CWP-835

Elastic full-waveform inversion for a synthetic VTI model fromValhall field

Nishant Kamath, Ilya Tsvankin & Esteban Dıaz

ABSTRACT

One of the main challenges for full-waveform inversion (FWI) is taking into accountboth anisotropy and elasticity. Here, we perform FWI for a synthetic 2D elastic VTI(transversely isotropic with a vertical symmetry axis) model based on the geologic sec-tion at Valhall field in the North Sea. Multicomponent data are generated by a finite-difference code for surface acquisition. The model is parameterized in terms of theP- and S-wave vertical velocities (VP0 and VS0) and the P-wave NMO and horizontalvelocities (Vnmo and Vhor). We apply FWI in the time domain using a multiscale ap-proach with three frequency bands. An approximate Hessian matrix, computed usingthe L-BFGS algorithm, is employed to scale the gradients of the objective functionand improve the convergence. In the absence of significant diving-wave energy, FWIallows us to update the model primarily using reflection data. The largest and mostaccurate updates are obtained for the velocities VP0 and Vhor. The vertical velocitiesVP0 and VS0 are updated both in the shallow and deeper parts of the model, whereasVnmo and Vhor change primarily in the near-surface horizons. Further improvement isexpected after extending the frequency range for the inversion.

1 INTRODUCTION

Full-waveform inversion (FWI), as originally proposed byTarantola (1984), is designed to estimate an earth model thatminimizes the difference between the observed and simulatedseismic wavefields. The main advantage of this method isthe possibility of achieving high resolution by employing thephase and amplitude information contained in the recordedwaveforms.

FWI typically operates with diving-wave energy to up-date the long-wavelength (or smoothly varying) component ofthe velocity model. Once an accurate background model hasbeen obtained using low frequencies, the FWI gradient be-comes more sensitive to reflection events, which can help up-date the reflectivity and improve the resolution (Bunks et al.,1995).

In most implementations of FWI, only the shallow partof the model probed by diving waves can be updated to obtainan accurate background velocity field. In the deeper horizons,where the background velocity is incorrect, the reflectors areimaged at the wrong depths, which hampers model updatesusing reflected waves. Hence, reflection data are insufficientto update the model without an accurate background velocityfield.

Most existing FWI algorithms assume the earth modelto be isotropic and acoustic (Pratt, 1999; Sirgue and Pratt,2004; Shin and Cha, 2008; Plessix et al., 2010; Baeten et al.,2013; Choi and Alkhalifah, 2013). Recent advances includemethods for mitigating the nonlinearity of the problem (Choiand Alkhalifah, 2013; Alkhalifah, 2015) and using the en-

tire bandwidth of the data at the same time (Biondi and Al-momin, 2014). When FWI is performed for anisotropic me-dia, it is typically done in the acoustic approximation. Plessixand Cao (2011), Gholami et al. (2013), and Alkhalifah andPlessix (2014) discuss the sensitivity of the objective func-tion to different parameters of acoustic VTI media. Theyapply FWI for various combinations of the P-wave verti-cal (VP0), NMO (Vnmo = VP0

√1 + 2δ), and horizontal

(Vhor = VP0

√1 + 2δ) velocities, and the anisotropy coeffi-

cients ε, δ, and η = (ε−δ)/(1+2δ). Optimal parameter com-binations for FWI depend on acquisition geometry, the dataused for the inversion (diving waves or reflection events), andtype of inversion (simultaneous or hierarchic). Although theacoustic approximation is computationally efficient, it cannotproperly model reflection amplitudes and handle shear- andmode-converted waves.

Barnes et al. (2008) conduct a feasibility study for elasticFWI of cross-well seismic data from VTI media. For noise-free data, although VP0, VS0, and density are well-constrained,the anisotropy parameters ε and δ are resolved only in the lat-erally homogeneous parts of the model. Lee et al. (2010) per-form FWI of synthetic surface data for 2D elastic VTI media.Parameterizing the model in terms of the stiffness coefficients,however, leads to ambiguity in their results.

