Elastic reflection waveform inversion with petrophysical model constraints

Daniel Rocha & Paul Sava, Center for Wave Phenomena, Colorado School of Mines

SUMMARY

Elastic wavefield tomography faces challenging pitfallsdue to its multiparameter and multicomponent charac-ter, while seeking a model that generates accurate andhigh-quality images of the subsurface. Inter-parametercrosstalk and absence of petrophysical constraints causeelastic inversion to fail, delivering unphysical and artifact-contaminated models. Therefore, we propose to use theelastic reflection waveform inversion (ERWI) method-ology, which inverts both for the background velocitymodel and for the reflectivity image, coupled with apetrophysical constraint term. We demonstrate thatconstraining ERWI leads to models that are more plau-sible, exhibit fewer artifacts, and obey the imposed con-straints. We alternate between smooth and rough modelupdates, keeping both data fitting, image focusing andpetrophysical constraints consistent with one anotherin a common objective function. Compared to uncon-strained inversion, our method delivers a higher-qualitymodel as well as improved convergence and accuracy.

INTRODUCTION

Wavefield tomography provides high-resolution velocityestimation by accurate wave extrapolation using the fullwaveform and bandwidth of seismic signals. Its im-plementations are classified as either data-domain, i.e.,full-waveform inversion (FWI) (Lailly, 1983; Tarantola,1984) and/or as image-domain, i.e., wave-equation mi-gration velocity analysis (WEMVA) (Sava and Biondi,2004; Symes, 2008; Yang and Sava, 2011; Diaz et al.,2013). However, FWI might fail and not deliver modelsthat improve image quality since it su↵ers from cycleskipping and does not explicitly constrain the migratedimages. Also, this technique is more suitable for div-ing and direct waves, although reflections dominate sur-face data and migrated images. Alternatively, WEMVAseeks the model that delivers the most focused image,but such model is generally deficient in high resolution.

As a mixed-domain wavefield tomography method, re-flection waveform inversion (RWI) has recently gainedinterest by using reflections in data-domain wavefieldtomography and also inverting for the migrated image(Hicks and Pratt, 2001; Xu et al., 2012; Wang et al.,2013; Guo and Alkhalifah, 2017). This technique ex-ploits an objective function in the data domain and sep-arately updates the smooth and rough (i.e., image) partsof the earth model, such that the smooth model onlyincorporates low-wavenumber updates. Least-squaresmigration robustly computes the rough model (Alvesand Biondi, 2016; Feng and Schuster, 2017; Duan et al.,2017), while optimization via adjoint-state method pro-vides the smooth updates (Tarantola, 1988; Sava, 2014).

For all wavefield tomography methods, a multiparame-

ter earth model via elastic extrapolation delivers morerealistic reflectivity and valuable subsurface informa-tion (Chang and McMechan, 1987; Yan and Sava, 2011;Ravasi and Curtis, 2013). However, multiparameter in-version su↵ers from crosstalk among model parameters(Operto et al., 2013; Pan and Innanen, 2016). Cou-pling between the multiple wave modes and their di↵er-ent illumination responses cause contamination of theinverted model. Radiation pattern analysis clarifies theambiguity among physical properties (Burridge et al.,1998; Kohn et al., 2012; Kamath and Tsvankin, 2016;Oh and Alkhalifah, 2016), but also shows that amplituderesponses overlap for di↵erent earth model contrasts,thus expressing the di�culty to isolate sensitivity ker-nels for the various parameters, especially with limitedacquisition coverage and highly irregular illumination.

To mitigate the inter-parameter crosstalk and thus ob-tain a more geological-plausible model, various authorspropose using known physical relationships between pa-rameters to constrain the elastic inversion (Baumstein,2013; Peters et al., 2015; Duan and Sava, 2016). Thismethodology delivers more realistic models since crosstalkand artifacts do not obey petrophysical properties. Wechoose to impose such physical constraints on the earthmodel and implement a scheme that simultaneously op-timizes the model and the migrated image. Our petro-physical model constraint uses a logarithmic barrier sim-ilarly to the method by Duan and Sava (2016), but ap-plied in the context of mixed-domain elastic wavefieldtomography. Therefore, our method avoids the com-bined harmful e↵ect of inter-parameter crosstalk (byphysical constraints) and high-wavenumber updates (bylow-wavenumber RWI gradients) into the inversion.

