Model Reduction Approaches to Forward and Inverse Elastic Wave

Model Reduction Approaches to Forward and Inverse Elastic Wave Problems with Active Sources in Fractured Reservoirs Vladimir Druskin Contributors: Alexander Mamonov, Dzevat Omeragic, Olga Podgornova, Andrew Thaler, Mikhail Zaslavsky, Smaine Zeroug

Outline

1

Motivations

2

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS ROMs via Stieltjes string/optimal grid Multi-scale mimetic reduced-order models with spectral accuracy

3

Nonlinear seismic imaging via ROM backprojection 1D model with SISO data 2D model with MIMO data

4

Conclusions

Motivations

Simple model with multiple fractures1

1

Courtesy of the U.S. Department of Energy’s National Energy Technology Laboratory

Motivations

Elastic wave propagation in model with hydraulic fractures

Waves in fractured reservoirs satisfying dynamic elasticity system σij,j

= dvi,j /dt,

Λijkl (vk,l + vl,k )/2 = dσij /dt. Red stars are seismic sources and receivers.

Motivations

Challenges for forward modeling and inversion Forward modeling: Fractures are sharp contrast layers O(1mm) thick and O(10m) long, that leads to fine (compared to the wavelength) grid and results in huge computational cost for low order approximations. High-order and spectral methods require adaptive gridding to conform fractures. That leads to prohibitively expensive time-stepping due to severe limitation on the CFL limit.

Inversion: Traditional seismic imaging methods (e.g., RTM) suffer from multiple reflections, do not produce true amplitude. Need time-consuming preprocessing to alleviate these artifacts. The optimization techniques in the framework of so-called full waveform inversion (FWI) may require weeks of processing time due to multiple forward solver calls.

Motivations

Reduced Order Models Reduced order models (ROMs) approximate transfer functions of large-scale linear dynamical systems by small equivalent ones. We hope to design ROMs for low dimensional approximation of scattering elastic waves by fine-scale fractured models. The hyperbolic elasticity system in our case is linear time-invariant (LTI) dynamical system. Model reduction field for such system is burgeoning field from 1980’s, with applications in optimal design of electrical circuits and mechanical structures. In particularly, the LTI model reduction apparatus was greatly benefited from discovery of Pad´e-Krylov connection in late 1980’s. Here we built ROM modeling and imaging techniques based on that approach.

Motivations

Mathematicians enlisted to help us:

From left to right: Thomas Joannes Stieltjes, 1856-1894; Mark Grigorievich Krein, 1907–1989; Israel Moiseevich Gelfand, 1913–2009.

Outline

1

Motivations

2

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS ROMs via Stieltjes string/optimal grid Multi-scale mimetic reduced-order models with spectral accuracy

3

Nonlinear seismic imaging via ROM backprojection 1D model with SISO data 2D model with MIMO data

4

Conclusions

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

ROMs via Stieltjes string/optimal grid

Two-point second order ODE problem We start with model problem a mixed boundary value problem on [0, 1] for ODE system (σux )x − λρu = 0,

σ

d u|x=0 = −1, u|x=1 = 0 dx

with σ(x) > 0, ρ(x) ≥ 0, regular enough functions on [0, 1], λ ∈ C \ R− E.g., this eqn. can be obtained after the Laplace transform of wave problem on [0, 1] × t. We define the Neuman-to-Dirichlet (NtD) map a.k.a. current-to-voltage, transfer, impedance or Weyl function f (λ) as f (λ) = u|x=0 .

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

ROMs via Stieltjes string/optimal grid

Rational approximations of Stieltjes functions f (λ) is a Stieltjes function, i.e., f (z) = probability measure µ(s).

R0

1 −∞ z+s

dµ(s) with

We denote fk a Stieltjes rational approximant of f that can be written in partial fraction form as fk (λ) =

k X i=1

yi λ + θi

with negative non-coinciding poles −θi and positive residues yi . We chose the poles and residues from 2k matching conditions, e.g., Simple Pad´e approximation di (f − fk )|λ=0 = 0 dλi

i = 0, . . . , 2k − 1.

