Localized time-lapse full-waveform inversion

Colton Kohnke and Paul Sava

Center for Wave Phenomena, Colorado School of Mines

ABSTRACTFull waveform inversion of seismic data requires building the gradient of an objectivefunction from simulated forward and adjoint wavefields to update the model. Aftereach model update, wavefields need to be recomputed for the next iteration. Typicalmodels are large and computing wavefields is computationally expensive. For time-lapse applications, the model is expected to change only in a local region, thereforemuch of the computing time is wasted in regions where the model does not need up-dating. The technique described in this paper outlines a process to localize the forwardand adjoint wavefield propagation to a local domain such that the wavefields simulatedlocally are exact compared with those simulated in the full domain. The gradient ofthe objective function is also accurately built inside the local domain. The wavefieldsand gradient at each iteration are computed locally at a cost orders of magnitude lowerthan in the full domain, enabling more computationally efficient inversions. Wavefieldlocalization facilitates more efficient FWI, while also constraining model updates tothe local domain.

Key words: 4C, 4D, computational efficiency, localization, FWI

1 INTRODUCTION

As geophysical problems get larger, our ability to compute so-lutions efficiently depends on computation speed, and compu-tational efficiency. While the first factor depend primarily onhardware advances, the second is more problem-dependent.An example is time-lapse full waveform inversion (FWI)(Virieux and Operto, 2009; Tarantola, 1984; Symes, 2007;Pratt, 1999). In this problem, seismic surveys are repeatedmultiple times, typically after injection or production, fol-lowed by iterative inversion to infer model changes betweenthe baseline and subsequent surveys. The model update at eachwaveform inversion iteration is computed from the gradientof an objective function, which is simply the cross-correlationof forward and adjoint wavefields, as in reverse-time migra-tion. The adjoint wavefield is computed using the data differ-ence at the receivers as its source term. After each model up-date, the forward and adjoint wavefields must be recomputedto build the gradient for the next model update. In general,these models are large and computing wavefields is expensive.This problem is further exaggerated by the fact that for time-lapse applications the model is expected to change in a local-ized region of interest, and therefore much of the time spentsimulating waves is wasted.

The technique described in this paper outlines a pro-cess to localize wavefield propagation and build FWI gradi-ents inside a small domain at lower computational cost than

using the full model. This is accomplished by using localiza-tion techniques (Vasmel and Robertsson, 2016; Broggini et al.,2017; Willemsen et al., 2016) that essentially move the sourcesand receivers to the local domain of interest by assemblingan equivalent localized experiment. Boundary conditions onthe local domain are applied such that wavefields simulatedcheaply in the local domain are the same as if they were com-puted using the full computational domain. The computationaldomain is thus restricted to the space of interest, making wave-fields simulations less expensive.

Techniques for acoustic wavefield localization are intro-duced by van Manen et al. (2007) who explore perturbedscattering problems to show exact reconstruction in a localdomain. Vasmel and Robertsson (2016) and Broggini et al.(2017) expand the theory with time-domain boundary condi-tion on the local domain to exactly reproduce interior wave-fields and show applications to seismic imaging. Willemsenet al. (2016) and Willemsen and Malcolm (2016) show bound-ary conditions in the frequency domain and FWI applied toupdating salt boundaries. Time-lapse FWI in the frequency do-main is discussed by Malcolm and Willemsen (2016). In thispaper we show how the forward and adjoint wavefields can belocalized in the time-domain, and how local gradients are builtfor accelerated time-lapse FWI. Reverse-time migration fol-lows a similar process to building FWI gradients and benefitsfrom wavefield localization.

Similar theory for localization exists for both elastic and

2 Colton Kohnke and Paul Sava

electromagnetic waves, meaning that localized inversion forthese methods is also possible. Since all time-varying (acous-tic, elastic, and electromagnetic) field methods can be solvedlocally using finite-differences, other computationally expen-sive processes, such as joint time-lapse inversion of seismicand electromagnetic data, can be explored in a local domain.