THEORY

We consider the elastic wave equation

⇢u�r [� (r · u)]�r ·hµ

⇣ru+ruT

⌘i= f , (1)

where u (x, t) is the wavefield, f (x, t) is the source, � (x)and µ (x) are Lame parameters, ⇢ (x) is the density ofthe medium, r represents spatial derivative operators,and the superscript dot indicates time di↵erentiation.Under the single-scattering assumption, one can con-sider both wavefield and medium parameters as com-posed of a background and a small perturbation:

⇢ = ⇢0 + �⇢ ,� = �0 + �� , µ = µ0 + �µ , (2)

u = u0 + �u , (3)

where � indicates small perturbed quantities, and thesubscript 0 indicates background quantities. Substitut-ing equations 2-3 in equation 1, and ignoring higher-order terms involving the product of the small earth

Elastic RWI with petrophysical constraints

(a) (b)

Figure 1: True (top), unconstrained (middle), and constrained (bottom) inverted models for (a) � and (b) µ. Thetop panels also show the 11 sources (red) and the line of multicomponent receivers (yellow), and the reflector (black).Adding the physical constraint term into the inversion mitigates the spurious artifacts from inter-parameter crosstalkand sparse acquisition, delivering a more plausible earth model.

model perturbations with �u, we obtain

⇢0�u�r[�0(r·�u)]�r·hµ0

⇣r�u+r�uT

⌘i= (4)

��⇢u0+r[��(r·u0)]+r·h�µ

⇣ru0+ruT

0

⌘i+f .

Equation 4 indicates the interaction between model per-turbations (�⇢, ��, �µ) and the background wavefield u0

in the source term for the scattered wavefield �u, whichleads to the scattered data �d = K

r

�u (Kr

is an ex-traction operator at receivers). Therefore, we write themodel perturbations generating scattered data as

�d = Kr

�u = L�m , (5)

where �m = [�⇢ �� �µ]T and L is the single-scatteringoperator, whose adjoint is migration:

�mmig

= LT�d . (6)

Least-squares migration leads to a more robust �m im-age by minimizing the following objective function

JLS

=12kL�m� �dk2 . (7)

Using L and LT, one minimizes equation 7 by

�mLS

=⇣LTL

⌘�1LT

�d . (8)

Equation 8 describes the least-squares inversion of themodel perturbation �m, while keeping the backgroundmodel m0 = [⇢0 �0 µ0]

T unchanged. To update m0,

we perform non-linear inversion based on the adjoint-state method, which formulates equation 1 as

A(m)u = f , (9)

where A(m) is the elastic wave-equation operator andm represents the total model. For data-misfit minimiza-tion, the objective function for total model inversion is

JD

=12kd� d

obs

k2 =12kK

r

u� dobs

k2 , (10)

where dobs

is observed data. To construct the modelupdate, the gradient of J

D

with respect to m is

@JD

@m=

@A@m

u ? a , (11)

where @A

@m

involves derivatives applied to the state wave-field (Tarantola, 1984; Plessix, 2006). The symbol ?

in equation 11 represents zero-lag crosscorrelation be-tween the state (u) and adjoint (a) wavefields. Theadjoint wavefield uses the adjoint wave-equation oper-ator and the objective function derivative with respectto the state wavefield as its source term:

AT(m)a =@J

D

@u= KT

r

(d� dobs

) . (12)

Conventional FWI uses the gradient expression in equa-tion 11. Alternatively, in RWI, both state and adjointwavefields are decomposed into background and scat-tered parts based on equation 3:

@JD

@m=

@A@m

(u0 + �u) ? (a0 + �a) , (13)

Elastic RWI with petrophysical constraints

Figure 2: Crossplot in �-µ space of inverted models:unconstrained (left) and constrained (right). The blackline corresponds to the true model. The logarithmic bar-rier function used in the constrained inversion is plottedin the background. The unconstrained inverted modelis spread throughout the model space without followinga trend, while the constrained model is restricted to theregion delimited by the barriers.

which leads to

@JD

@m=

@A@m

[u0 ? �a+ �u ? a0 + u0 ? a0 + �u ? �a] .

(14)The separation between background and scattered wave-fields provides low-wavenumber content for the wave-form inversion gradient. The first term in equation 14involves crosscorrelation of waves propagating from thesource to the image point, while the second correlateswaves propagating from the receivers to the image point.Hence, such events coincide in space and time alongtheir similar propagation paths, resulting in low wavenum-bers. The other two terms involve crosscorrelation ofwaves that only coincide at the image point, producingreflectivity that is characterized by high wavenumbers.Therefore, RWI provides the smooth update (m0) by

@JD

@m0=

@A@m

[u0 ? �a+ �u ? a0] . (15)

Auxiliary terms in the objective function that act inthe model space avoids inversion results with unphysi-cal models, which are common with data-only objectionfunctions. For instance, one can exploit known physicalrelationships between two sets of model parameters bya logarithmic penalty function (Peng et al., 2002; Gassoet al., 2009; Duan and Sava, 2016):

JC

= �⌘

X

x

[log(hu

) + log(hl

)] , (16)

where ⌘ is a weighting scalar parameter relative to theother objective function terms, while h

u

and h

l

definefunctions in the dual-model space determining its upperand lower bounds, which for � and µ are

h

u

= ��+c

u

µ+b

u

= 0 , h

l

= ��c

l

µ�b

l

= 0 , (17)

Figure 3: Objective functions in logarithmic scale forunconstrained (dashed) and constrained (solid) ER-WIs, which alternate between least-squares migrationof model perturbations (red) and waveform inversion ofbackground model (blue). Note how constrained ERWIperforms better at the final iterations for both smoothmodel (at iteration 32) and image (final) inversions.

where c

u,l

and b

u,l

are the slopes and intercepts of thelines. For model pairs that fall inside the region boundedby h

u

and h

l

, the distance to the barrier lines determinesthe value of the constraint. Model samples that are closeto one of the barrier lines during inversion lead to largeJ

C

, and are thus forced to move away from that barrier.

