Elastic Full-Waveform Inversion with Facies Information

Sagar Singh1, Ilya Tsvankin1 & Ehsan Zahibi Naeini21Center for Wave Phenomena, Colorado School of Mines2 Ikon Science, UK

ABSTRACTConventional reservoir-characterization techniques utilize amplitude-variation-with-offset (AVO) analysis to invert for the elastic parameters or directly for the physicalproperties of reservoirs. However, errors in velocity models, inaccurate amplitudes,and structural complexity often degrade the quality of AVO-inversion results. Full-waveform inversion (FWI), which directly estimates the medium parameters fromseismic traces, can provide high-resolution velocity models and represents a poten-tially powerful reservoir characterization tool. However, FWI relies on the accuracy ofthe initial model and suffers from parameters trade-offs. Here, we use a probabilisticBayesian framework to supplement data fitting with rock-physics constraints based ongeological facies obtained from well logs. The advantages of the facies-based FWI aredemonstrated on a structurally complicated isotropic elastic model. In particular, ouralgorithm does not require ultra-low-frequency data and reduces cross-talk between themedium parameters. The developed methodology can be directly extended to realisticanisotropic models.

Key words: Full-waveform inversion, Bayesian algorithm, elastic media, facies, rock-physics constraints.

1 INTRODUCTION

Amplitude-variation-with-offset analysis remains the mostcommon tool for estimation of reservoir properties from seis-mic data (Buland and Omre, 2003; Coleou et al. 2005; Saus-sas and Sams, 2012; Grana, 2016; Zabihi Naeini and Exley,2017). Elastic parameters or reservoir properties can be eval-uated by deterministic or stochastic inversion of the AVO re-sponse (Russell, 1988). Conventional algorithms usually gen-erate a reservoir model by iteratively reducing the differencebetween migrated data and synthetic traces generated usingpetroelastic models. Saussas and Sams (2012) employ geo-logic facies to constrain this inversion workflow, which yieldsa more plausible reservoir model. However, as discussed byZabihi Naeini et al. (2016), the AVO-based approach suffersfrom such inherent problems as high sensitivity to velocity er-rors and distortion due to model complexity. Also, AVO anal-ysis operates with only reflection amplitudes rather than fullwaveforms.

FWI has been successfully used for building high-resolution velocity models that improve seismic images(Tarantola, 1984; Virieux and Operto, 2009) and could beapplied to characterization of petroleum reservoirs (ZabihiNaeini et al. 2015). However, computational demands forlarge-scale implementation of such a method are yet to be eval-uated. Despite the potential advantages of FWI over conven-

tional methods for reservoir characterization, waveform inver-sion involves a number of challenges. In the absence of lowfrequencies in the recorded data, FWI often fails to converge tothe global minimum if the initial model is not sufficiently accu-rate. Parameter trade-offs, even in a purely isotropic acousticmedium, can lead to substantial parameter distortions (Opertoet al., 2013; Alkhalifah and Plessix, 2014; Oh and Alkhalifah,2016). One option to reduce the parameter trade-offs withoutrelying on ultra-low frequencies is to include in the FWI work-flow prior model constraints based on geological facies (com-monly derived from well-logs).

To incorporate facies-based rock-physics constraints, onehas to know the spatial distribution of the facies. Attempts toconstrain FWI by including a facies distribution have shownsome promise (Zhang et al. 2017). In particular, Kamath etal. (2017) use a two-stage process to incorporate facies infor-mation into FWI. The results of the conventional FWI (firstinversion stage) are employed to build the facies which arecompared to the a priori facies model obtained from a welllog. Each grid point in the model is assigned to one of the fa-cies depending on the match between the synthetic and actualfacies distribution, thereby yielding constraints for the secondinversion stage. Zhang et al. (2017) determine the spatial dis-tribution of the facies by calculating at each grid point themaximum posterior probability of the inverted parameter forall facies in the a priori model. Both techniques use informa-

Singh et al.

tion from a single borehole to improve the inversion resultsfor relatively simple horizontally layered models. The com-mon shortcoming of both methods is that the wrong facies canbe assigned to some grid points if the model is structurallycomplicated with pronounced lateral heterogeneity.

To make the facies analysis more robust, Zhang et al.(2018) employ P-wave radiation patterns to determine the re-gions where each parameter is well constrained by the data.When calculating the facies map, a more significant weightis assigned to the parameter which has a smaller uncertainty,thereby yielding a modified facies map. Although this tech-nique enhances resolution for structurally complicated media,the initial models are assumed to be highly accurate.

