image-domain elastic waveﬁeld tomography for passive data

CWP-954

Image-Domain Elastic Wavefield Tomography for Passive Data

Can Oren & Jeffrey ShraggeCenter for Wave Phenomena and Dept. of Geophysics, Colorado School of Mines, Golden CO [email protected]

ABSTRACTAn accurate estimation of microseismic event locations plays an important role inthe success of fluid-injection programs. Wavefield-based elastic time-reverse imag-ing (TRI) offers a robust approach to locate microseismic events that occur due toinduced seismicity. The event location accuracy, though, is greatly dependent on theveracity of the elastic velocity models. In this study, we propose a methodology for mi-croseismic image-domain wavefield tomography using the elastic wave equation andextended source images. Gradients are computed using the adjoint-state method andbuild on the extended image residuals that satisfy the differential semblance optimiza-tion (DSO) criterion. The objective function is designed to optimize the focusing oftime-reversed microseismic energy at zero lag in extended source images. The func-tion applies a penalty operator to the extended images to annihilate the energy at zerolag and highlight residual energy at non-zero lags caused by backpropagation througherroneous velocity models. Minimizing the objective function leads to a model opti-mization problem aimed at improving the image-focusing quality. Realistic syntheticexperiments demonstrate that one can compute accurate velocity model gradients usingthe proposed method, which can significantly improve the focusing of imaged events,leading to enhanced fluid-injection programs.

Key words: wavefield tomography, elastic, extended images, microseismic

1 INTRODUCTION

One of the main objectives in seismic monitoring of fluid injection programs including hydraulic fracturing and waste-water disposalis to infer the locations of (micro) earthquakes caused by induced or triggered seismicity (Maxwell and Urbancic, 2001). Accu-rate event locations provide useful information when determining fracture lengths and heights for hydraulic stimulation (Maxwell,2014). Full-wavefield time-reverse imaging (TRI) methods based on wave-equation migration have recently gained popularity toserve this purpose (Artman et al., 2010; Chambers et al., 2014; Douma and Snieder, 2015; Witten and Shragge, 2015; Nakata andBeroza, 2016; Oren and Shragge, 2019; Rocha et al., 2019). These types of imaging methods require accurate velocity models to ob-tain reliable microseismic event locations. In most cases, though, such information is not available at injection sites due to a limitednumber of well logs and perforation shots. In particular, the aforementioned limitations could possibly prevent an accurate velocitymodel construction, especially for surface-recorded data, which could significantly degrade the accuracy of event location estimates.

In microseismic event location applications, full-wavefield migration-based imaging typically involves two steps: (1) extrapolat-ing recorded microseismic event in reverse time through a subsurface model (McMechan, 1982), and (2) evaluating an imagingcondition (i.e., zero-lag auto/crosscorrelation), which results in a zero-lag image (Artman et al., 2010). If the subsurface model issatisfactorily accurate and a judicious preprocessing treatment is applied to the recorded event (e.g., source radiation pattern and/ornoise elimination), the peak amplitude of the resulting zero-lag image can be inferred as the correct spatial source location. Whenthese conditions are not met, the zero-lag image exhibits erroneous misfocusing energy. One notable caveat of zero-lag images isthat they generally do not provide sufficient information to overcome the model inaccuracy. To address this problem, different typesof extended imaging conditions, which involve an extension of zero-lag images calculated at the estimated image point, have beenproposed to update migration velocities (Witten and Shragge, 2015; Oren and Shragge, 2019; Rocha et al., 2019).

2 C. Oren & J. Shragge

Extended images are successfully used as a quality control tool to update velocity models using image misfocusing criteria foractive and passive seismic surveys. The update can be achieved by differential semblance optimization (DSO) (Symes and Caraz-zone, 1991), which is one of the most common model building approaches in exploration seismology. The principle of DSO is tominimize differences between neighboring lags or angles associated with a given reflection (Shen and Symes, 2008). In reflectionseismology, common-image-point space- and time-lag gathers can be effectively used to reconstruct velocity models for complexgeologic structures in acoustic media (Yang and Sava, 2015; Dıaz and Sava, 2017). Similarly, space-lag common-image-gathers canbe constructed through a crosscorrelation-based converted-phase imaging condition to perform elastic migration velocity analysis(Shabelansky et al., 2015). For microseismic data, Witten and Shragge (2017a) propose a pseudo-acoustic image-domain inver-sion method to invert for P- and S-wave velocity models and successfully apply their approach to a 3-D field data set (Witten andShragge, 2017b). There are several advantages of such microseismic image-domain tomography methods. They require neitherfirst-arrival picking nor the origin time of events, which can be problematic to determine for surface-recorded data exhibiting lowsignal-to-noise (S/N) ratios. Also, unlike the recently published full-waveform inversion (FWI) approaches (Sun et al., 2016; Wangand Alkhalifah, 2018), image-domain inversion methods hold less stringent requirements for initial velocity model accuracy toachieve a successful optimization.