Kamath and Tsvankin (2013) perform elastic FWI ofmulticomponent data from a horizontally layered VTI model.They show that it is possible to estimate VP0, VS0, ε, and δeven for offset-to-depth ratios close to unity, primarily becausethe algorithm accurately models reflection coefficients. The

108 N. Kamath, I. Tsvankin & E. Dıaz

gradients of the FWI objective function for elastic anisotropicmedia are derived by Kamath and Tsvankin (2014a). Theyalso develop a methodology for FWI in 2D elastic VTImedia for different parameterizations and test it on modelswith Gaussian anomalies in the Thomsen parameters. Kamathand Tsvankin (2014c) present general expressions for “radia-tion” (i.e. sensitivity) patterns of model parameters for elasticanisotropic FWI and use this formalism to explain the FWIresults for transmission data in 2D VTI media.

Here, we invert multicomponent surface data generatedfor a 2D elastic VTI model based on the geologic sectionat Valhall field. First, we review the inversion methodologybased on the algorithm of Kamath and Tsvankin (2014a). Thenwe describe the model used to generate the “observed” wave-field and the processing applied to the simulated data. A multi-scale approach is employed to perform FWI in three frequencybands. Finally, we discuss the preliminary results obtained foreach range of frequencies.

2 METHODOLOGY

2.1 Inversion methodology

We carry out FWI in the time domain by minimizing the fol-lowing objective function:

F =1

2

N∑r=1

∫ T

0

‖u(xr, t)− d(xr, t)‖2dt , (1)

where N is the number of receivers, T is the trace length,u(xr, t) is the displacement computed for a trial model at re-ceiver location xr , and d(xr, t) is the recorded displacement.The gradient of F with respect to the stiffness coefficientscijkl is obtained using the adjoint-state method (Kamath andTsvankin, 2014a):

∂F∂cijkl

= −∫ T

0

∂ui

∂xj

∂ψk

∂xldt , (2)

where u andψ are the forward and adjoint displacement fields,respectively. The gradient for a chosen model parameter mn

is obtained as:∂F∂mn

=∑ijkl

∂F∂cijkl

∂cijkl∂mn

. (3)

Here, we define the model in terms of the velocities VP0, VS0,Vnmo, and Vhor (parameterization I in Kamath and Tsvankin,2013). The gradients of the objective function F with respectto the velocities are given in Appendix A.

The 2D elastic finite-difference modeling code from theMadagascar package is used to generate the data. To obtainthe model update at each iteration, the gradient is scaled by afactor which can be chosen in different ways. In the widelyused steepest-descent method, the gradient is scaled by thestep length. This approach, however, does not account for theenergy loss due to geometric spreading. In the absence ofa Hessian term (or its approximation), the gradient has toohigh amplitudes near the sources and receivers, which com-

plicates velocity estimation. Therefore, the gradients at thesource and receiver locations have to be masked to ensuremeaningful model updates. In addition, the steepest-descenttechnique could result in slow convergence if the objectivefunction has a long and narrow valley.

The problems associated with the steepest-descent algo-rithm can be circumvented by applying the Gauss-Newtonmethod and computing the model update ∆m from

∆m = [H]−1 G, (4)

where H is the Hessian operator and G is the gradient ofthe objective function. Computation of the Hessian, how-ever, is extremely expensive. The Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm or its low-memory equivalent (L-BFGS) is often used to obtain an approximate Hessian. Weuse the L-BFGS package of Nocedal (1980) to compute theoperatorH and update the model from equation 4.

2.2 Synthetic model and data processing

The synthetic model used here is based on the parametersestimated for Valhall field in the North Sea (Munns, 1985).The fields of VP0, VS0, ε, and δ were provided to us byRomain Brossiere (Universite Joseph Fourier, Grenoble) andOlav Barkved (BP, Norway). The VP0 field (Figure 1(a)) in-cludes low-velocity gas layers above the reservoir, which is lo-cated at a depth of 2.5 km. The original model is sampled every3.125 m in the horizontal and vertical directions. However, toreduce computational time in the finite-difference modeling,the grid spacing is increased to 20 m. Although the originalValhall model has a water column on top, we make the entiresection elastic and clip the shear-wave velocity VS0 so that itsminimum value near the surface is 700 m/s.

Multicomponent synthetic data are generated by a hor-izontal array of sources placed with a 80 m increment at adepth of 20 m. The receivers are located at every grid point atthe same depth (20 m). The wavefield is generated by a ver-tical point force; the source signal is a Ricker wavelet with apeak frequency of 5 Hz.

