Seismic imaging and multiple removal via model order reduction Alexander V. Mamonov 1 , Liliana Borcea 2 , Vladimir Druskin 3 , and Mikhail Zaslavsky 3 1 University of Houston, 2 University of Michigan Ann Arbor, 3 Schlumberger-Doll Research Center Support: NSF DMS-1619821, ONR N00014-17-1-2057 A.V. Mamonov ROMs for imaging and multiple removal 1 / 26
Motivation: seismic oil and gas exploration Problems addressed: Imaging : qualitative 1 estimation of reflectors on top of known velocity model Multiple removal : from 2 measured data produce a new data set with only primary reflection events Common framework : data-driven Reduced Order Models (ROM) A.V. Mamonov ROMs for imaging and multiple removal 2 / 26
Forward model: acoustic wave equation Acoustic wave equation in the time domain u tt = Au in Ω , t ∈ [ 0 , T ] with initial conditions u | t = 0 = B , u t | t = 0 = 0 , sources are columns of B ∈ R N × m The spatial operator A ∈ R N × N is a (symmetrized) fine grid discretization of A = c 2 ∆ with appropriate boundary conditions Wavefields for all sources are columns of √ − A ) B ∈ R N × m u ( t ) = cos( t A.V. Mamonov ROMs for imaging and multiple removal 3 / 26
Data model and problem formulations For simplicity assume that sources and receivers are collocated , receiver matrix is also B The data model is √ D ( t ) = B T u ( t ) = B T cos( t − A ) B , an m × m matrix function of time Problem formulations : Imaging : given D ( t ) estimate “reflectors”, i.e. discontinuities of c 1 Multiple removal : given D ( t ) obtain “Born” data set F ( t ) with 2 multiple reflection events removed In both cases we are provided with a kinematic model , a smooth non-reflective velocity c 0 A.V. Mamonov ROMs for imaging and multiple removal 4 / 26
Reduced order models Data is always discretely sampled , say uniformly at t k = k τ The choice of τ is very important, optimally τ around Nyquist rate Discrete data samples are � � √ D k = D ( k τ ) = B T cos B = B T T k ( P ) B , k τ − A where T k is Chebyshev polynomial and the propagator (Green’s function over small time τ ) is � � √ ∈ R N × N P = cos τ − A B ∈ R mn × m should A reduced order model (ROM) � P ∈ R mn × mn , � fit the data D k = B T T k ( P ) B = � B T T k ( � P ) � B , k = 0 , 1 , . . . , 2 n − 1 A.V. Mamonov ROMs for imaging and multiple removal 5 / 26
Projection ROMs Projection ROMs are of the form P = V T PV , � B = V T B , � where V is an orthonormal basis for some subspace What subspace to project on to fit the data? Consider a matrix of wavefield snapshots U = [ u 0 , u 1 , . . . , u n − 1 ] ∈ R N × mn , u k = u ( k τ ) = T k ( P ) B We must project on Krylov subspace K n ( P , B ) = colspan [ B , PB , . . . , P n − 1 B ] = colspan U Reasoning : the data only knows about what P does to wavefield snapshots u k A.V. Mamonov ROMs for imaging and multiple removal 6 / 26
ROM from measured data Wavefields in the whole domain U are unknown , thus V is unknown How to obtain ROM from just the data D k ? Data does not give us U , but it gives us inner products ! Multiplicative property of Chebyshev polynomials T i ( x ) T j ( x ) = 1 2 ( T i + j ( x ) + T | i − j | ( x )) Since u k = T k ( P ) B and D k = B T T k ( P ) B we get i u j = 1 ( U T U ) i , j = u T 2 ( D i + j + D i − j ) , i Pu j = 1 ( U T PU ) i , j = u T 4 ( D j + i + 1 + D j − i + 1 + D j + i − 1 + D j − i − 1 ) A.V. Mamonov ROMs for imaging and multiple removal 7 / 26
ROM from measured data Suppose U is orthogonalized by a block QR (Gram-Schmidt) procedure U = VL T , equivalently V = UL − T , where L is a block Cholesky factor of the Gramian U T U known from the data U T U = LL T The projection is given by P = V T PV = L − 1 � � � U T PU L − T , where U T PU is also known from the data Cholesky factorization is essential, (block) lower triangular structure is the linear algebraic equivalent of causality A.V. Mamonov ROMs for imaging and multiple removal 8 / 26
Problem 1: Imaging ROM is a projection, we can use backprojection If span ( U ) is suffiently rich, then columns of VV T should be good approximations of δ -functions , hence P ≈ VV T PVV T = V � PV T As before, U and V are unknown We have an approximate kinematic model , i.e. the travel times Equivalent to knowing a smooth velocity c 0 For known c 0 we can compute everything, including � U 0 , V 0 , P 0 A.V. Mamonov ROMs for imaging and multiple removal 9 / 26