2 THE VECTOR-ACOUSTIC OPERATOR

Modeling seismic data is a game of tradeoffs. In an idealworld, one would use the most accurate elastic wave equationsolver to simulate wavefields in a model. However, this pro-cess is computationally expensive and the number of neededmodel parameters grows with increasing model complexity.Inversion using the elastic wave equation is costly and needs torecover a multitude of physical properties. In isotropic media,one can employ an acoustic wave equation solver as an ap-proximation to the elastic wave equation. This approximationloses the sensitivity to elastic properties, but gains computa-tional speedup as wavefield simulations in acoustic media aresignificantly faster than in elastic. The number of model pa-rameters recovered for acoustic inversion is also significantlysmaller.

Standard marine seismic acquisition systems record onlythe acoustic pressure wavefield. However, recent technologyadvancements in marine seismic acquisition allows for therecording of vector acoustic seismic data, comprised of bothpressure and particle velocity wavefields (Robertsson et al.,2008). The systems are also capable of using monopole anddipole sources (Halliday et al., 2012). Having access to vec-tor acoustic wavefields provides opportunities to improve bothreverse-time migration (RTM) depth-imaging and full wave-form inversion (FWI) (Fleury and Vasconcelos, 2013). Tomodel wavefields produced by these systems one must use thefirst order vector-acoustic wave equation, as discussed next.

To confine wavefields to a local domain, the pressureand particle velocity on the boundary of the domain needs tobe computed. To accomplish this, we consider the first ordervector-acoustic wave equations

ρ∂v

∂t+∇p = f (1)

1

ρc2∂p

∂t+∇ · v = q, (2)

which couple the particle velocity v(x, t) with the pres-sure p(x, t) reconstructed from dipole f(x, t) and monopoleq(x, t) sources. The wavefields propagate in a mediumwith spatially variable density ρ(x) and velocity c(x). Thevector-acoustic equations are solved iteratively using finite-differences on a staggered grid in space and time, similar toa Yee lattice in electromagnetics (Yee, 1966). Implementingthe staggering in space and time greatly increases the numeri-cal stability of the system compared to the second order waveequation at the cost of computing an additional wavefield vari-able (v) at each time step. The pressure sits on the main grid atinteger time steps and the particle velocity is shifted in spaceand exists at half time steps. The equations are solved in a

p Vx

Vz

Figure 1. 2D finite-difference grids for equation 1. The pressure gridis shown in black. The grid for the x-component of particle velocityis shown in blue and is shifted by dx/2. The z-component of par-ticle velocity is shown in green and is shifted by dz/2. In 3D, they-component of particle velocity is shifted by dy/2. Furthermore, theparticle velocity grids are shifted from the pressure grid in time bydt/2.

leapfrog scheme: particle velocity is derived from the preced-ing pressure, and pressure is derived from the preceding par-ticle velocity. A layout of the pressure and particle velocitygrids is shown in Figure 1. The black grid is for pressure andthe blue and green grids are for particle velocity. The lattergrids are shifted in space by their respective offsets and existhalf a time step ahead of the pressure grid. The monopole qsource term exists at half time steps, at the same times as theparticle velocity grids. The dipole f source term exists at fulltime steps with the pressure grid.

We can use the system of equations 1 and 2 to define alinear operator L linking source wavefield ws(x, t), and dataat the source locations ds(t)

Lws = ds. (3)

The forward wave propagation operator L, source wavefieldws(x, t), and data at the source locations ds(t) are defined as

L =

ρ∆t

0 0 . . . G 0 0 . . .− ρ

∆tρ

∆t0 . . . 0 G 0 . . .

0 − ρ∆t

ρ∆t

. . . 0 0 G . . ....

......

. . ....

......

. . .D 0 0 . . . 1

ρc2∆t0 0 . . .

0 D 0 . . . − 1ρc2∆t

1ρc2∆t

0 . . .

0 0 D . . . 0 − 1ρc2∆t

1ρc2∆t

. . .