(1)

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

ROMs via Stieltjes string/optimal grid

Stieltjes continued fraction (S-fraction) Theorem (Thomas Joannes Stieltjes, 1893) Any partial fraction fn (λ) with positive residues yi and non-coinciding poles −θi ∈ R− can be equivalently presented as S-fraction n X i=1

yi2 = λ + θi

1 1

hˆ1 λ +

1

h1 +

1

hˆ2 λ + . . . hn−1 +

1 hˆn λ +

1 hn

with real positive coefficients hˆi , hi , i = 1, . . . , n via a O(n) direct algorithm (e.g., Lanczos).

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

ROMs via Stieltjes string/optimal grid

Krein’s observation, (Gantmakher and Krein, 1950) fk = w 1 ,

1 ˆ hi



 wi+1 − wi wi − wi−1 − − λwi = 0, i = 1, . . . , k, hi hi−1   w1 − w0 = −1, wk+1 = 0. h0

For ρ(x) ≡ σ(x) ≡ 1 coefficients hi and hˆi become respectively primary and dual steps of optimal finite-difference grid.

(2)

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

ROMs via Stieltjes string/optimal grid

Pad´e optimal grid, k = 10

Figure: Pad´e grid for L = 1, exponential convergence.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

λ⇒

∂2 ∂t 2

ROMs via Stieltjes string/optimal grid

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

λ⇒

∂2 ∂t 2 ,

perturbed

ROMs via Stieltjes string/optimal grid

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

ROMs via Stieltjes string/optimal grid

Takeaway We can achieve spectral convergence of the NtD map using the simplest second order finite-difference (FD) scheme with the three-point stencil. Adding one grid layer far from source and receiver leads to a significant perturbation of the wavefield. Hints our imaging approach.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

Multi-scale mimetic reduced-order models (MSMROM) We target the most general large scale time domain elastic problem, including multiple scales, fractures, liquid&air inclusions, anisotropy, etc. Objectives: spectral convergence (approaching 2 ppw) with model independent rectangular tensor-product grid and relax the CFL limit; sparsity pattern with the communication cost of the simplest second order finite-difference scheme, suitable to the high performance computing environment, in particular, to GPUs. Related work: multiscale ROMs by Efendiev, 2013; multiscale FE.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

MSMROM algorithm outline 3D computational domain Ω for wave equation with the variable coefficients is split into subdomains (with yellow boundaries). Offline preprocessing A sparse ROM of the Schur complement of every subdomain is computed via Stieltjes string with matrix coefficients. It accurately represents interaction of the subdomain with neighbors. Costly but embarrassingly parallel. Online computing Obtained reduced order sparse system is solved via time-stepping The communication cost between subdomain is minimal. Ideal for HPC, in particular for GPU.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

Two subdomains, acoustic wave equation

For simplicity, consider nonsingular frequency-domain problem (A + λI )u = q, A = ∇ · (σ∇u) on Ω ∈ R2 with Dirichlet conditions. Ω = Ω1 ∪ Ω2 , Γ = Ω1 ∩ Ω2 , q is a distribution supported on Γ.

∂ ∂ Conjugation conditions u|Γ−0 − u|Γ+0 = 0, σ ∂ν u|Γ+0 − σ ∂ν u|Γ−0 = q.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

Spectral approximation of the NtD

(A + λI )u = 0 on Ω1 ⊂ Ω with Dirichlet conditions on ∂Ω1 \ Γ, ∂ u|Γ = u|Γ . F1 σ ∂ν Pad´e approximation by S-fraction with operator coefficients:

F1 (λ) ≈ F1n (λ) =

1 Hˆ01 λ +

.

1 H11 +

1 ..