The actual parameter fields are smoothed to generate thestarting model. The smoothing is applied directly to the ve-locities VP0, VS0, Vnmo, and Vhor (Figure 2). The L-BFGSimplementation requires putting bounds on the model param-eters, so we restrict each velocity to lie between the maximumand minimum values for the actual model. The density is fixedat the correct value and not updated during the inversion. FWIis carried out using a multiscale approach, with 20 iterationsfor each of the following frequency bands: 0 − 1.5 Hz, 0 − 3Hz, and 0− 5 Hz.

3 PRELIMINARY RESULTS

Because of the lack of significant diving-wave energy in theobserved (modeled) data, the updates are based primarily onreflections. Comparison of the difference between the actualand starting velocity fields (Figure 3) with the velocity updates

Elastic FWI for 2D VTI media 109

(a) (b) (c) (d)

Figure 1. Parameters (a) VP0, (b) VS0, (c) ε, and (d) δ of a synthetic VTI model based on sections from the Valhall field. The velocities have unitsof km/s.

(a) (b) (c) (d)

Figure 2. Starting model for FWI: (a) VP0, (b) VS0, (c) Vnmo, and (d) Vhor.

(a) (b) (c) (d)

Figure 3. Difference between the actual and starting models: (a) VP0, (b) VS0, (c) Vnmo, and (d) Vhor. The units for the velocity differences/updateshere and in the plots below are m/s.

(a) (b) (c) (d)

Figure 4. Model updates (with respect to the starting model in Figure 2) for the frequency range 0−1.5 Hz: (a) VP0, (b) VS0, (c) Vnmo, and (d)Vhor.

110 N. Kamath, I. Tsvankin & E. Dıaz

(a) (b) (c) (d)

Figure 5. Model updates (also with respect to the starting model) for the frequency range 0−3 Hz: (a) VP0, (b) VS0, (c) Vnmo, and (d) Vhor.

(a) (b) (c) (d)

Figure 6. Improvement in waveform matching after the inversion in the second frequency band (0−3 Hz). The difference between the observedhorizontal displacement and that computed for the (a) starting (for this band) and (b) final model. The difference between the observed verticaldisplacement and that computed for the (c) starting and (d) final model. The shot is located at 4.4 km; all four displacements are plotted on the samescale.

for the first frequency band (Figure 4) indicates the magnitudeof the improvements. The VP0- and VS0-fields are updatedthroughout the section, although the most pronounced changesare above 2.5 km. The updates in the parameters Vnmo andVhor, however, are restricted to the shallow part of the model(down to 1.5 km). Although the largest updates amount to onlyabout 10% of the required change, they have the correct sign(e.g., compare figures 3(a) and 4(a)).

The radiation patterns (Kamath and Tsvankin, 2014b)of the model parameters help explain the obtained results.A VP0-anomaly on a horizontal reflector scatters P-wave en-ergy mostly along the vertical symmetry axis. In the case ofan anomaly in Vhor, most of the energy is scattered near theisotropy plane, which explains why the updates in the horizon-tal velocity are concentrated in the shallow part of the model.The maximum energy scattered by a Vnmo-anomaly is onlyabout 25% of that for VP0, and is focused at angles near 45◦.Therefore, the algorithm cannot update Vnmo at depths below2 km. A scatterer in VS0 has a pattern similar to that in Vnmo,but the maximum energy is higher (comparable to that of VP0,assuming VP0/VS0 = 2). Hence, there is a trade-off betweenthe P- and S-wave vertical velocities.

The inversion result for the first frequency band is usedas the starting model for the second one (0−3 Hz). Includinghigher frequencies improves model resolution, with the low-velocity updates in Figure 5 helping identify the gas clouds(Figure 1). Also, the magnitude of the updates is substantiallyincreased (compare Figures 4 and 5). On the whole, the inver-sion updates the VP0- and Vhor-fields more significantly thanthose of VS0 and Vnmo.

The horizontal and vertical components of the data misfit

computed for the starting and final models demonstrate sig-nificant improvement in waveform matching (Figure 6). Thedecrease in data misfit is particularly visible at the far offsetson the horizontal component (Figures 6(a) and 6(b)) and thenear offsets on the vertical component (Figures 6(c) and 6(d)).

The L-BFGS algorithm allows the user to specify the de-sired number of calculations of the gradients, but the actualnumber of model updates depends on the values of the objec-tive function and the gradients. Hence, although we compute50 and 70 gradients in the first and second frequency bands,respectively, the number of model updates is only 18 and 31(Figure 7).