We present a methodology on how to compute the isotropic elastic model gradients of the image-domain objective function forsurface-recorded microseismic data. To achieve this goal, we use the elastic TRI approach as well as adjoint-state tomography tosimultaneously construct P- and S-wave velocity model gradients. Distinct from the pseudo-acoustic approach proposed by Wittenand Shragge (2017a), our methodology is fully elastic and explicitly allows for an extension to multiparameter anisotropic inversion.To generate image-domain residuals, we use penalized results from the extended PS energy imaging condition (Oren and Shragge,2019), which form the basis of our adjoint-state inversion formalism. This imaging condition precludes wave-mode decompositionduring backpropagation, which offers an advantage over its crosscorrelation counterpart outlined in Shabelansky et al. (2015).

We begin by presenting the PS energy imaging condition along with the penalty function, the combination of which forms the ob-jective function we seek to minimize. Next, we derive expressions for the elastic model gradients through adjoint-state tomography(Plessix, 2006). To demonstrate the effectiveness of our inversion methodology, we present a number of 2-D synthetic examples.We first show the key ingredients of the proposed method using smooth velocity models, and then present two examples usingstructurally complex anisotropic models with a realistic source and receiver geometry as more challenging settings for estimatingelastic model gradients. Finally, we conclude with a discussion of the advantages and shortcomings of the proposed methodology.

2 THEORY

We consider the source-free elastic wave equation (EWE) in a slowly varying isotropic medium:

u = α∇(∇ · u)− β∇× (∇× u), (1)

where u(x, t) is the displacement field as a function of space (x) and time (t); α = (λ + 2µ)/ρ and β = µ/ρ are the model pa-rameters; and λ(x), µ(x), and ρ(x) are the two Lame parameters and density, respectively. A superscript dot on the displacementfield denotes first-order time differentiation.

To create images of microseismic events, we use an elastic TRI procedure involving the backpropagation of the multicomponentdata and the application of an imaging condition. We prefer to use the PS energy imaging condition due to its enhanced sensitivityto velocity model perturbations (Oren and Shragge, 2019). This imaging condition involves the difference/sum of the kinetic andpotential wavefield energy terms, which correspond to the differential and total wavefield energy, respectively. Because we initiallyseek to consider isotropic elastic inversion, we compute our zero-lag and extended source images using only the kinetic energy termto simplify the derivation of the gradient terms:

I(x,λλλ, e) =

∫ T

0

ρ(x + λλλ) u†α(x + λλλ, t, e) · u†β(x− λλλ, t, e) dt, (2)

where the subscripts α and β represent P- and S-wave data components separated beforehand through data-domain preprocessing;

Passive Elastic Wavefield Tomography 3

the dagger symbol † denotes adjoint; u†(x, t, e) is the adjoint particle velocity field (state variable); λλλ is the vector space-lag ex-tension (Sava and Vasconcelos, 2011); and e is the event index. In practice, the integral evaluation in equation 2 starts from themaximum time of the data window (T ) and progresses in reverse time back to the window origin time of t = 0 s using the adjointisotropic EWE operator. In the following numerical experiments section, we demonstrate that the zero-lag and extended PS energyimages constructed using the kinetic term in equation 2 are sufficient to accurately compute the model gradients in isotropic media.

For the inversion procedure, we define the objective function to be minimized as

J =1

2

∑e

∫ ∫ λλλ

−λλλP 2(λλλ) I2(x,λλλ, e) dλλλ dx, (3)

where P (λλλ) is a to-be-specified image-domain penalty operator (e.g., DSO) (Symes and Carazzone, 1991) designed to annihilatethe energy focused at and about zero lag; what remains is considered to be the image-domain residual. There are a number ofdifferent penalty operators proposed to serve this purpose (Shragge et al., 2013; Yang and Sava, 2015; Dıaz and Sava, 2017), butherein we use a Gaussian function centered at zero lag to penalize the extended image volumes:

P (λλλ) = 1− exp(− λ2

x + λ2z

2σ2

), (4)

where σ controls the variance of the Gaussian function in the shift dimensions. To calculate the gradients (sensitivity kernels) ofthe objective function in equation 3 with respect to the model parameters α and β (see equation 1), we use perturbation theory withthe goal of obtaining an expression like