. +

1 1 λ+ 1 Hˆn−1 Hn1

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

Discretization via Steiltjes string with matrix elements


∂ ˆ1 1 (H11 )−1 u11 − u|Γ−0 + σ ∂ν u|Γ−0 =  −λH0 u0 ,
1 )−1 u 1 1 1 −1 u 1 − u 1 ˆ1 1 2 (Hj+1 j+1 − u j − (Hj ) j j−1 = −λHj uj . 2

Here and below fictitious variables are in black.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

MSMROM: Stieltjes strings connected via conjugation condition

Using conjugation
cond. we attach the
Stieltjes strings together ˆ1 ˆ2 (H11 )−1 u11 − u|Γ + (H12 )−1 u12 −  u|Γ = −λ(  H0 + H0 )u0 − q,
i )−1 u i i i −1 u i − u i ˆi i (Hj+1 j+1 − u j − (Hj ) j j−1 = −λHj uj , i = 1, 2 The above system mimics second-order FD scheme, that’s why we call it mimetic.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

MSMROM in the time domain 2

d By substituting − dt 2 in place of λ obtain block-tridiagonal ODE system



d2 (H11 )−1 u11 − u|Γ + (H12 )−1 u12 − u|Γ = (Hˆ01 + Hˆ02 ) 2 u0 − q, dt

d2 i i = Hˆji 2 uji , i = 1, 2 (Hj+1 )−1 u i j+1 − u i j − (Hji )−1 uji − uj−1 dt In general multi-domain setting the MMSROM yields graphs with the operator-valued Stieltjes strings as the edges. Such a structure makes it very efficient for HPC, especially, parallel processing and GPU thanks. Complemented with initial conditions such systems can be solved by explicit time-stepping (on line part). Work in progress: corners; balanced truncation for boundary basis functions; CFL→ Nyquist; full elastic anisotropy.

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

3D elasticity example, three-phase model

−3

6

x 10

Fine grid solution Multi−scale solution

5 4 4

2 0

Y

Solution

3

2

−2 −4

source

receiver

1

−6 −8

0 0

1

2

3 X

4

5

−10 0

2

4

6 Time

3D composite, elasticity system: solid matrix with solid, liquid&air inclusions. Dimensionality reduction is factor 40. Serial computing speedup (online part) is factor 4.

8

10

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS

Multi-scale mimetic reduced-order models with spectral accuracy

Scattering by 3D liquid filled fractures in elastic background

Left: elastic model with water filled fracture; distances in meters; pulse Ricker wavelet. Right: simulation results for full and reduced models. Dimensionality reduction is 1800 vs 192000, speedup = 10 on serial processor. Discrepancy due to not properly capturing the physics by

Outline

1

Motivations

2

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS ROMs via Stieltjes string/optimal grid Multi-scale mimetic reduced-order models with spectral accuracy

3

Nonlinear seismic imaging via ROM backprojection 1D model with SISO data 2D model with MIMO data

4

Conclusions

Nonlinear seismic imaging via ROM backprojection

ROM as direct imaging algorithm In the first part of my talk I addressed the high computational cost of the forward solvers by introducing their fast proxies constructed via ROMs. They can be indeed applied for FWI algorithms with optimization framework. Here we pursue another inversion approach, first emerged for the solution of the EIT and CSEM problems 3 The data are fit by a reduced order model via a direct algorithm, and then the unknown PDE coefficients are estimated from the matrix of the ROM state equation. The latter must be transformed to the form mimicking discretization of the underlying PDE. Such an approach can be used as a direct imaging algorithm or as a preconditioner for nonlinear least square fitting. Here we develop projection framework inspired by connection of the optimal FD grids and spectral-Galerkin. 4 3

Borcea,Dr., Guevara Vasquez &Mamonov, Inside Out, 2012; Borcea,Dr., Mamonov&Zaslavsky, Inverse Problems, 2014. 4 Dr., Moskow, Math. Comp., 2003.

