Source estimation for Wave Equations with uncertain parameters

Sergiy Zhuk, Stephen Moore, Alberto Costa Nogueira Junior, Andrew Rawlinson,Tigran Tchrakian, Lior Horesh, Aleksandr Aravkin and Albert Akhriev

Abstract— Source estimation is a fundamental ingredient ofFull Waveform Inversion (FWI). In such seismic inversionmethods wavelet intensity and phase spectra are usually es-timated statistically although for the FWI formulation as anonlinear least-squares optimization problem it can naturallybe incorporated to the workflow. Modern approaches for sourceestimation consider robust misfit functions leading to the wellknown robust FWI method. The present work uses syntheticdata generated from a high order spectral element forwardsolver to produce observed data which in turn are used toestimate the intensity and the location of the point seismicsource term of the original elastic wave PDE. A min-maxfilter approach is used to convert the original source estimationproblem into a state problem conditioned to the observationsand a non-standard uncertainty description. The resultingnumerical scheme uses an implicit midpoint method to solve,in parallel, the chosen 2D and 3D numerical examples runningon an IBM Blue Gene/Q using a grid defined by approximatelysixteen thousand 5th order elements, resulting in a total ofapproximately 6.5 million degrees of freedom.

I. INTRODUCTION

Source estimation is an essential ingredient of the FullWaveform Inversion (FWI) technique since the inferredshape and location of the seismic wavelet may stronglyinfluence the seismic inversion results and the subsequentassessments of the quality of the reservoir model. In seismicinversion methods like FWI, wavelet intensity/amplitudeand phase spectra are usually estimated statistically fromseismic data alone or from a combination of seismic dataand available well measurements such as sonic and densitycurves. After the seismic wavelet is estimated, it is usedto compute seismic reflection coefficients in the inversionprocess. A reliable wavelet estimate allows for better traceinversion, spike deconvolution, line-mistying correction aswell as forward modeling [7]. In the context of FWI for-mulation as a nonlinear least-squares optimization problem,source estimation can naturally be incorporated by choosingthe source weights for a given model to be a minimizer of aleast-squares misfit function [8]. Most recent approaches forsource estimation consider variable projection for nonlinearleast-squares [4] and robust misfit using twice differentiablemisfit functions [1], [9].

Sergiy Zhuk, Tigran Tchrakian and Albert Akhriev are with IBMResearch - Ireland, Server 3, Damastown Ind. Park, Dublin 15, Ireland,sergiy.zhuk,tigran,albert.akhriev@ie.ibm.com

Stephen Moore and Andrew Rawlinson are with IBM Research - Aus-tralia, stevemoore,arawlins@au1.ibm.com

Alberto Costa Nogueira Junior is with IBM Research - Brasil,albercn@br.ibm.com

Lior Horesh and Aleksandr Aravkin are at T.J.Watson IBM ResearchCenter, lhoresh,saravkin@us.ibm.com

In this work, synthetic data generated by a high orderspectral element forward solver produce observed data whichare used to estimate the intensity and the location of thepoint seismic source term of the original elastic wave PDE.The elastic wave equation in displacement formulation isdiscretized by using the spectral element method and an Or-dinary Differential Equation (ODE) is derived to approximatethe dynamics of the projection coefficients. The latter are,in fact, approximations of the values of the displacementfiled over the set of grid points. It is then assumed thatthe source term is absolutely continuous function of timewith unknown but bounded derivative. We also assume thatat initial time the source intensity equals zero. The latterassumptions allow us to introduce an additional differentialequation for the source term. A min-max filter approach [2],[6], [5] is then applied to this extended ODE in order toderive the estimate of the artificial state representing thesource term. The equations for the filter and the minimaxgain are then discretized by using an implicit mid-pointmethod which preserves quadratic invariants. The numericalexperiments conducted for smooth (Ricker wavelet) and non-smooth (rectangular function) sources, and 2D and 3D waveequations prove the efficacy of our approach.

II. NOTATION

Let Ω ⊂ R3 denote a computational solid domain withboundary Γ and set ΩT := Ω×[tmin, tmax]. L2(ΩT ) denotesthe space of square integerable functions over ΩT . In whatfollows we employ Einstein summation notation, that issummation is implied over the repeated index. For instance,∂xj

σij =∑

j ∂xjσi,j where σij(x) is a component of stress

tensor field σ.