......

.... . .

......

.... . .

,

(4)

Localized time-lapse full-waveform inversion 3

ds dr

L

LT

ws

wrf,q v,p

s r

Figure 2. Visual representation of the forward and adjoint operators inthe full domain. The forward operator L creates the source wavefieldws from the data ds at the source s, and is measured at the receiversr. The adjoint creates the adjoint wavefield wr from the data dr atthe receivers r and is measured at the sources s.

ws =

v0

v1

v2

...p0

p1

p2

...

, and ds =

f0f1f2...q0q1q2...

. (5)

Numeral subscripts in ws and ds denote the timestep. G isthe gradient operator (∇) and D is the divergence operator(∇·). The gradient operator performs spatial derivatives onpressure and shifts information spatially from the pressure gridto the particle velocity grid. The divergence operator performsspatial derivatives on particle velocity and shifts informationspatially from the velocity grid to the pressure grid. The topleft and bottom right blocks on L perform the finite-differencetime derivative of the particle velocity and pressure wavefields.These derivates allow for temporal movement from pressure toparticle velocity (bottom right) and particle velocity to pres-sure (top left). Through these operations, all quantities areplaced on the correct grids as required by the wave equations1 and 2. The system in equation 3 is solved forward in time forws starting with the first particle velocity time step v0, thenp0, v1, p1, etc. The resulting wavefield ws is the state vari-able needed for waveform inversion. The data this wavefieldproduces at receiver locations dr(t) is found by spatially re-stricting the wavefield to the receivers. Figure 2 depicts howforward problem is solved. The source term ds is injected atthe source location and is propagated forward in time using L,producing the source wavefield ws. The wavefield ws is savedat the receiver locations to produce dr .

The adjoint wavefield is found using the adjoint modelingoperator

dr = L†wr (6)

to solve for the adjoint wavefield wr given the data at the re-ceiver locations. L† is the adjoint of the wave propagation op-

erator L and is defined as

L† =

ρ∆t

− ρ∆t

0 . . . D† 0 0 . . .

0 ρ∆t

− ρ∆t

. . . 0 D† 0 . . .

0 0 ρ∆t

. . . 0 0 D† . . ....

......

. . ....

......

. . .G† 0 0 . . . 1

ρc2∆t− 1ρc2∆t

0 . . .

0 G† 0 . . . 0 1ρc2∆t

− 1ρc2∆t

. . .

0 0 G† . . . 0 0 1ρc2∆t

. . .

......

.... . .

......

.... . .

.

(7)

D† and G† are the adjoint of the corresponding operators.D† describes the negative gradient operator (−∇) and G† de-scribes the negative divergence operator (−∇ · ). This systemis solved for the adjoint wr wavefields in reverse time, startingwith qN , then for fN , qN−1, fN−1, etc. The receiver wavefield(the adjoint state variable) is restricted spatially to obtain thedata at the source ds.

The forward and adjoint wavefields can be used to solvetomographic problems using the adjoint-state method Plessix(2006); Tromp et al. (2004); Fichtner and Trampert (2011),which is an efficient way of computing model gradients. In thismethod the state variable (the forward wavefield ws) is com-puted using equation 3. The adjoint source is derived by takingthe difference between the observed and predicted wavefieldsat the receiver locations (dr = dobs

r −dprer ). The adjoint source

is used to create the adjoint variable (the adjoint wavefield wr)using equation 6. The gradient of the model, which can beused for FWI, is the zero-lag crosscorrelation of the state andadjoint variables. The gradient requires two wavefield simu-lations, one for each state variable at every iteration, whichcan be computationally expensive depending on the size of themodel.

3 WAVEFIELD LOCALIZATION

In 4D FWI, the model is expected to change in a localized re-gion of interest due to a known production or injection event.This means that much of the time spend computing the for-ward and adjoint wavefields is in a portion of the model thatdoes not change between model updates. It is significantlymore efficient to localize the wavefields to the region of in-terest, with the constraint that the waves that propagate insidethe local region are the same as the waves simulated in the fulldomain.