Considering all facies present in the model should pro-vide higher resolution compared to that for a single-facies re-lationship between the elastic parameters for the entire section(Kemper and Gunning, 2014; Zabihi Naeini and Exley, 2017).Here, we use a probabilistic approach based on the Bayesianframework to estimate the spatial distribution of the facies andconvert them into the elastic properties by creating a confi-dence map. That map is employed as a constraint in the in-version workflow. Implementing this technique for structurallycomplex models such as Marmousi (Martine et al., 2006) is notstraightforward. Building a sufficiently accurate a priori modelrequires dense well coverage to properly account for lateralheterogeneity. In the example below, we use multiple well logsfor the Marmousi model, but they are still sparsely located inspace. Therefore, our algorithm employs image-guided inter-polation (Hale, 2010) to reconstruct the model parameters be-tween the wells.

After describing the method, we apply it to the struc-turally complex Marmousi model. The test shows that facies-based constraints can help recover a high-resolution model inthe absence of ultra-low frequencies in the seismic data. Ad-ditionally, the initial model can significantly differ from theactual parameter distribution. The current implementation islimited to isotropic elastic media, but the method is being ex-tended to VTI (transversely isotropic with a vertical symmetryaxis) models.

2 THEORY

FWI is an iterative data-fitting technique that typically requiresconstraints to resolve the model parameters. Our objectivefunction E(m) is based on the approach described by ZabihiNaeini et al. (2016), who propose to use FWI as a reservoir-characterization tool (Figure 1):

E(m) = Ed(m)+βEprior(m) (1)

Ed(m) = ||Wd(dsim−dobs)||2,

Eprior(m) = ||Wm(minv−mc)||2.

The Ed(m)-term represents the standard l2 data-misfit

Update Forward model

Adjoint

Rock-physicsconstraints

ComparisonSyntheticshot gathers

Recordedshot gathers

Objective Function

Computefacies-basedgradients

Figure 1. Flowchart for the proposed reservoir-oriented FWI (adaptedfrom Zabihi Naeini et al., 2016).

norm, where dobs is the recorded seismic data, dsim is the sim-ulated synthetic data, and Wd and Wm are weighting opera-tors. The term βEprior(m) incorporates a priori model rock-physics constraints based on the geologic facies. The vectormc represents the confidence map updated at each iteration,and β is a scaling factor.

Probabilistic approach based on the Bayesian frameworkis used to estimate the a priori spatial distribution of the facies.We compute the posterior probability of the facies as follows:

P(f|m) =P(f) ·P(m|f)∫

P(f)P(m|f), (2)

where f is the a priori facies vector, P(m|f) is the likelihoodfunction, and m represents the inverted model obtained by theconventional FWI. A Gaussian model is used to describe theuncertainty in the model space (Tarantola, 1984):

P(m|f) = exp[− γ(m− f) · (m− f)

], (3)

where γ is the resolution parameter of the confidence map. Theposterior probability is computed for each facies at all gridpoints of the model. The facies with the maximum posteriorprobability at a certain grid point determines the correspond-ing value on the confidence map.

The gradient of the data misfit with respect to themodel parameters is obtained from the adjoint-state method(Plessix, 2006; Kamath and Tsvankin, 2016). The bounded,low-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) al-gorithm of Byrd et al. (1995) is employed along with a multi-scale approach to iteratively update the model.

Accounting for lateral heterogeneity becomes highlychallenging for complicated models such as Marmousi. There-fore, as mentioned above, we use multiple wells and build theprior model with image-guided interpolation based on the con-ventional RTM (reverse time migration) image, which is up-dated for each frequency band.

Facies-based FWI

(a) (b)

(c)

Figure 2. Parameters of the Marmousi model: (a) P-wave veloity VP, (b) S-wave velocity VS, and (c) density ρ.

(a) (b)

(c)

Figure 3. Initial models of: (a) VP, (b) VS, and (c) ρ.

3 NUMERICAL EXAMPLES

The algorithm is tested on the elastic Marmousi model, whichis 10 km wide and 3.48 km thick (Figure 2). The modelingoperator uses an 8th-order finite-difference scheme with PMLboundary conditions to solve the isotropic elastic wave equa-tion. The data are generated by 100 shots and recorded by 400receivers. The sources and receivers are placed 40 m and 80m, respectively, beneath the surface. To build the initial model,well-log data at location x = 0.8 km are extrapolated throughthe section with mild smoothing in both the horizontal andvertical directions (Figure 3). The input data are the verticaland horizontal particle velocities. The medium is parameter-ized by the P-wave (VP) and S-wave (VS) velocities and densityρ (Kohn, 2015).