III. PROBLEM STATEMENT

The elastic wave equation is a statement of Newton’ssecond law of motion applied to a continous solid materialand is defined in ΩT as:

ρ∂ttui = ∂xjσij + fi (1)

where i, j ∈ [1− 3], t ∈ [tmin, tmax], x ∈ Ω, ui(x, t) is thei-th component of the displacement vector field, ρ(x) is thedensity, fi(x, t) is the seismic source, modeled in the formof a point source:

fi(x, t) = θ(t)δ(xi − xi,source)

and σij is the stress tensor field, modeled with the linearconstitutive relation

σij = cijklεkl = λεkkδij + 2µεij

where cijkl denotes a general 4th order elasticity tensor andεij the strain tensor, and specifically for a linear elastic solid,λ(x) and µ(x) are the first and second Lame parametersrespectively. The elastic wave equation is subject to thestandard initial conditions ui = ∂tui = 0, free surfaceboundary conditions

σijnj = 0,

and Clayton and Enquist absorbing boundary conditions

σijnj = −ρv∂tui

respectively, where nj denotes the normal to the boundaryΓ and v denotes the elastic wave speed (which will be Pwave speed for displacement components parallel to nj andS wave speed for components perpendicular to nj).

Assume that a function Y is observed in the form:

Y (x, y, z, t) =

∫ΩT

g(x− x′, y − y′, z − z′, t− t′)×

× u(x′, y′, z′, t′)dx′dy′dz′dt′ + η(x, y, t) ,(2)

where g ∈ L2(ΩT ) is an averaging kernel and η modelsan observation noise. Given the data Y , we would like toestimate the position xi,source and intensity θ of the sourceterm f .

IV. REVIEW OF THE SPECTRAL ELEMENT METHOD FORELASTIC WAVE EQUATION

Using the Spectral Element Method, the computationaldomain is defined as a tesselation of hexahedral elementswith the displacement field approximated as ui(x, t) ≈ui,pqr(t)ψpqr(x), which combined with Galerkin projectiongives the weak form:

∫Ωe

ψabcρψpqr∂ttui,pqrdΩ +

∫Γe

ψabρvψpq∂tui,pq dΓ+

∫Ωe

∂xjψabccijkl

1

2(∂xl

ψpqruk,pqr + ∂xkψpqrul,pqr) dΩ =

∫Ωe

ψabcfidΩ (3)

In this specific implementation the basis functions ψpqr aretaken to be a tensor product of the family of N th orderLagrange polynomials ψpqr = `p`q`r, which integrating overa reference element with Gauss-Legendre-Lobatto (GLL)

quadrature, gives the elemental matrices:

Mei,j,abc,pqr =

N+1∑f,g,h=1

wfwgwhψabcρψpqrJ∣∣ξ1,f ,ξ2,g,ξ3,h

Kei,j,abc,pqr =

N+1∑f,g,h=1

wfwgwh∂ξkψabc

(∂ξk∂xj

)1

2cijkl(

∂ξmψpqr

(∂ξm∂xl

)+ ∂ξnψpqr

(∂ξn∂xk

))J∣∣ξ1,f ,ξ2,g,ξ3,h

Cei,j,ab,pq =

N+1∑f,g=1

wfwgψabρviψpqJ∣∣ξ1,f ξ2,g

(4)

sei,abc =

N+1∑f,g,h=1

wfwgwhψabcfiJ∣∣ξ1,f ,ξ2,g,ξ3,h

where ξi ∈ [−1,+1] denote the coordinates of the referenceelement, J defines the Jacobian of the transformation map-ping a given element to the reference, and wi denote theweights associated with the GLL quadrature. The elementalmatrices may be assembled into their global counterparts,defining the system of ODEs

M u + Cu +Ku = s (5)

where M , C, K, and s are the global mass, damping,stiffness matrices, and source vector. Intial conditions aregiven as follows: s(tmin) = s(tmax) = 0. We stress thatu models projection coefficents representing the solution ofthe wave equation (1) in the basis of the finite dimensionalspace spanned by the Lagrange polynomials ψpqr = `p`q`r.In fact, u “lives” in the physical space and represents theapproximation of the values of the displacement field ui overthe set of chosen GLL quadrature points.

It was recognized in [11], [12] that the approximationerror of the spectral element method, as well as any Galerkinprojection method, may be modelled as an uncertain butbounded input g in the following form:

M u + Cu +Ku = s + g (6)

Accounting for the projection error allows one to increasethe approximation quality of the model (5).

