝑐𝐹𝑏 = 𝑗ωμ0𝜏𝑏(𝐹𝑏 + 𝐹𝑐). 𝐹𝑏 × 𝜉 = 0. Here 𝐹𝑏 is unknown, while layered Earth resposne 𝐹𝑐 can be easily computed quasi- analytically. We will discuss efficient solution of the finite- difference (FD) discretization of the later equations. 𝜏𝑐 𝜏𝑐 + 𝜏𝑏

𝐸 𝑊 𝐹 = 𝐹𝑏 + 𝐹𝑐

FD System - 1

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We use edge-based electric fields and edge-sampled conductivities on a non-uniform grid, 𝑂𝑦 × 𝑂𝑧 × 𝑂𝑨 . The total number of unknowns is 𝑜 ≈ 3𝑂𝑦𝑂𝑧𝑂𝑨 .

FD System - 2

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We will use the following notations for FD operators and unknowns, 𝐹𝑐 ≈ 𝑓𝑐, 𝐹𝑏 ≈ 𝑓𝑏 𝜏 𝑦, 𝑧, 𝑨 ≈ Σ, (diagonal matrices) 𝜏𝑐 𝑨 ≈ Σ𝑐, 𝜏𝑏 𝑦, 𝑧, 𝑨 ≈ Σ𝑏, 𝑠𝑝𝑢 𝑠𝑝𝑢 − 𝑗ωμ0𝜏𝐽 ≈ 𝐵, 𝑠𝑝𝑢 𝑠𝑝𝑢 − 𝑗ωμ0𝜏𝑐𝐽 ≈ 𝐵𝑐 FD secondary field formulation: 𝐵𝑓𝑏 = 𝑗𝜕𝜈0 Σ𝑏𝑓𝑐 or 𝐵𝑐𝑓𝑏 = 𝑗𝜕𝜈0Σ𝑏(𝑓𝑏 + 𝑓𝑐). This problems have typically 1 to 10 million unknowns. Their efficient solution is of major importance.

Major Preconditioning Approaches

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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.

FD GF Preconditioner – 1

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Matrix 𝐵𝑐 can be efficiently factorized, so that complexity to compute 𝐵𝑐

−1𝑣 is at most 𝑃(𝑂𝑦𝑂𝑧𝑂𝑨 𝑂𝑦 + 𝑂𝑧 )

• perations and auxiliary memory 𝑃 𝑜 .

Consequently, we may used as a preconditioner, 𝐵𝑐

−1 𝐵 𝑓𝑏 = 𝑗𝜕𝜈0 𝐵𝑐 −1 Σ𝑏𝑓𝑐 or 𝑓𝑏 = 𝑗𝜕𝜈0 𝐵𝑐 −1 Σ𝑏(𝑓𝑏 + 𝑓𝑐)

This is pretty much an equivalent of the IE formulation since 𝐵𝑐

−1 is the FD Green’s function (GF) of the layered media.

How good will be this preconditioner?

FD GF Preconditioner – 2

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We studied the respective eigenvalue problem, 𝐵𝑐

−1 𝐵 𝑤 = 𝜇 𝑤,

to understand properties of this preconditioner. 𝑑𝑝𝑜𝑒 𝐵𝑐

−1𝐵 ≈ |𝜇max |

|𝜇min | ≤ 𝛾 𝛽

Contraction Operator – 1

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Let us try to improve the later result. Our formulation of the Maxwell’s equations imply the energy equality, ‍

𝑊

𝜏𝑐 𝐹𝑏 2𝑒𝑊 + 𝑆𝑓 ‍

𝑊

𝐹𝑏

∗ ⋅ 𝐾𝑏𝑒𝑊 = 0.

It also holds at the discrete level, ∥ Σ𝑐

1 2𝑓𝑏 ∥2 +𝑆𝑓(𝑓𝑏 ∗, 𝑘𝑏) = 0.

The equality can be used to transform the FD system. Introduce, 𝐿1 = 1 2 Σ + Σ𝑐 Σ𝑐

−1/2,

𝐿2 = 1 2 Σ − Σ𝑐 Σ𝑐

−1/2,

𝑓 𝑏 = 𝐿1𝑓𝑏.

Contraction Operator – 2

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Then 𝑓𝑏 will satisfy, 𝐽 − 𝐷 𝑓 𝑏 = 𝑔, where, 𝐷 = 2𝑗ωμ0 Σ𝑐

1 2 𝐵𝑐 −1 Σ𝑐 1 2 + 𝐽 𝐿2𝐿1 −1,

𝑔 = 𝑗ωμ0Σ𝑐

1 2𝐵𝑐 −1Σ𝑏𝑓𝑐.

Interestingly, 𝐷 < 1. Thus we will refer this transformation as the contraction operator (CO) preconditioner.

Contraction Operator – 3

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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.

𝐽 − 𝐷 𝑣 = 𝜇 𝑣. 𝐷 ≤ 1 − 2 min (𝛽, 1 𝛾) =: 𝛿. 1

Marine resistivity model of a hydrocarbon deposit

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200

0.3 1 2 5 9 16 100 4 7 12

Ohm·m indicated

Sampled model/ towed source and receiver array

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Source Receiver array Colors indicate Ohm·m

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: We observed a speed up of 2.5 times!

FD GF Preconditioner Contraction

• perator

Iterations/ time, s Iterations/ time, s 78 / 445 31 / 180

Towed Streamer Data Sensitivity – 1

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We modeled responses at 32 setup locations for models with and without deposit. vs.

Towed Streamer Data Sensitivity – 2

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Below is the ratio the responses. We good data sensitivity: 48% amplitude anomaly, 32° phase anomaly. common mid-point half-offset

Summary

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• 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.