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

1D wave problem We start with 1D wave equation on [0, 1] × t

−uxx +

1 utt = 0, v2

ut |t=0 = 0, u|t=0 = b,

ux |x=0 = 0, u|x=1 = 0.

with regular enough variable wave speed v (x) > 0. After transition to the travel time coordinates and the Liouville transform, this equation can be equivalently transformed to Au + utt = 0,

ut |t=0 = 0, u|t=0 = b, −1/2 5 .

with Sch¨odinger operator Au = uxx + qu, where q = v 1/2 vxx 5

Here and below we abuse notation by using the same variable notations for the transformed equation.

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Inverse problem √ The solution can be written as u = cos(t A)b. We consider single input/single output (SISO) data Z f (t) =

1

√ u(x, t)b(x) dx = hu(t), bi = hcos(t A)b, bi,

0

assuming that b is an approximation of δ(x + 0) with exponentially decaying spectral distribution. Inverse problem: f 7→ v . We choose time discretization step τ corresponding the Nyquist limit of 2ppw and find a ROM interpolating f (τ j) for j = 0, . . . , 2n − 1.

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Formulation in terms of Chebyshev polynomials The data can be considered in the context of the Chebyshev moment problem via identity f (τ j) = hTj (P)b, bi,

j = 0, . . . , 2n − 1,

where Tj √ are the Chebyshev polynomials of the first kind, 6 P = cos(τ A) is so called transition operator. √ We denote snapshots ui = u(iτ ) = cos(iτ A)b, i = 0, . . . , n − 1. We denote U ∈ R∞,n the semiinfinite matrix with columns ui , and U∗ U, U∗ PU ∈ Rn×n matrices with elements ui∗ uj = hui , uj i, ui∗ Puj = hui , Puj i resp. √ √ ∗ u = b ∗ T (P)T (P)b = b ∗ cos(iτ A) cos(jτ A)b = Obviously, u j i j i n o √ √ 0.5 b ∗ cos[(i + j)τ A)]b + b ∗ cos[(i − j)τ A)]b = 0.5 {f [(i + j)]τ + f [(i − j)τ ]}, etc for U∗ PU, i.e., all matrices can be computed from the sum of Toeplitz and Hankel matrices of the data. 6

The √ Chebyshev polynomials appear from identities cos(τ j A)b ≡ cos[j arccos(P)]b ≡ Tj (P)b.

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Galerkin framework, tri-diagonalization Let R be the upper triangular matrix obtained from the Cholesky decomposition RR ∗ = (U∗ U)−1 with natural row indexes from 1 to n. It is the QR transform of U yielding V = UR ∈ R∞,n , V∗ V = I . Lemma The QR transform generates tridiagonal matrix T = R ∗ (U∗ PU)R = V∗ PV, that is the projection of P on V.

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Galerkin projection in terms of A √ Using P = cos τ A we obtain A = − τ22 (1 − P) + O(τ 2 ). By abusing notation, below we use instead of T its affine transform τ22 (1 − T ), i.e., obtain T = V∗ AV + O(τ 2 ). So, neglecting O(τ 2 ) term we conclude that T is the Ritz matrix (the Galerkin projection) of A in the orthogonal basis V . The next step: T ,→ v . Tridiagonal T can be interpreted as the finite-difference approximation of A, so v can be imaged on a special optimal grid.7 The optimal grid approach shows good results for 1d, but not enough resolution in 2D.

7

Dr.,Mamonov, Thaler, Zaslavsky, 2015

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Galerkin approximation of A Galerkin back-projection formula gives as A ≈ VT V∗ , We can compute T directly from the data (Hankel+Toeplitz,then Cholesky), but don’t know V. Let assume that V ≈ V0 , where V0 is computed for some background wave-speed distribution v0 . If this assumption holds, then A ≈ V0 T V0∗ . It turns out, that this assumption is the core of famous Marchenko-Gelfand-Levitan inversion algorithm.