Now, in order to relate the observed data Y (x, y, z, t) tothe projection coefficients u described by (6) we also projectthe observation equation (2) onto the span of the Lagrangepolynomials. As a result we arrive at the following system:

Y (t) = Hu + e(t) , (7)

where e(t) denotes the observation error and the matrix Hdescribes the locations of the receivers in Ω. We furtherassume that e(t) is a realisation of a random process withzero mean and a given covariance function R(t).

Now, we would like to estimate the position and intensityof the source term s assuming that

s = Bf

and f = g(t) where B is a matrix representing a possiblespatial localisation of the source term. We stress that, ingeneral, we do not require any knowledge about source

location and in that case B = I, where I is the identitymatrix, equal in dimension to the total number of GLLquadrature points Np. However, if this data is available onemay incorporate it into B which will improve the estimationresults. We further assume that vector function g has boundedenergy, and together with the projection error term satisfiesthe following inequality:∫ tmax

tmin

‖g(t)‖2 + ‖g‖2dt ≤ µ2 , (8)

where µ is a given scalar.

V. MIN-MAX SOURCE ESTIMATOR

Now, let us describe the estimation procedure. To this end,we extend (6) with an additional equation:

M u = Bf + g − Cu−Ku , u(0) = u = 0 ,

f = g(t) , f(0) = 0 .(9)

so that the source estimation problem now becomes a stateestimation problem: estimate Bf(t∗), taking into account ob-servations, Y (t), and a non-standard uncertainty description(random noise in observations and deterministic projectionerror together with uncertain but bounded-in-energy timederivative of the source term in the state equation (9)). Westress that each component of s represents the value of thecontinuous source term f at a certain GLL point and so wesimultaneously estimate position and intensity of f over thegrid imposed on Ω.

Taking into account the aforementioned uncertainty de-scription (ellipsoid (8)), a minimax state estimation [2], [6],[5] is a natural choice in order to construct a robust estimateof f(t∗). Assume that t∗ = tmax. We construct a minimaxfilter to estimate f(t∗). Namely, we define an extended statevector x(t) = (u, u, f(t))>, extended state transition andobservation matrices

A :=(

0 I 0−M−1K −M−1C M−1B

0 0 0

), He =

(H 0 0

)Then (9) is equivalent to the following system of ODEs:

x = Ax +Wg ,

where W = [0, I]. Now, we remind that R corresponds tothe error in the observations (roughly speaking: high eigenvalues of R correspond to low error and on the contrary).Also we note that µ defines the energy of the model errorg: big µ corresponds to large model error g.

Following [10] we introduce the following equations forthe estimator x:˙x = Ax + PRHe(Y (t)−Hex) , x(tmin) = 0 ,

P = AP + PA> + µWW> − PH>e RHeP , P (tmin) = 0 .(10)

Note that P is an analogue of the state error covariancematrix of the Kalman Filter, and for each time instant t itdefines an ellipsoid centered around x(t) so that any “truesolution” of (9) which is compatible with the observed dataand uncertainty description belongs to this ellipsoid. As a

result we obtain a guaranteed estimate of the state vectorx(t) = (u, u, s(t))>.

If t∗ = tmin, then one needs to apply a variational methodto estimate x(t∗). Namely, we need to solve the 2-pointboundary value problem:

˙x = Ax + PHeR(Y (t)−Hex) , z(tmax) = 0 ,

z = −AT z−HeR(Y (t)−Hex) , z(t∗) = x(t∗) .(11)

If tmin < t∗ < tmax then one may first apply filtering to getthe estimate of x(t∗) and then improve the obtained estimatex(t∗) by using it in (11). This allows us to take into accountthe entire dataset and leads, in general, to improved estimatesof s(t∗).

In what follows we concentrate on the case t∗ = tmax forsimplicity.

VI. STRUCTURE PRESERVING TIME INTEGRATION FORTHE MIN-MAX ESTIMATOR

In order to construct a numerical approximation for (10),we note that (9) is stiff and so implicit methods are prefer-able. In addition, the eigenvalues of P (t) define the sizeof the ellipsoid containing x(t) and so to get reliable errorestimates, the chosen numerical method should preservequadratic invariants (for instance, the Frobenius norm ofP (t)). We recall that the solution P of the Riccati equationmay be derived from Hamiltonian system:(

dU/dtdV/dt

)=

[0 I−I 0

] [−µWW> AT

A H>e RHe

](UV

).

