Coda-wave decorrelation sensitivity kernels in 2-D elastic media: a numerical approach

(1)

Article

Reference

Coda-wave decorrelation sensitivity kernels in 2-D elastic media: a numerical approach

DURAN, Alejandro, PLANES, Thomas, OBERMANN, Anne

DURAN, Alejandro, PLANES, Thomas, OBERMANN, Anne. Coda-wave decorrelation sensitivity kernels in 2-D elastic media: a numerical approach. Geophysical Journal International , 2020, vol. 223, no. 2, p. 934-943

DOI : 10.1093/gji/ggaa357

Available at:

http://archive-ouverte.unige.ch/unige:146551

Disclaimer: layout of this document may differ from the published version.

1 / 1

(2)

Geophys. J. Int.(2020)223,934–943 doi: 10.1093/gji/ggaa357 Advance Access publication 2020 July 30

GJI Seismology

Coda-wave decorrelation sensitivity kernels in 2-D elastic media: a numerical approach

Alejandro Duran,

¹

Thomas Plan`es

²

and Anne Obermann

¹

1Swiss Seismological Service, ETH Zurich,8006Zurich, Switzerland. E-mail:[email protected]

2Department of Earth Sciences, University of Geneva,1205Geneva, Switzerland

Received 2020 June 3; in original form 2020 February 24

S U M M A R Y

Probabilistic sensitivity kernels based on the analytical solution of the diffusion and radiative transfer equations have been used to locate tiny changes detected in late arriving coda waves.

These analytical kernels accurately describe the sensitivity of coda waves towards velocity changes located at a large distance from the sensors in the acoustic diffusive regime. They are also valid to describe the acoustic waveform distortions (decorrelations) induced by isotrop- ically scattering perturbations. However, in elastic media, there is no analytical solution that describes the complex propagation of wave energy, including mode conversions, polarizations, etc. Here, we derive sensitivity kernels using numerical simulations of wave propagation in heterogeneous media in the acoustic and elastic regimes. We decompose the wavefield intoP- andS-wave components at the perturbation location in order to construct separatePtoP,Sto S,PtoSandStoPscattering sensitivity kernels. This allows us to describe the influence ofP- andS-wave scattering perturbations separately. We test our approach using acoustic and elastic numerical simulations where localized scattering perturbations are introduced. We validate the numerical sensitivity kernels by comparing them with analytical kernel predictions and with measurements of coda decorrelations on the synthetic data.

Key words: Coda waves; Seismic interferometry; Wave propagation; Wave scattering and diffraction.

1 I N T R O D U C T I O N

Besides static imaging, seismic methods have the ability to monitor the dynamic evolution of the subsurface. The evolving targets—

for example, magma chambers, fault zones, reservoirs, etc.—

sometimes generate only minute perturbations of the propagating medium. Detecting such perturbations with first arrival waves can be challenging. Multiply scattered wave trains, forming coda waves, offer a very sensitive alternative for monitoring purposes (see the review of Obermann & Hillers2019). Poupinetet al.(1984) first demonstrated such sensitivity using the coda of earthquake multiplets to monitor subtle crustal velocity changes related to the activity of the Calaveras Fault, California.

Since then, the detection of temporal changes in the coda—

from repeating earthquakes or more recently from ambient noise correlations—has been successfully applied in various settings such as: CO2and geothermal fluid injection studies (Ugaldeet al.2013;

Hillerset al.2015; Obermannet al.2015; Tairaet al.2018; S´anchez- Pastoret al.2019), volcanoes (e.g. Poupinetet al.1996; Grˆetet al.

2005; Sens-Sch¨onfelder & Wegler 2006; Brenguier et al. 2016;

Donaldson et al. 2017; S´anchez-Pastor et al. 2018), fault zones (e.g. Poupinetet al.1984; Schaff & Beroza2004; Peng & Ben-Zion 2006; Brenguieret al.2008; Wuet al.2009; Minatoet al.2012;

Takagiet al.2012; Roux & Ben-Zion2014) and civil/geotechnical engineering (e.g. Olivieret al.2015,2017; Salvermoseret al.2015).

Furthermore, not only is the temporal evolution of the changes of interest, but also their spatial distribution within the medium, as well as their physical origin (subject to interpretation). Retrieving the spatial distribution of the changes using coda waves is not a straightforward problem given the complexity of the multiply scattered wave paths. Therefore, coda-wave measurements are often limited to detection purposes only. However, Pacheco & Snieder (2005) showed that the phase shift of coda waves induced by a localized velocity change could be modeled using a ‘probabilistic’

sensitivity kernel. This kernel describes the proportion of scattered- wave energy that interacts, on average, with the velocity change.