First, we apply conventional (unconstrained) FWI assum-ing that ultra-low frequencies are available (Figure 4); the in-

version operates with four frequency bands (0-2Hz, 2-5Hz, 2-10Hz, 2-20Hz). A vertical profile at x = 6 km further illus-trates the accuracy of the inversion (Figure 5). Although theconventional inversion is able to reconstruct the model, suchlow frequencies are usually absent in seismic data. This result(hereafter called the benchmark section) is used for compari-son with the output of our facies-based FWI.

Next, we remove the ultra-low frequencies (0-2Hz) fromthe data and apply conventional FWI with a multiscale ap-proach for the following frequency bands: 2-5Hz, 2-8Hz, 2-13Hz and 2-20Hz. The final inverted models are shown in Fig-ure 6 and the vertical parameter profiles in Figure 7. In theabsence of ultra-low frequencies, the parameters in the deeperpart of the section are significantly distorted.

To find out whether information about facies can com-pensate for the lack of low frequencies, we apply our facies-constrained algorithm for the same frequency content of the

Singh et al.

(a) (b)

(c)

Figure 4. Results of the conventional FWI with ultra-low frequencies in the data: (a) VP, (b) VS, and (c) ρ.

0 2.5 5

1

2

3

z (k

m)

0 2 4

1

2

3

1 2 3

1

2

3

VpVpVp VsVsVs ρρρ

km/s km/s g/cm333

Figure 5. Vertical profiles at x = 6 km: the actual (blue line), initial (red), and inverted (yellow) parameters.

(a) (b)

(c)

Figure 6. Results of the conventional FWI in the absence of ultra-low frequencies (0-2Hz): (a) VP, (b) VS, and (c) ρ.

Facies-based FWI

0 2.5 5

1

2

3

z (k

m)

0 2 4

1

2

3

1 2 3

1

2

3

VpVpVp VsVsVs ρρρ

km/s km/s g/cm333

Figure 7. Vertical profiles at x = 6 km: the actual (blue line), initial(red), and inverted (yellow) parameters.

Figure 8. Locations of the well logs (red triangles) used for image-guided interpolation.

data as in the previous experiment. Rock-physics constraintsare incorporated in the inversion workflow using the verticalparameter profiles (”well logs”) at several locations along theline. Because the model is strongly heterogeneous, multiplewell logs (Figure 8) are required to produce a sufficiently de-tailed confidence map.

The facies information for the well logs is obtained fromMartin et al. (2006). A confidence map for each parameter iscomputed from the facies data using the Bayesian frameworkdiscussed above. We adjust the weighting factor β of the a pri-ori model-based constraints in the objective function in orderto guide FWI towards the actual model. The facies-based FWI(Figure 9) improves the resolution and estimates the model pa-rameters with higher accuracy than the conventional FWI.

The normalized misfit function for differecnt frequencybands is shown in Figure 10. The misfit function convergesfaster in the lowest-frequency band (2-5Hz) than for high fre-quencies (2-20Hz). The truncation of the misfit function at in-termediate and high frequencies is caused by the algorithm’sfailure to find an appropriate gradient direction.

A vertical profile at x = 6 km (Figures 11 and 12) illus-trates the advantages of the facies-based FWI over its conven-tional counterpart. Although our algorithm facies-FWI sub-stantially outperforms the conventional FWI, the medium pa-rametes still deviate from the benchmark results between 3 kmand 3.2 km. This indicates the need for further development ofrobust prior-model constraints.

4 CONCLUSIONS

We presented an efficient methodology for incorporating rock-physics constraints into the FWI workflow. A confidence mapwith the facies information, based on the Bayesian framework,helps resolve the elastic parameters with high spatial resolu-tion. A synthetic test on the elastic Marmousi model demon-strates the advantages of the facies-based FWI approach overthe conventional algorithm and its potential as a reservoir-characterization tool. Multiple well logs, sparsely located inspace, are employed in combination with image-guided inter-polation to create an a priori elastic model. A confidence mapis updated at each iteration and used as a constraint in the FWIworkflow. Despite the model complexity, the facies-based FWIwas able to reduce parameter trade-offs and produce a high-resolution model without the ultra-low frequencies (0-2Hz) inthe data.

5 ACKNOWLEDGMENTS

We thank the members of the A(anisotropy)-Team at the Cen-ter for Wave Phenomena (CWP) for useful discussions. Thiswork was supported by the Consortium Project on SeismicInverse Methods for Complex Structures at the Center forWave Phenomena (CWP) and competitive research fundingfrom the King Abdullah University of Science and Technol-ogy (KAUST).

REFERENCES

Alkhalifah, T., and R.-E. Plessix, 2014, A recipe for practi-cal full-waveform inversion in anisotropic media: An ana-lytical parameter resolution study: Geophysics, 79, no. 3,R91–R101.