δJ =

∫ (Kα(x)δα+Kβ(x)δβ

)dx, (5)

whereKα(x) andKβ(x) are the gradients of the objective function, which is perturbed with respect to the model parameters α andβ, respectively. Following the adjoint-state method (Plessix, 2006), we obtain the gradient terms as

Kα(x) = −∑e

∫ T

0

u†α(x, t, e) · υυυα(x, t, e) dt (6)

and

Kβ(x) =∑e

∫ T

0

u†β(x, t, e) · υυυβ(x, t, e) dt, (7)

where the adjoint-state variables υυυα(x, t, e) and υυυβ(x, t, e) are calculated using the forward propagation of the adjoint sources:

υυυα(x, t, e) = aaa

∫ λλλ

−λλλρ(x) u†β(x− 2λλλ, t, e)R(x− λλλ,λλλ, e) dλλλ (8)

and

υυυβ(x, t, e) = bbb

∫ λλλ

−λλλρ(x + 2λλλ) u†α(x + 2λλλ, t, e)R(x + λλλ,λλλ, e) dλλλ, (9)

where aaa = ∇∇ · L−1 ; bbb = ∇ × ∇ × L−1 ; L−1 is the inverse of the forward isotropic EWE operator; and R(x,λλλ, e) =

P 2(λλλ) I(x,λλλ, e) is the penalized (residual) extended image volume. Appendix A presents the full derivation and definitions of thegradient terms Kα and Kβ as well as the adjoint-state variables υυυα and υυυβ .

3 NUMERICAL EXPERIMENTS

In this section, we describe several synthetic experiments that illustrate the effectiveness of our method. In each experiment, wemodel microseismic sources using an Ormbsy wavelet specified by the four corner frequencies [f1, f2, f3, f4] = [2, 3, 20, 25] Hzwith events characterized by a moment-tensor stress-source mechanism (i.e., nonzero stress components of τxx= −1 and τzz= 1).


(a) (b)

Figure 1. Experiment 1: True (a) P- and (b) S-wave velocity models with homogeneous background gradients.

We use a graphics processing unit (GPU)-based finite-difference time-domain (FDTD) solver with the second-order temporal andeighth-order spatial accuracy stencil (Weiss and Shragge, 2013).

Using the true respective velocity models, we first forward model synthetic elastic 2-D multicomponent microseismic data. Prior toimaging, we separate the direct P- and S-wave arrivals by applying hyperbolic mute functions to the data. We then perform elasticsource imaging by backpropagating the receiver wavefield and applying the PS energy imaging condition (equation 2) using theincorrect velocity models. Finally, we compute the elastic model gradients for each event and stack over all sources to obtain thefinal results using equations 6 and 7. To stabilize the gradient computation, we also apply a band-pass filter between 2 Hz and 6 Hzto the state and adjoint-state wavefields. After calculating each individual gradient, we apply illumination compensation (Warneret al., 2013; Yang et al., 2013) leading to more accurate results by reducing the artifacts at and around source locations. Suchan illumination compensation approach includes dividing the gradient by a stabilized measure of the total adjoint-state wavefieldenergy for each event. This procedure is followed by the application of a 2-D smoothing operator with a length of 25 samples inboth vertical and horizontal directions.

3.1 Experiment 1

In the first numerical experiment, we illustrate our method using the P- and S-wave velocity models shown in Figure 1 in isotropicmedia. The models share the same kinematics and includes smooth background P- and S-wave velocities (VP (z) = 3 + z/2 km/sand VS(z) =

√3 + z/2 km/s) along with two Gaussian low and high velocity variations with maximum and minimum pertur-

bations of ∆VP = ∆VS = ±0.3 km/s. We also use a constant density model (ρ = 2.0 g/cm3). The numerical setup consists ofa computational domain of dimension [Nx, Nz] = [608, 224], Nt = 5001 time steps, temporal and spatial sampling intervals of∆t = 0.5 ms and ∆x = ∆z = 0.01 km, respectively, with no free-surface boundary condition applied. We create the incorrect(starting) velocity models by removing the low- and high-velocity Gaussian anomalies from the background linear gradient trends.

Figures 2a and 2b respectively display the vertical and horizontal components of the surface-recorded 2-D elastic data simulatedfrom a single event at [x, z] = [4.0, 1.9] km using the true velocity models in Figure 1. As a requirement of the PS energy imagingcondition, we apply a mask around the direct P- and S-wave arrivals prior to imaging. This operation is followed by the backprop-agation of the separated wave modes (Figures 2c and 2d) using the EWE operator. In the data domain, the wave-mode separationprocedure (completed by hyperbolic mute operators) is repeated for all microseismic events.