Consider the schematic geometry in Figure 3. Usingequations equation 3 and 6 we can compute the forward andadjoint wavefields between the sources and receivers and ob-tain a gradient to update the model. Assuming that perturba-tions only occur in the region Ω, we can simulate wavefieldsfrom data recorded on ∂Ω using Rayleigh’s Reciprocity The-orem (de Hoop, 1995; Vasmel and Robertsson, 2016):

4 Colton Kohnke and Paul Sava

f,q v,p

Ω

∂Ω n

local domain

fulldomain

ds dr

L

wsd

ws

(a)

f,q v,p

Ω

∂Ω n

local domain

fulldomain

ds dr

LT

wrkwr

(b)

Figure 3. Localization process for the (a) source and (b) receivers.On the source side, the data at the source ds is propagated using L

and measured on ∂Ω. The local source term d is derived on ∂Ω andpropagated using L to produce the local source wavefield ws. On thereceiver side, the data at the receiver dr is propagated using L† andmeasured on ∂Ω. The local adjoint source term k is derived on ∂Ω

and propagated using L† to produce the local adjoint wavefield wr .

p =

∮∂Ω

[pA ∗ vB − vA ∗ pB ]ndS = (8)∫Ω

[pA ∗ qB − vA ∗ fB − qA ∗ pB + fA ∗ vB ]dV. (9)

The symbol ∗ denotes temporal convolution. Rayleigh’sReciprocity Theorem links the acoustic states, A and B, ofa region Ω bounded by ∂Ω with outward pointing normal n.The acoustic states can have different source terms inside Ω,but as long as the wavefields on the boundary are known, thewavefields inside the region can be computed. If we replacethe B acoustic state source terms with the Green’s function,we obtain

p =

∮∂Ω

[p ∗Gf + v ∗Gq] · ndS (10)

as the boundary condition to reconstruct the wavefields insideΩ (Broggini et al., 2017). G is the Green’s function, where thesuperscript denotes the source type. The left-hand side repre-

sents the pressure at any location inside Ω, and the right-handside values exist on the boundary. The Green’s function acts asa propagator that moves energy from the boundary to the in-side of the domain, and can be replaced with weighted sourceterms on the boundary. Gf is replaced with dipole sourcesweighted by pn and Gq is replaced with monopole sourcesweighted by v · n. Therefore, equation 10 states that if thepressure p and particle velocity v wavefields are known on ∂Ωin the full domain, the ∂Ω sources q and f can be derived suchthat the localized pressure and particle velocity wavefields areequivalent to those computed in the full domain. The sourceson ∂Ω are

f∂Ω = p∂Ωn

q∂Ω = v∂Ω · n.(11)

We can also write the new localized source term on ∂Ωas

d∂Ω = Nd∂Ω. (12)

The operator N acts on d∂Ω, the measured data on ∂Ω in thefull domain, to produce the localized source terms d∂Ω on ∂Ω.Our convention is that the overbar denotes derived terms thatare used for localization, whereas the subscript denotes wherethe quantity exists. Therefore, the operator N is defined as

N =

[0 nn· 0

](13)

Localizing the adjoint wavefield mirrors the forward. Wesimulate the adjoint wavefield by time reversal and capture iton ∂Ω to get the adjoint source terms k∂Ω. Therefore, the newadjoint sources are

Nk∂Ω = k∂Ω. (14)

Here k∂Ω is the measured adjoint wavefields on ∂Ω in the fulldomain and k∂Ω is the new source term on ∂Ω necessary tosimulate the adjoint wavefields in the local domain. Substitut-ing the local forward and adjoint source terms into the forwardand adjoint equations yields the linear system

Lws = d∂Ω

k∂Ω = L†wr

(15)

required to solve for the localized forward ws and adjoint wr

wavefields. By using the derived sources on the boundary ofthe local region, the wavefields inside the region will be un-changed, and are confined to a small region of the model, thusreducing computational cost considerably.