The gradient of JC

with respect to either � or µ is

@JC

@�

=�⌘

�� c

u

µ� bu

� ⌘

�� c

l

µ� bl

, (18)

@JC

@µ

=⌘c

u

�� c

u

µ� bu

+⌘c

l

�� c

l

µ� bl

. (19)

Equations 18 and 19 show that the gradient tends to 1if any of its denominators tends to 0. The total objectivefunction involving data-misfit term and petrophysicalconstraint is

J = JD

+ JC

. (20)

EXAMPLE

We illustrate our ERWI method with the synthetic elas-tic model shown in Figure 1, where the top of Figures 1aand 1b show the acquisition geometry, the reflector loca-tion and the negative Gaussian anomalies for the true �

and µ models, respectively. We use as initial model theconstant background from the true model. After 38 in-version iterations, which alternate between smooth andrough model updates, we obtain the models with uncon-strained (middle of Figure 1) and constrained (bottomof Figure 1) ERWIs. Note how the constrained ERWImitigates the spurious artifacts around the anomaly inthe unconstrained inverted model, which are caused by acombination of sparse geometry, limited bandwidth andcrosstalk between model parameters. The constraintterm uses two barrier lines defined by h

l

= ��1.5µ�1.98and h

u

= ��+ µ+ 5.43 according to equation 17. Thebackground of Figure 2 shows a map of J

C

with two

Elastic RWI with petrophysical constraints

(a) (b)

Figure 4: RTM (top) and LSRTM (middle) images obtained with the initial velocity, and LSRTM (bottom) imagesobtained with constrained inverted velocities for (a) �� and (b) �µ. Note the sparse acquisition artifacts in RTM, theirregularity of the imaged flat reflector for RTM and LSRTM as imprint of wrong velocity, and the improved reflectorflatness and focusing for LSRTM with the constrained inverted velocity.

visible convergent barrier lines, and all model samplesfrom unconstrained and constrained inversions. Theunconstrained model samples are spread broadly intothe model space, while the constrained samples are con-fined within the barriers and closer to the line of truemodel samples. The point in this line that is closestto the bottom-right corner is the background modelvalue, where both constrained and unconstrained inver-sions start from. Both ERWIs alternate between image(rough model) and background (smooth model) inver-sions, and their objective functions decrease over iter-ations, as shown in Figure 3. The unconstrained in-version has a smaller objective function for the first 20iterations. However, for the remaining iterations, theconstrained ERWI has better performance for the objec-tive function, finalizing with smaller residuals for bothsmooth and rough model inversions relatively to the un-constrained ERWI.

To show how ERWI with physical constraints delivers amodel that e↵ectively increases image quality, we showthe model perturbation �� and �µ images in Figures 4aand 4b, respectively. The top reverse time migration(RTM) images show the cross-cutting artifacts due tosparse acquisition and a false subsidence of the flat re-flector due to imaging with the wrong initial velocity.Performing least-squares migration (LSRTM) with theinitial inversion background model provides image im-provement (middle images in Figure 4) by mitigating thecross-cutting artifacts and sharpening the imaged reflec-

tor, but still exhibits the imprint of the wrong velocity atthe center of the reflector for ��, and at [x, z] = [1, 0.8]km and [x, z] = [2, 0.8] km for �µ. Finally, using theconstrained ERWI model, LSRTM delivers the bottomimages in Figure 4, which show a flatter reflector closerto the true one.

CONCLUSIONS

We use the mixed-domain ERWI framework coupledwith petrophysical constraints to seek both high-qualityimages and plausible earth models. Smooth updatesare possible for the background model by implementingseparation of wavefields into their background and scat-tered constituents, whose proper correlation provideslow-wavenumber gradients. Our model constraints arebased on a linear trend between two model parameters,impeding the inversion to deliver an unphysical modelwith crosstalk artifacts. Although our constrained ERWImethod has an additional objective function term basedon the imposed constraints, we achieve improved conver-gence compared to conventional data-misfit minimiza-tion.

ACKNOWLEDGEMENTS

We thank the sponsors of the Center for Wave Phenom-ena, whose support made this research possible. Thesynthetic examples in this paper use the Madagascaropen-source software package (Fomel et al., 2013) freelyavailable from http://www.ahay.org.