Figures 3a and 3b show the zero-lag PS energy images calculated using the correct and incorrect velocity models, respectively.Figure 3a shows that the energy (red dot) is collapsed at the true source location (green dot), which is expected when imaging withthe correct velocity models. Figure 3b shows a focus (red dot) smeared due to the incorrect imaging velocities. We also computethe extended PS energy images at the maximum amplitude points of the zero-lag PS energy images (Figures 3a and 3b) using thecorrect and incorrect velocity models for evaluating their sensitivity to velocity inaccuracy. Figure 3c displays an extended imagefocusing at zero lag due to the correct velocities while Figure 3d displays an extended image exhibiting energy shifted away fromzero lag due to the inaccurate velocities. The extended image in Figure 3d reveals the directions of the required velocity updates.

As we define our objective function in equation 3, we apply a penalty function to the extended images to annihilate the energy atzero lag and highlight residual energy at non-zero lags. Figure 4a depicts the 2D Gaussian penalty operator (equation 4) appliedto the extended images shown in Figures 3c and 3d. Figures 4b and 4c display the penalized extended images (i.e., the residuals)


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

Figure 2. Experiment 1: Simulated (a) uz and (b) ux components of 2-D microseismic data associated with a single event located at [x, z] =

[4.0, 1.9] km computed using the true velocity models in Figure 1. (c) P- and (d) S-wave modes in the ux component separated in the data domainusing hyperbolic time muting. Although not shown here, the corresponding wave modes in the uz component are similarly separated.

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

Figure 3. Experiment 1: (a)-(b) Zero-lag and (c)-(d) space-lag extended PS energy images computed using (a) and (c) the correct and (b) and(d) incorrect velocity models. The green dot denotes the true source location while the red dot denotes the estimated source location based on themaximum image magnitude. Extended images are evaluated only at the estimated source locations, which correspond to the zero-lag image maxima.

calculated using the correct and incorrect velocities, respectively. The extended image in Figure 4b naturally exhibits much lessresidual energy whereas the extended image in Figure 4c exhibits a significant amount of residual energy, which we use to determinethe required velocity updates. The penalized extended image in Figure 4c is then used for the computation of the adjoint sources asdescribed in equations 8 and 9.

Figure 5a depicts the low- and high-velocity Gaussian perturbations that we aim to construct as well as the receivers (red dots)located at each computation grid point (∆r = 0.025 km) at the surface and the nine modeled microseismic events (green dots)located at [x, z] = [1.0 − 5.0, 1.9] km. Figures 5b and 5c respectively display the constructed VP and VS gradients computedusing equations 6 and 7. Both the VP and VS gradients show the target Gaussian perturbations with different spatial resolutions.Considering the gradient formulations in equations 6 and 7, the VS gradient typically features a higher-resolution recovery comparedto the VP gradient. We examine the potential reasons that could affect the gradient resolution in the following discussion section.

3.2 Experiment 2

Our second experiment illustrates the proposed method using the Barrett unconventional model (Regone et al., 2017), which in-cludes complex geological structures and a reservoir region consisting of two shale layers (located between 2.5− 3.3 km in depth)separated by thick chalk layers. The Barrett model was designed to exhibit vertical transverse isotropy (VTI) but here we use it totest our ability to invert for isotropic (vertical) velocity perturbations assuming the correct background parameters. We use Thomsen(1986) notation for our parameterization, which is characterized by the vertical P- and S-wave velocities [VP0, VS0] along with theanisotropy coefficients [ε, δ] (Figure 6). Although not shown here, the VP0 and ρ models show similar structural complexity to the


(a) (b) (c)

Figure 4. Experiment 1: (a) Penalty operator used to annihilate the energy at zero lag of the extended images in Figures 3c and 3d. The blue colorrepresents unity while the white color represents zero. (b)-(c) Penalized (residual) space-lag extended PS energy images computed using the correctand incorrect velocities.

(a) (b)

(c)

Figure 5. Experiment 1: (a) True velocity variations, which are the target of the image-domain elastic inversion, in both VP and VS models alongwith the nine source (green dots) and surface receiver (red dots) geometry at ∆r = 0.025 km. (b) VP and (c) VS gradients constructed through theproposed method.

VS0 model while the coefficient δ is described with smooth linear trends with depth, which is kinematically similar to the coefficientε. In this experiment, the true VP0 and VS0 models additionally include two Gaussian low and high velocity variations of skewedaspect ratio (∆VP0 = ∆VS0 = ±0.3 km/s) that are respectively located at [x, z] = [4.0, 1.5] km and [x, z] = [6.0, 1.0] km.For the purposes of the gradient calculations, the anisotropy parameter fields [ε, δ] are assumed to be known. We create the incor-rect background velocity models by removing the Gaussian anomalies, which represent the inversion targets. The numerical setupconsists of a computational domain of dimension [Nx, Nz] = [400, 148], Nt = 4100 time steps, temporal and spatial samplingintervals of ∆t = 1 ms and ∆x = ∆z = 0.025 km with no free-surface boundary condition applied.