4 EXAMPLES

To illustrate this concept of localization, consider a 300 ×200 × 200 gridpoint subset of the SEAM model and a 1003

gridpoints subdomain centered around a 0.90km radius sphere(Ω), Figure 4. The large velocity model is the full domain andthe region highlighted by the black circles form the smallervelocity model and the local domain. The local domain is anorder of magnitude smaller than the full domain, which is in-

Localized time-lapse full-waveform inversion 5

Figure 4. Subset of the SEAM velocity model used for the full domainsimulations. The local domain boundary ∂Ω is highlighted in black.

dicative of an order of magnitude speedup for wavefield prop-agation.

Figure 5 shows the comparison of the forward operatorin the full and local domains for the pressure wavefields. Thewavefield reconstructed inside Ω matches the one computedin the full domain. Similarly, the adjoint pressure wavefieldsin the full and local domain are shown in Figure 6. The ad-joint wavefields in the local domain also match inside the lo-cal domain as if they were computed in the full domain. Theboundaries of the local domain cancel out the wavefields andconfine them to propagating inside ∂Ω for both forward andthe adjoint wavefields.

This process of wavefield localization relocates sourcesand receivers to an interior surface ∂Ω in a process similar toredatuming. The sources on ∂Ω are able to confine propagat-ing waves to the small region; any outgoing waves are can-celed out by these boundary sources. The derived forward andadjoint sources therefore form the boundary condition on Ω. Ifthe velocity model inside Ω changes, the total wavefield prop-agates inside Ω and the boundary condition cancels out thewavefield that propagate in the original model (the backgroundwavefield). We are then left with the scattered wavefield on∂Ω, which can be compared (subtracted) from the observeddata relocated to ∂Ω to form the adjoint source. This impliesthat once the source and receiver wavefields are localized, wehave access to all of the necessary information for FWI re-stricted to the domain Ω.

Consider a velocity model that has a perturbation con-fined to Ω, simulating the time-lapse change in the model (Fig-ure 7). The gradient in the full domain is built by crosscorrela-tion of the forward ws and adjoint wr wavefields. In the localdomain, the gradient is computed from the crosscorrelation ofthe local forward ws and adjoint wr wavefields. To computethe local adjoint source k we relocate the observed data at thereceiver to ∂Ω and take the data difference with the predictedwavefields on ∂Ω. Once the observed and predicted wavefieldsare moved to ∂Ω, the full domain can be ignored because thelocalized forward operator computes the scattered wavefield

on ∂Ω. Figure 8 shows the result of computing the pressuregradient in the full and the local domains. Since the wavefieldsare accurately localized, the gradients are also localized, andthe full and local domain solutions match.

5 CONCLUSION

Time-lapse seismic FWI can be computationally expensivedue to the size of the models used for wavefield propagation.We propose a method to localize the forward and adjoint wave-fields required by FWI to compute gradients at significantlylower computational cost (orders of magnitude in 3D) in a lo-cal domain. The localization is enabled by boundary conditionderived from Rayleigh’s Reciprocity Theorem. The wavefieldsand gradients in the local domain match exactly their counter-parts computed in the full model.

6 ACKNOWLEDGEMENTS

We thank the sponsors of the Center for Wave Phenom-ena, whose support made this research possible. The syn-thetic examples in this paper use the Madagascar open-sourcesoftware package Fomel et al. (2013) freely available fromhttp://www.ahay.org.

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6 Colton Kohnke and Paul Sava

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 5. The forward pressure wavefields in the (left) full domain and (right) local domain.

Localized time-lapse full-waveform inversion 7

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 6. The adjoint pressure wavefields in the (left) full domain and (right) local domain.

8 Colton Kohnke and Paul Sava

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(a)

(b)

Figure 8. Gradient of objective function built from the crosscorrela-tion of forward and adjoint pressure wavefields in (a) the full domainand (b) the local domain.

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