Buland, A., and H. Omre, 2003, Bayesian linearized AVOinversion: Geophysics, 68, 185198.

Coleou, T., F. Allo, R. Bornard, J. Hamman, and D. Cald-well, 2005, Petrophysical seismic inversion: SEG TechnicalProgram Expanded Abstracts, 1355–1358.

Grana, D., 2016, Bayesian linearized rock-physics inversion:Geophysics, 81, no. 6, 625–641.

Hale, D., 2010, Image–guided 3D interpolation of boreholedata: SEG Technical Program Expanded Abstracts 2010,1266–1270.

Kamath, N., Tsvankin, I and Zabihi Naeini, E., 2017, Facies-constrained FWI: Toward application to reservoir character-ization, The Leading Edge, 36, no. 11, 924–930.

Kemper, M., and Gunning, J., 2014, Joint impedance and fa-cies inversionseismic inversion redefined: First Break, 32,no. 9, 89–95.

Kohn, D., De Nil, D., Kurzmann, A., Przebindowska, A. andT. Bohlen, 2012, On the influence of model parametrizationin elastic full waveform tomography: Geophysical JournalInternational, 191, 325–345.

Martin, G., Wiley, R and Marfurt, K. J., 2006, Marmousi2:

Singh et al.

(a) (b)

(c)

Figure 9. Results of the facies-based FWI in the absence of ultra-low frequencies (0-2Hz) in the data: (a) VP, (b) VS, and (c) ρ.

0 20 40 60 80 100 120 140Iteration number

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Nor

mal

ized

mis

fit

2-5 Hz2-8 Hz2-13 Hz2-20 Hz

Figure 10. Normalized objective function in different frequencybands.

0 2.5 5

1

2

3

z (k

m)

0 2 4

1

2

3

1 2 3

1

2

3

VpVpVp VsVsVs ρρρ

km/s km/s g/cm333

Figure 11. Vertical profiles at x = 6 km: the actual (blue line), initial(red), and inverted (yellow) parameters.

0 2.5 5

1

2

3

z (k

m)

0 2 4

1

2

3

1 2 3

1

2

3

VpVpVp VsVsVs ρρρ

km/s km/s g/cm333

Figure 12. Vertical profiles at x = 6 km: the benchmark results (green),and the parameters obtained by conventional FWI (black) and thefacies-based FWI (magenta).

An elastic upgrade for Marmousi: The Leading Edge, 25,no. 2, 156–166.

Modrak, R. and Tromp, J., 2016, Seismic wavefield inversionbest practices: regional, global and exploration test cases:Geophysical Journal International, 206, 1864–1889.

Oh, J.-W., and T. Alkhalifah, 2016, Elastic orthorhombicanisotropic parameter inversion: An analysis of parameteri-zation: Geophysics, 81, no. 6, C279–C293.

Operto, S., Y. Gholami, V. Prieux, A. Ribodetti, R. Brossier,L. Metivier, and J. Virieux, 2013, A guided tour of multipa-rameter full-waveform inversion with multicomponent data:From theory to practice: The Leading Edge, 32, 1040–1054.

Russell, B. H., 1988, Introduction to seismic inversion meth-ods: SEG.

Saussus, D., and M. Sams, 2012, Facies as the key to usingseismic inversion for modelling reservoir properties: FirstBreak, 30, no. 7, 45–52.

Tarantola, A., 1984, Linearized inversion of seismic reflec-tion data: Geophysical Prospecting, 32, 998–1015.

Zhang, Zhen-dong., Alkhalifah, T and Zabihi Naeini, E.,

Facies-based FWI

2017, Multiparameter elastic full waveform Inversion withfacies constraints: SEG Technical Program Expanded Ab-stracts 2017: 1551–1555.

Zhang, Zhen-dong., T. Alkhalifah, E. Zabihi Naeini, and B.Sun, 2018, Multiparameter Elastic Full Waveform Inversionwith Facies-based Constraints: Geophysical Journal Inter-national, 213, 3, 2112-2127.

Virieux, J., and S. Operto, 2009, An overview of full wave-form inversion in exploration geophysics: Geophysics, 74,no. 6, 1–26.

Zabihi Naeini, E., and R. Exley, 2017, Quantitative interpre-tation using facies-based seismic inversion: Interpretation,5, 1–8.

Zabihi Naeini, E. and Hale, D., 2015, Image- and horizon-guided interpolation: Geophysics, 80, no. 3, 47–56.

Zabihi Naeini, E., T. Alkhalifah, I. Tsvankin, N. Kamath, andJ. Cheng, 2016, Main components of full waveform inver-sion for reservoir characterization: First Break, 34, 37–48.