Figures 7a and 7b show the vertical and horizontal components of the data modeled from a single event at [x, z] = [5.6, 3.2] kmusing the true models in Figure 6. Compared to the modeled data in Experiment 1, the Barrett modeling results exhibit more com-plicated wave phenomena due to the model structural complexity. Using the same wave-mode separation strategy described earlier,we effectively separate the direct P- and S-wave arrivals as shown in Figures 7c and 7d, which are then used for imaging.

Figures 8a and 8b depict the results from the zero-lag PS energy imaging of the data in Figure 7 computed with the correct andincorrect velocity models, respectively. Figure 8a shows well-focused energy (red dot) near the true source location (green dot)


(a) (b)

Figure 6. Experiment 2: True (a) VS0 and (b) ε models for the Barrett unconventional model. The VP0 and δ models (not shown) are structurallysimilar to the models in (a) and (b), respectively.

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

Figure 7. Experiment 2a: Simulated (a) uz and (b) ux components of 2-D microseismic data associated with a single event located at [x, z] =

[5.6, 3.2] km computed using the true models in Figure 6. (c) P- and (d) S-wave modes in the ux component separated in the data domain usinghyperbolic time muting prior to imaging.

whereas Figure 8b shows an image focused (red dot) away from the true source location (green dot) due to the incorrect velocitymodels. To evaluate the sensitivity of the zero-lag images to velocity errors, we also compute the extended PS energy images at themaximum amplitude points of the zero-lag PS energy images (Figures 8a and 8b) using the correct and incorrect velocity models.Figure 8c displays focused energy at zero lag while Figure 8d displays unfocused energy shifted away from zero lag due to theerroneous velocities. The extended image in Figure 8d clearly indicates the directions of the required velocity updates.

Figure 9a displays the low- and high-velocity Gaussian variations that we aim to recover along with the receivers (red dots) locatedat each computation grid point (∆r = 0.025 km) at the surface and the modeled nine microseismic events (green dots) locatedat the reservoir interval [x, z] = [2.5 − 7.5, 3.2] km. Figures 9b and 9c respectively show the recovered VP0 and VS0 gradientsfeaturing the target Gaussian anomalies. Overall, the results show that both gradients are accurately constructed using the proposedmethod. In addition, the VS0 gradient exhibit a slightly higher-resolution recovery compared to the VP0 gradient.

The previous experiments have so far used an idealized surface receiver geometry as well as a regularly spaced microseismic eventdistribution. This scenario, though, is far from what would be expected in the field experiments. Therefore, we next demonstrate theeffectiveness of the proposed image-domain elastic inversion algorithm using an irregular source distribution and a sparser receivergeometry. Figure 11a displays the target low- and high-velocity Gaussian variations along with the receivers (red dots) located atthe surface with an interval of ∆r = 0.25 km and the 15 modeled microseismic events (green dots) located irregularly along thereservoir interval shown in Figure 6. The zero-lag and extended PS energy images computed using the correct and incorrect modelsare shown in Figure 10. The results show that the sparse receiver geometry does not influence the PS energy images compared tothe results (Figure 8) obtained with the dense receiver geometry. In addition, the extended PS energy image suggests the directionof the required velocity updates.


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

Figure 8. Experiment 2a: (a)-(b) Zero-lag and (c)-(d) space-lag extended PS energy images computed using (a) and (c) the correct and (b) and(d) incorrect velocity models. The green dot denotes the true source location while the red dot denotes the estimated source location based on themaximum image magnitude. Extended images are evaluated only at the estimated source locations, which correspond to the maxima of the zero-lagimages.

(a) (b)

(c)

Figure 9. Experiment 2a: (a) True velocity variations, which are the target of the image-domain elastic inversion, in both VP0 and VS0 modelsalong with the source (green dots) and every 10th receiver (red dots) geometry. (b) VP0 and (c) VS0 gradients constructed through the proposedmethod.

Figures 11b and 11c show that the Gaussian perturbations are fairly well constructed in the VP0 and VS0 gradients. Given thereasonable number of receivers as well as the irregular source distribution, the quality of the Gaussian perturbation recovery doesnot seem to be reduced due to the aforementioned realistic conditions. However, the irregular distribution of the events may causethe high-amplitude artifacts at deeper levels shown in red around [x, z] = [5.0, 2.9] km in Figure 11c. These artifacts, though, couldbe reduced using a more effective adjoint-source illumination compensation strategy.