ROM backprojection PV T and make another approximation: Take backprojection P ≈ V � replace unknown V with V 0 P ≈ V 0 � PV T 0 For the kinematic model we know V 0 exactly P 0 ≈ V 0 � P 0 V T 0 Approximate perturbation of the propagator P − P 0 ≈ V 0 ( � P − � P 0 ) V T 0 is essentially the perturbation of the Green’s function δ G ( x , y ) = G ( x , y , τ ) − G 0 ( x , y , τ ) But δ G ( x , y ) depends on two variables x , y ∈ Ω , how do we get a single image ? A.V. Mamonov ROMs for imaging and multiple removal 10 / 26
Backprojection imaging functional Take the imaging functional I to be I ( x ) ≈ δ G ( x , x ) = G ( x , x , τ ) − G 0 ( x , x , τ ) , x ∈ Ω In matrix form it means taking the diagonal � � V 0 ( � P − � P 0 ) V T I = diag ≈ diag ( P − P 0 ) 0 Note that � � [ V 0 V T ] P [ VV T 0 ] − [ V 0 V T 0 ] P 0 [ V 0 V T I = diag 0 ] Thus, approximation quality depends only on how well columns of VV T 0 and V 0 V T 0 approximate δ -functions A.V. Mamonov ROMs for imaging and multiple removal 11 / 26
Simple example: layered model ROM backprojection image I True c A simple layered model, p = 32 RTM image sources/receivers (black × ) Constant velocity kinematic model c 0 = 1500 m / s Multiple reflections from waves bouncing between layers and reflective top surface Each multiple creates an RTM artifact below actual layers A.V. Mamonov ROMs for imaging and multiple removal 12 / 26
Snapshot orthogonalization Snapshots U Orthogonalized snapshots V t = 10 τ t = 15 τ t = 20 τ A.V. Mamonov ROMs for imaging and multiple removal 13 / 26
Snapshot orthogonalization Snapshots U Orthogonalized snapshots V t = 25 τ t = 30 τ t = 35 τ A.V. Mamonov ROMs for imaging and multiple removal 14 / 26
Approximation of δ -functions Columns of V 0 V T Columns of VV T 0 0 y = 345 m y = 510 m y = 675 m A.V. Mamonov ROMs for imaging and multiple removal 15 / 26
Approximation of δ -functions Columns of V 0 V T Columns of VV T 0 0 y = 840 m y = 1020 m y = 1185 m A.V. Mamonov ROMs for imaging and multiple removal 16 / 26
High contrast example: hydraulic fractures True c RTM image Important application: hydraulic fracturing Three fractures 10 cm wide each Very high contrasts: c = 4500 m / s in the surrounding rock, c = 1500 m / s in the fluid inside fractures A.V. Mamonov ROMs for imaging and multiple removal 17 / 26
High contrast example: hydraulic fractures True c ROM backprojection image I Important application: hydraulic fracturing Three fractures 10 cm wide each Very high contrasts: c = 4500 m / s in the surrounding rock, c = 1500 m / s in the fluid inside fractures A.V. Mamonov ROMs for imaging and multiple removal 18 / 26
Large scale example: Marmousi model A.V. Mamonov ROMs for imaging and multiple removal 19 / 26
Problem 2: multiple removal Introduce Data-to-Born (DtB) transform: compute ROM from original data, then generate a new data set with primary reflection events only Born with respect to what? Consider wave equation in the form � c � u tt = σ c ∇ · σ ∇ u , where acoustic impedance σ = ρ c Assume c = c 0 is a known kinematic model Only the impedance σ changes Above assumptions are for derivation only , the method works even if they are not satisfied A.V. Mamonov ROMs for imaging and multiple removal 20 / 26
Born approximation Can show that P ≈ I − τ 2 2 L q L T q , where L q = − c ∇ · + 1 q = c ∇ + 1 L T 2 c ∇ q · , 2 c ∇ q , are affine in q = log σ Consider Born approximation (linearization) with respect to q around known c = c 0 Perform second Cholesky factorization on ROM 2 τ 2 ( � I − � P ) = � L q � L T q Cholesky factors � L q , � L T q are approximately affine in q , thus the perturbation L q − � � L 0 is approximately linear in q A.V. Mamonov ROMs for imaging and multiple removal 21 / 26
Data-to-Born transform P 0 from D 0 corresponding to q ≡ 0 ( σ ≡ 1) Compute � P from D and � 1 Perform second Cholesky factorization , find � L q and � L 0 2 Form the perturbation 3 � L ε = � L 0 + ε ( � L q − � L 0 ) , affine in ε q Propagate the perturbation 4 � � I − τ 2 k = � B T T k � � L ε � L T � D ε B 2 ε Differentiate to obtain DtB transformed data 5 � � k + d D ε F k = D 0 k � , k = 0 , 1 , . . . , 2 n − 1 � d ε ε = 0 A.V. Mamonov ROMs for imaging and multiple removal 22 / 26
Example: DtB seismogram comparison Impedance σ = ρ c Velocity c Original data D k − D 0 DtB transformed data F k − D 0 k k A.V. Mamonov ROMs for imaging and multiple removal 23 / 26
Example: DtB+RTM imaging Impedance σ = ρ c Velocity c RTM image from original data RTM image from DtB data A.V. Mamonov ROMs for imaging and multiple removal 24 / 26
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