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Marchenko-Gelfand-Levitan in nutshell Snapshots (τ = σ)

Orthogonalized Snapshots (τ = σ)

0.2

0.2

0.4

0.4

x

0

x

0

0.6

0.6

0.8

0.8

1

20

40

60

Snapshot #

80

1

20

40

60

80

Snapshot #

Left. Matrix U for a layered model. It is upper triangular due to causality, with pronounced multiple reflections. Right. Matrix V obtained after the Gram-Schmidt orthogonalization of U’s columns (QR). The orthogonalization suppresses multiples. Basic linear algebra: full rank upper triangular matrix + QR = identity

Nonlinear seismic imaging via ROM backprojection

1D model with SISO data

Galerkin back-projection algorithm We denote T0 = V0∗ A0 V0 , where A0 is the Schroedinger operator corresponding the background wave-speed v0 . Then we define the Galerkin back-projection algorithm as δq = diag [V0 (T − T0 )V0∗ ]. d −0.5 d −0.5 By definition A − A0 = v 0.5 dx v − v00.5 dx v0 , so

δq ≈ v 0.5

d −0.5 d v − v00.5 v0−0.5 . dx dx

Nonlinear seismic imaging via ROM backprojection

2D model with MIMO data

2D setting

We consider 2D inverse problem for acoustic wave eq. with an array of m receivers. The shots are fired by moving the transmitter consequently at the receiver positions, so the data are the elements of the matrix-valued multi-input/multi-output (MIMO) transfer function F (t) = F (t)∗ ∈ Rm×m , sampled at t = τ l, l = 0, . . . , 2n − 1.

Nonlinear seismic imaging via ROM backprojection

2D model with MIMO data

MIMO back-projection imaging algorithm All SISO linear algebra is automatically extended to the MIMO case by using m × m matrices instead of numbers, i.e., instead of tridiagonal T ∈ Rn×n we will have block-tridiagonal matrix T ∈ Rmn×mn with m × m blocks, etc. Thus, the MIMO imaging algorithm can be similarly to the SISO case written as δq = diag [V0 (T − T0 )V0∗ ]. However, transformation to travel time coordinates generally is not possible in the multidimensional setting. Still, the algorithm works, if the background solution have qualitatively correct kinematic properties, e.g., if V0 can be computed with the help of one-way equations.

Nonlinear seismic imaging via ROM backprojection

2D model with MIMO data

Backprojection vs reverse time migration (RTM)

Nonlinear seismic imaging via ROM backprojection

2D model with MIMO data

2D multiple fractures image:

Left: fracture acoustic model; distances in meters; p-wave speed in km/sec; pulse Ricker wavelet with cutoff 5 kHz, sources/receivers are ×. Right: image (perturbation) with homogeneous background. Width of the imaged fractures ≈ half of the wavelength in the background. Hole in the rightmost fracture is possibly due to lack of aperture. Remains of suppressed multiple reflections manifest by rough spots in the images.

Outline

1

Motivations

2

MULTI-SCALE MIMETIC REDUCED-ORDER MODELS ROMs via Stieltjes string/optimal grid Multi-scale mimetic reduced-order models with spectral accuracy

3

Nonlinear seismic imaging via ROM backprojection 1D model with SISO data 2D model with MIMO data

4

Conclusions

Conclusions

Summary Model reduction (of linear time-invariant dynamical systems) applied to solve scattering elastic waves by fractures: work very much in progress. Forward problems: Developed multiscale mimetic ROM of elastic wave propagation – can be viewed as spectral approximation of the elastic media with fractures by composite of sparse elastic networks Targeting HPC, significant acceleration on serial processors

Inverse problems: We construct ROM from data and then image media directly via the equivalent reduced order state variable problem Can be viewed as a data-driven finite-dimensional projection formulation of Marchenko-Gelfand-Levitan methods Cost comparable to simple migration methods

Conclusions

To do: A lot!

Conclusions

To do:

For example, forward modeling and possibly imaging of complex multiple fractures in full 3D elasticity framework.