4 DISCUSSION

The numerical experiments demonstrate that the resulting VS /VS0 gradients naturally exhibit a higher-resolution recovery com-pared to the VP /VP0 gradients due to the S-wavefield having shorter wavelength relative to the P-wavefield. Other essential factorsinfluencing the resolution of the gradient construction include the data frequency content, spatial sampling of the medium by thestate and adjoint-state variables, and surface acquisition aperture. The results also show that the gradients include several artifactsdue to the use of only a few microseismic events during inversion. However, using additional events could possibly result in fewerartifacts and better recovery of the target area. As the final step of the inversion, one can apply a multiparameter gradient-based op-


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

Figure 10. Experiment 2b: (a)-(b) Zero-lag and (c)-(d) space-lag extended PS energy images computed using (a) and (c) the correct and (b) and(d) incorrect velocity models. The green dot denotes the true source location while the red dot denotes the estimated source location based on themaximum image magnitude. Extended images are evaluated only at the estimated source locations, which correspond to the maxima of the zero-lagimages.

(a) (b)

(c)

Figure 11. Experiment 2b: (a) True velocity variations, which are the target of the image-domain elastic inversion, in both VP0 and VS0 modelsalong with the nine source (green dots) and receiver (red dots) geometry at ∆r = 0.025 km. For plotting purposes, we display every secondreceiver. (b) VP0 and (c) VS0 gradients constructed via inversion.

timization scheme to find an appropriate step length using the gradients of the objective function when determining the magnitudeof the multiparameter model update (Nocedal and Wright, 2006). This stage, though, is beyond the scope of this paper.

The proposed inversion methodology is typically less prone to the well-known cycle skipping problem of FWI, but generallyproduces lower resolution inversion results relative to FWI (Shabelansky et al., 2015; Witten and Shragge, 2017a). Additionally,because stacking-based migration methods enhance the S/N ratio of microseismic data, image-domain inversion generally does notsuffer from low S/N data as quickly as data-domain methods. Furthermore, our inversion approach can be extended to anisotropicmedia by incorporating the potential wavefield energy term in the PS energy imaging condition as described by Oren and Shragge(2019). Perturbing the stiffness tensor present in the potential energy term with respect to anisotropy parameters can allow for suchan extension. Therefore, conducting a sensitivity analysis of the PS energy imaging condition to anisotropy parameters would beimportant for determining the prospectus for image-domain anisotropic elastic inversion.


5 CONCLUSIONS

We develop a fully automatic adjoint-state wavefield tomography method for passive data to jointly construct elastic velocitygradients. Our method is based upon the space-lag PS energy extended image residuals that satisfy the DSO criterion. We conductseveral synthetic numerical experiments that gradually present more complex models along with more realistic source and receiverconfigurations. The experiments show that one can generate accurate elastic model gradients that are well suited for tomographicvelocity model building using our developed inversion framework. Using an iterative multiparameter inversion scheme, the initialmodels ideally will converge to the actual perturbations given these accurately recovered gradients. Finally, although the syntheticexperiments we present herein are 2-D examples, our inversion methodology as theoretically described is equivalently applicablefor 3-D implementations.

6 ACKNOWLEDGEMENTS

We thank the Center for Wave Phenomena consortium sponsors for their financial support. We also thank the Reservoir Character-ization Project for providing access to the Barrett unconventional model. The reproducible examples in this paper were generatedon the CSM Wendian HPC facilities using the Madagascar software package (http://www.ahay.org).

APPENDIX A - DERIVATION OF THE GRADIENTS

To derive the gradients Kα(x) and Kβ(x) in equations 6 and 7, we perturb the objective function (equation 3) with respect to theimage and obtain

δJ =

∫ ∫δI(x,λλλ)R(x,λλλ) dλλλ dx, (A.1)

where R(x,λλλ) = P 2(λλλ) I(x,λλλ) and

δI(x,λλλ) =

∫ρ(x + λλλ)

(δu†α(x + λλλ, t) · u†β(x− λλλ, t) + u†α(x + λλλ, t) · δu†β(x− λλλ, t)

)dt. (A.2)

If we substitute equation A.2 into equation A.1, we obtain

δJ := δJα + δJβ , (A.3)

where

δJα :=

∫ ∫ ∫ (ρ(x + λλλ) δu†α(x + λλλ, t) · u†β(x− λλλ, t)R(x,λλλ)

)dt dλλλdx (A.4)

and

δJβ :=

∫ ∫ ∫ (ρ(x + λλλ) u†α(x + λλλ, t) · δu†β(x− λλλ, t)R(x,λλλ)