Indeed, by straightforward differentiation it is easy to checkthat P = V U−1. By noting that the latter equation for Uand V is Hamiltonian, we apply implicit midpoint methodin order to construct numerical approximations for U , V andP . Namely, we define: Pn = VnU

−1n for n > 0 and P0 = 0

and set Pin = VinU−1in where Un, Vn and Uin, Vin are

defined from the following equations:

Un+1 = Un +h

2

2∑i=1

δUin ,

Vn+1 = Vn +h

2

2∑i=1

δVin ,

Vin = Vn +h

2

2∑j=1

δVjn ,

Uin = Un +h

2

2∑j=1

δUjn ,

δUin = AUin +H>e RHeVin ,

δVin = −ATVin + µWW>Uin .

(12)

To make the above procedure numerically stable (note thatto get Pn+1 we have to invert Un+1) we employ thereinitialization trick proposed in [3]: namely, we set Vn =Pn, Un = I so that Un+1 obtained through (12) is well-conditioned numerically. The advantage of the reinitialized

0 500 1000 1500 2000 2500 3000 35000

10

20

30

40

50

60

70

80

90

100

Gain 1

Gain 2

Gain 3

Fig. 1. 1st element of the diagonal of the Riccati matrix P (t) computedby means of the reinitialized method (Gain 1) and standard midpoint (Gain2) for the first 2500 steps.

0 500 1000 1500 2000 250010

−2

10−1

100

101

102

103

Gain 1

Gain 2

Fig. 2. 1st element of diagonal of the Riccati matrix P (t) computed bymeans of the reinitialized method (Gain 1) and standard midpoint (Gain 2)for the last 1000 steps.

method over conventional non-reinitialized one is demon-strated by Figure VI. The implicit mid-point method for thestate estimator x takes the following form.

x− h

2Ax = xn +

h

2P1nH

Te R−1(Y (tn+ 1

2)−Hex1n) ,

xn+1 = 2x− xn , x(tmin) = 0 , tn+ 12

:= tn +h

2.

(13)

VII. EXPERIMENTAL RESULTS

In this section we present results of the numerical experi-oments on source estimation for 2D acoustic and 3D elasticwave equations. Figure 3 represents so called crash-testdescribing the case when the assumption about smoothnessof the source intensity over time is violated. The simulationwas performed for 2D acoustic wave equation and it wasassumed that the position of the source is known. So, infact, the problems of source tracking was solved.

Figure 4 shows the estimation results for the case of2D acoustic wave equation and given source position whensubject to a standard Ricker wavelet source function. Thesource estimate (bottom left) is almost exact if the recieversare quite close to the actual source location (top left) and

0.0 0.1 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

t [s]

f

true source

estimate

Fig. 3. Non-smooth source intencity Comparisons between true andestimated acoustic source functions for SEM approximation of 2D acousticwave equation with incomplete observations.

0 2000 4000 6000 8000 10000

0

2000

4000

6000

x

y

Locations of source HredL and 37 receivers HgreenLSource = 84500, 500< HGLL point = 93L

0 200 400 600 800 1000-2 ´ 107

-1 ´ 107

0

1 ´ 107

2 ´ 107

3 ´ 107

Time

Am

plit

ud

e

Source = 84500, 500<HGLL point = 93L

0 2000 4000 6000 8000 10000

0

2000

4000

6000

x

y

Locations of source HredL and 37 receivers HgreenLSource = 84500, 1500< HGLL point = 241L

0 200 400 600 800 1000-2 ´ 107

-1 ´ 107

0

1 ´ 107

2 ´ 107

3 ´ 107

Time

Am

plit

ud

e

Source = 84500, 1500<HGLL point = 241L

Fig. 4. Source estimation. In the upper plots the red dot is the locationof true source and the greens dots are the locations of 37 detectors. Inthe lower plots, the grey and blue lines are the true and estimated sourcesrespectively.

there is a time delay in the intensity tracking (bottom right)if the recievers are further away (top right) from the source.

Figure 5 displays the source estimation results for the caseof 3D elastic wave equation (top left) and assuming thatthe source position is unknown but is in either of 3 givenlocations. The receivers are located close to the source. Thealgorithm properly identifies actual source location (middle)out of 3 possible locations and tracks the intensity of thesource. The other two locations receive low weights (com-pared to the one in the middle).