The gradients are smoothed before each model update,with the same smoothing operator applied to all four veloci-ties. Because VS0 is much smaller than the three P-wave ve-locities, the updates in VS0 have larger wavenumbers. The arti-facts in the inverted VS0-field (Figures 4(b) and 5(b)) might bedue to the high shear-wave energy near the source. The Hes-sian computed by the L-BFGS algorithm is approximate andmay be unable to effectively suppress the energy at the sourcelocation.

4 CONCLUSIONS

We presented preliminary results of elastic FWI for a 2D syn-thetic VTI model fashioned after the geologic section at Val-hall field in the North Sea. The model includes low-velocitylenses above the reservoir, which are supposed to representgas clouds.

The inversion is carried out in the time domain using the

Elastic FWI for 2D VTI media 111

(a)

(b)

Figure 7. Change in the normalized objective function with iterationsfor the (a) first and (b) second frequency bands.

gradients of the objective function derived with the adjoint-state method. The gradients are scaled by an approximateHessian estimated using the L-BFGS algorithm. The startingmodel is generated by smoothing the actual parameter fields,and a multiscale approach is employed to mitigate cycle-skipping.

After the inversion in the first frequency band (0−1.5Hz), the vertical velocities VP0 and VS0 are updated downto 2.5 km, whereas the improvements in Vnmo and Vhor arerestricted to depths of less than 1.5 km. The relatively lowdata misfit for this frequency band likely suppresses the mag-nitude of the updates. Including higher frequencies (0−3 Hz)improves the resolution and accuracy of the updates in VP0

and Vhor. In addition, higher frequencies produce a more ac-curate Vnmo-field in the deeper regions. The offset range wasnot sufficient to record diving-wave energy illuminating thedeeper part of the model. Incorporating diving waves shouldincrease the magnitude of the updates, especially for the hori-zontal velocity at depth.

Ongoing work includes inverting data for a broader rangeof frequencies and performing FWI for other parameteriza-tions introduced in our previous publications. Also, extend-ing the model laterally should help record diving waves withdeeper penetration.

5 ACKNOWLEDGMENTS

We are grateful to the members of the A(nisotropy)- andi(maging)- teams at CWP and to Chunlei Chu and Phuong Vu(BP, Houston) for fruitful discussions. We would also like tothank Romain Brossiere (Universite Joseph Fourier, Grenoble)and Olav Barkved (BP, Norway) for providing the syntheticmodel. This work was supported by the Consortium Projecton Seismic Inverse Methods for Complex Structures at CWPand by the CIMMM Project of the Unconventional NaturalGas Institute at CSM. The reproducible numerical examplesin this paper are generated with the Madagascar open-sourcesoftware package freely available from http://www.ahay.org.

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Elastic FWI for 2D VTI media 113

APPENDIX A: GRADIENTS FOR DIFFERENT PARAMETERIZATIONS

The gradient of the objecive function with respect to the stiffness coefficient cijkl is given in equation 2. Expressing the stiffnessesin terms of chosen model parameters mn makes it possible to obtain the gradients with respect to mn from equation 3. Kamath andTsvankin (2014a) derive the following gradients for VTI media with respect to the velocities VP0, VS0, Vnmo, and Vhor:

∂F∂VP0

= −2ρ VP0

T∫0

∂ψ3

∂x3

∂u3

∂x3+

1

2

√V 2nmo,P − V 2

S0

V 2P0 − V 2

S0

(∂ψ1

∂x1

∂u3

∂x3+∂ψ3

∂x3

∂u1

∂x1

) dt, (A1)

∂F∂VS0

= −2ρ VS0

T∫0

2V 2

S0 − V 2P0 − V 2

nmo,P

2√

(V 2nmo,P − V 2

S0)(V 2P0 − V 2

S0)− 1

(∂ψ1

∂x1

∂u3

∂x3+∂ψ3

∂x3

∂u1

∂x1

)

+

(∂ψ1

∂x3+∂ψ3

∂x1

)(∂u1

∂x3+∂u3

∂x1

)}dt, (A2)

∂F∂Vnmo

= −ρ Vnmo

T∫0

√V 2P0 − V 2

S0

V 2nmo,P − V 2

S0

(∂ψ1

∂x1

∂u3

∂x3+∂ψ3

∂x3

∂u1

∂x1

)dt, (A3)

∂F∂Vhor

= −2ρ Vhor

T∫0

∂ψ1

∂x1

∂u1

∂x1dt. (A4)

114 N. Kamath, I. Tsvankin & E. Dıaz