)dt dλλλ dx. (A.5)

To find δu†α and δu†β in equations A.4 and A.5, we rewrite the isotropic elastic wave equation (equation 1) using linear operatornotation:

L† u†α = dα, (A.6)

where dα is the separated P-wave data vector and L† is the adjoint isotropic EWE operator:


L† = α∇∇ · − β∇×∇×− ∂tt. (A.7)

We perturb equation A.6 with respect to the model parameter α and obtain

δL† u†α + L† δu†α = 0, (A.8)

and solve for δu†α

δu†α = −(L†)−1δL† u†α, (A.9)

where

δL† = δα∇∇ · − δβ∇×∇× . (A.10)

Introducing δL† into equation A.9 yields

δu†α = −(L†)−1(δα∇∇ · − δβ∇×∇×)u†α. (A.11)

Because the adjoint displacement vector field u†α is curl-free in isotropic elastic media and if we take the first-order time derivativeof each side, equation A.11 becomes

δu†α = −∂t(

(L†)−1δα∇∇ · u†α). (A.12)

If we substitute this expression into equation A.4

δJα = −∫ ∫ ∫

ρ(x + λλλ) ∂t(

(L†)−1δα∇∇ · u†α(x + λλλ, t))· u†β(x− λλλ, t)R(x,λλλ)dtdλλλdx. (A.13)

Using the inner product rule[i.e.,

(ρ(x) ∂t (L†)−1∇∇ ·

)†= ∇∇ · L−1 ∂t ρ(x)

], we remove the operator dependence on δα

(Shabelansky et al., 2015; Witten and Shragge, 2017a):

δJα = −∫δα

∫u†α(x + λλλ, t) ·

(∇∇ · L−1

∫ρ(x + λλλ) u†β(x− λλλ, t)R(x,λλλ) dλλλ

)dt dx, (A.14)

where L−1 is the inverse of the forward isotropic elastic wave equation operator. To further simplify the calculation of equa-tion A.14, we apply a shift in the spatial coordinates (Shen and Symes, 2008) and rearrange the terms as the following:

δJα = −∫δα

∫u†α(x, t) ·

(∇∇ · L−1

∫ρ(x) u†β(x− 2λλλ, t)R(x− λλλ,λλλ) dλλλ

)dtdx

= −∫δα

∫u†α(x, t) · υυυα(x, t) dt dx

=

∫δαKα(x) dx,

(A.15)

where

υυυα(x, t, e) = ∇∇ · L−1

∫ρ(x) u†β(x− 2λλλ, t, e)R(x− λλλ,λλλ, e) dλλλ. (A.16)

Similarly, we rewrite the isotropic EWE (equation 1) using linear operator notation to find u†β in equation A.5:


L† u†β = dβ , (A.17)

where dβ is the separated S-wave data vector. We perturb equation A.17 with respect to the model parameter β and obtain

δL† u†β + L† δu†β = 0, (A.18)

and solve for δu†β

δu†β = −(L†)−1δL† u†β . (A.19)

If we introduce δL† in equation A.10 into equation A.19

δu†β = −(L†)−1(δα∇∇ · − δβ∇×∇×)u†β . (A.20)

Because the adjoint S-wavefield u†β is divergence-free in isotropic elastic media and if we take the first-order time derivative ofeach side, equation A.20 becomes

δu†β = ∂t(

(L†)−1δβ∇×∇× u†β

). (A.21)

If we substitute this expression into equation A.5

δJβ =

∫ ∫ ∫ (ρ(x + λλλ) u†α(x + λλλ, t) · ∂t

((L†)−1δβ∇×∇× u†β(x− λλλ, t)

)R(x,λλλ)

)dt dλλλdx. (A.22)

Using the inner product rule, we remove the operator dependence on δβ:

δJβ =

∫ ∫ ∫ (∇×∇×L−1ρ(x + λλλ) u†α(x + λλλ, t) · δβ u†β(x− λλλ, t)R(x,λλλ)

)dtdλλλdx. (A.23)

To further simplify the calculation of equation A.23, we similarly apply a shift in the spatial coordinates and rearrange the terms asthe following:

δJβ =

∫ ∫ (∇×∇×L−1

∫ρ(x + 2λλλ) u†α(x + 2λλλ, t)R(x + λλλ,λλλ) dλλλ

)· δβ u†β(x, t)dt dx

=

∫δβ

∫u†β(x, t) · υυυβ(x, t) dtdx

=

∫δβKβ(x) dx,

(A.24)

where

υυυβ(x, t, e) = ∇×∇×L−1

∫ρ(x + 2λλλ) u†α(x + 2λλλ, t, e)R(x + λλλ,λλλ, e) dλλλ. (A.25)

REFERENCES

Artman, B., I. Podladtchikov, and B. Witten, 2010, Source location using time-reverse imaging: Geophysical Prospecting, 58,861–873.