The simulations were performed on an IBM Blue Gene/Qusing on a grid defined by approximately sixteen thousand5th order elements, resulting in a total of approximately6.5 million degrees of freedom. The parallelization strategyis based upon decomposition of the grid across multipleprocesses such that each process contains a unique set of the

global GLL points and hence contiguous rows of the globalmass, damping, stiffness matrices. The explicit constructionof a stiffness matrix implies that the time marching updatecan be performed by a distributed matrix vector multiplica-tion at each time step and to to so the implementation makesuse of the Watson Sparse Matrix Package (WSMP) in orderto provide a scalable code allowing for hybrid parallelizationbased on multithreading and the message passing interface(MPI).

VIII. CONCLUSION

The paper presents a simple yet powerful source estima-tion method based on min-max state estimation technique.The aproach taken here is hybrid, namely the elastic waveequation is first discretized by using spectral element method,and a source estimator is then derived for the continuousin time 2nd order linear ordinary differential equation withuncertain parameters modelling the projection error. The esti-mator is discretized by means of the implict midpoint methodwhich preserves quadratic invariants. Different numericalexperiments illustrate efficacy of the proposed technique.

REFERENCES

[1] A. Y. Aravkin, T. van Leeuwen, H. Calandra, and F. J. Herrmann.Source estimation for frequency-domain fwi with robust penalties. InEAGE. EAGE, EAGE, 06 2012.

[2] F. L. Chernousko. State Estimation for Dynamic Systems. Boca Raton,FL: CRC, 1994.

[3] J. Frank and S. Zhuk. Symplectic mobius integrators for lq optimalcontrol problems. In Proc. IEEE Conference on Decision and Control(to appear), 2014.

[4] G. Golub and V. Pereyra. Separable nonlinear least squares: thevariable projection method and its applications. In Institute of Physics,Inverse Problems, 2002.

[5] Alexander Kurzhanski and Istvan Valyi. Ellipsoidal Calculus for Esti-mation and Control. Systems & Control: Foundations & Applications.Birkhauser Boston Inc., Boston, MA, 1997.

[6] A. Nakonechny. A minimax estimate for functionals of the solutionsof operator equations. Arch. Math. (Brno), 14(1):55–59, 1978.

[7] M. O. Osman and E. A. Robinson. Seismic Source SignatureEstimation and Measurement. Society of Exploration Geophysicists,1996.

[8] R. G. Pratt. Seismic waveform inversion in the frequency domain,part 1: Theory and verification in a physical scale model. Geophysics,1999.

[9] B. Sjorgreen and N. A. Petersson. Source estimation by full waveform inversion. Journal of Scientific Computing, 2014.

[10] S. Zhuk. Kalman duality principle for a class of ill-posed minimaxcontrol problems with linear differential-algebraic constraints. AppliedMathematics and Optimisation, 68(2):289–309, 2013.

[11] S. Zhuk. Minimax projection method for linear evolution equations. InProc. IEEE Conference on Decision and Control, ieeexplorer.ieee.org,2013.

[12] S. Zhuk, J. Frank, I. Herlin, and R. Shorten. Data assimilation forlinear parabolic equations: minimax projection method. SIAM J. Sci.Comp., to appear.

0 2000 4000 6000 8000 10000

0

2000

4000

6000

x

y

possibleSourceLocations = 1Locations of source HredL and 37 receivers HgreenL

Actual source location = 84500, 500< HGLL point = 93L

0 200 400 600 800 1000

-2000

-1000

0

1000

2000

TimeAm

plitude

Source estimate

Source = 84172.67, 500<HGLL point = 92L

0 200 400 600 800 1000

-1¥107

0

1¥107

2¥107

Time

Amplitude

Source estimate HblueL and true source HgrayLSource = 84500, 500<HGLL point = 93L

0 200 400 600 800 1000

-2000

-1000

0

1000

2000

Time

Amplitude

Source estimate

Source = 84827.33, 500<HGLL point = 94L

Fig. 5. An illustration of a sample forward simulation on the SEG/EAGEsalt model dataset, illustrating the magnitude of the elastic displacmentfield and the receiver traces when subject to a standard Ricker waveletsource function, and the source estimates for 3 locations at (4172.67, 500),(4500, 500) and (4827.33, 500). True source is located at (4500, 500).Note the amplitude of the middle plot is much larger than the upper andlower plots.