Chambers, K., B. D. Dando, G. A. Jones, R. Velasco, and S. A. Wilson, 2014, Moment tensor migration imaging: GeophysicalProspecting, 62, 879–896.

Dıaz, E., and P. Sava, 2017, Cascaded wavefield tomography and inversion using extended common-image-point gathers: A casestudy: Geophysics, 82, no. 5, S391–S401.

Douma, J., and R. Snieder, 2015, Focusing of elastic waves for microseismic imaging: Geophysical Journal International, 200,390–401.

Maxwell, S., 2014, Microseismic imaging of hydraulic fracturing: Society of Exploration Geophysicists.Maxwell, S. C., and T. I. Urbancic, 2001, The role of passive microseismic monitoring in the instrumented oil field: Leading Edge,

20, 636–639.McMechan, G. A., 1982, Determination of source parameters by wavefield extrapolation: Geophysical Journal International, 71,

613–628.Nakata, N., and G. C. Beroza, 2016, Reverse time migration for microseismic sources using the geometric mean as an imaging

condition: Geophysics, 81, no. 2, KS51–KS60.Nocedal, J., and S. Wright, 2006, Numerical optimization: Springer Science and Business Media.Oren, C., and J. Shragge, 2019, 3D anisotropic elastic time-reverse imaging of surface-recorded microseismic data: 89th Annual

International Meeting, SEG, Expanded Abstracts, 3076–3080.Plessix, R. E., 2006, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications:

Geophysical Journal International, 167, 495–503.Regone, C., J. Stefani, P. Wang, C. Gerea, G. Gonzalez, and M. Oristaglio, 2017, Geologic model building in SEAM Phase II-Land

seismic challenges: Leading Edge, 36, 738–749.Rocha, D., P. Sava, J. Shragge, and B. Witten, 2019, 3D passive wavefield imaging using the energy norm: Geophysics, 84, no. 2,

KS13–KS27.Sava, P., and I. Vasconcelos, 2011, Extended imaging conditions for wave-equation migration: Geophysical Prospecting, 59, 35–

55.Shabelansky, A. H., A. E. Malcolm, M. C. Fehler, X. Shang, and W. L. Rodi, 2015, Source-independent full wavefield converted-

phase elastic migration velocity analysis: Geophysical Journal International, 200, 952–966.Shen, P., and W. Symes, 2008, Automatic velocity analysis via shot profile migration: Geophysics, 73, no. 5, VE49–VE59.Shragge, J., T. Yang, and P. Sava, 2013, Time-lapse image-domain tomography using adjoint-state methods: Geophysics, 78, no.

4, A29–A33.Sun, J., Z. Xue, S. Fomel, T. Zhu, and N. Nakata, 2016, Full waveform inversion of passive seismic data for sources and velocities:

SEG Technical Program Expanded Abstracts, 35, 1405–1410.Symes, W., and J. Carazzone, 1991, Velocity inversion by differential semblance optimization: Geophysics, 56, 654–663.Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954–1966.Wang, H., and T. Alkhalifah, 2018, Microseismic imaging using a source function independent full waveform inversion method:

Geophysical Journal International, 214, 46–57.Warner, M., A. Ratcliffe, T. Nangoo, J. Morgan, A. Umpleby, N. Shah, V. Vinje, I. Stekl, L. Guasch, C. Win, G. Conroy, and A.

Bertrand, 2013, Anisotropic 3D full-waveform inversion: Geophysics, 78, no. 2, R59–R80.Weiss, R. M., and J. Shragge, 2013, Solving 3D anisotropic elastic wave equations on parallel GPU devices: Geophysics, 78, no.

2, F7–F15.Witten, B., and J. Shragge, 2015, Extended wave-equation imaging conditions for passive seismic data: Geophysics, 80, no. 6,

WC61–WC72.——–, 2017a, Image-domain velocity inversion and event location for microseismic monitoring: Geophysics, 82, no. 5, KS71–

KS83.——–, 2017b, Microseismic image-domain velocity inversion: Marcellus Shale case study: Geophysics, 82, no. 6, KS99–KS112.Yang, T., and P. Sava, 2015, Image-domain wavefield tomography with extended common-image-point gathers: Geophysical

Prospecting, 63, 1086–1096.Yang, T., J. Shragge, and P. Sava, 2013, Illumination compensation for image-domain wavefield tomography: Geophysics, 78, no.

5, U65–U76.

image-domain elastic waveﬁeld tomography for passive data

Documents