Laroseet al.(2010) and Rossettoet al.(2011) showed that a similar sensitivity kernel can describe the coda-wave decorrelation (also called decoherence) induced by a local scattering change (or structural change). The similarities and differences between these kernels are described in detail in Plan`eset al.(2014) and Margerinet al.

(2016) and will be briefly summarized in Section 2.1.

The sensitivity kernels allow for the modeling of the decorrelation (phase shift, respectively) of the coda, given a known spatial distribution of scattering (velocity, respectively) changes;

this forms the forward problem. Conversely, retrieving the spatial 934 ^CThe Author(s) 2020. Published by Oxford University Press on behalf of The Royal Astronomical Society.

Downloaded from https://academic.oup.com/gji/article/223/2/934/5878873 by University de Geneve user on 16 December 2020

(3)

distribution of scattering (velocity, respectively) changes from a set of coda-wave decorrelation (phase shift, respectively) measurements amounts to solving an inverse problem. This was successfully applied to: numerical simulations (Plan`es2013; Plan`eset al.2015;

Kanu & Snieder2015b; Obermannet al.2019), locating precursors of volcanic eruptions (Obermannet al.2013a; Lesageet al.2014;

S´anchez-Pastoret al.2018), assessing fault zone damages (Fro- ment2011; Obermannet al.2014), monitoring subsurface changes due to high-pressure fluid injections (Hillerset al.2015; Obermann et al.2015; S´anchez-Pastoret al.2019) and monitoring crack/fissure growth in concrete blocks (Laroseet al.2010,2015; Zhanget al.

2016).

These kernels are based on the description of the intensity of the wave propagation in the scattering medium. In the aforementioned applications, the coda waves are assumed to be composed of a single effective propagation mode, either surface waves or body waves.

Due to the equipartition of seismic energy (Henninoet al.2001),S- wave energy has the largest body-wave contribution allowing us to consider an effective body-wave mode mainly composed ofSwaves.

This assumption of a single propagation mode allows the use of an analytic expression of the multiply scattered intensity propagator.

Hence, either a 2-D sensitivity kernel describing the 2-D surface wave intensity propagation, or a 3-D sensitivity kernel describing the 3-D body-wave intensity propagation was used. However, coda waves are effectively composed of different propagation modes, namely,P,Sand surfaces waves, in evolving proportions that depend on the degree of heterogeneity, coda lapse-time, source mechanism, etc. (Margerinet al.1999; Henninoet al.2001; Pertonet al.2009;

Obermannet al.2013b). Building sensitivity kernels that describe the full complexity of the coda-wave composition is a challenging task, as the elastic radiative transfer equation does not have an analytical solution, even for the half-space. For the single-scattering regime, Maedaet al.(2008) proposed a method to synthesize coda envelopes taking into account the coupling of body and Rayleigh waves. Recently, Margerinet al.(2019) followed a similar approach for the multiple scattering regime, proposing a system of scalar radiative transfer equations that describe the coupling of body and surface waves. The numerical solutions of these equations could be used to construct elastic kernels for uniform elastic half-space media.

Obermannet al.(2016b,2019) proposed to tackle the problem of elastic kernel construction as a linear combination of surface- and body-wave sensitivity kernels, allowing for the description of the sensitivity of multiply scattered waves as a function of depth and lapse time in the coda, in an elastic half-space. This approach is, however, restricted to the description of a subsurface with uniform background velocity and uniform heterogeneity level.

Kanu & Snieder (2015a) proposed a numerical computation of the intensity propagator, and hence the sensitivity kernel, through an ensemble average of the envelope of the coda wavefield obtained in different realizations of the disorder. They successfully tested this approach by imaging velocity variations in numerical 2-D acoustic heterogeneous media (Kanu & Snieder2015b). This approach has the advantage of intrinsically taking into account the full complexity of the wavefield (inhomogeneous velocity/scattering model, topography, etc.). Sniederet al.(2019) proposed a theoretical basis to extend this numerical approach in order to describe the effect of a weak velocity variation in the case of an elastic medium. The main idea for tackling the elastic case is to decompose the wavefield into P- andS-wave components at the location of the medium change, in order to construct separateP- andS-wave sensitivity kernels.

In this paper, we build on this work to address the modeling of the coda-wave decorrelation induced by a scattering change in an elastic medium. In contrast to the case of the velocity change, we demonstrate how the mode conversions occurring at the scattering change location imply additional cross-term kernels (Section 2).

We describe the implementation of the numerical kernels (Section 3) and test their validity through a set of 2-D numerical simulations (Section 4).

2 C O D A - WAV E D E C O R R E L AT I O N S E N S I T I V I T Y K E R N E L S

2.1 Kernels for decorrelation versus kernels for phase shift Small velocity perturbations of the propagation medium induce phase shifts (or apparent traveltime changes) of coda waves. In contrast, strong and localized scattering perturbations (also called structural changes) induce a decorrelation (or distortion) of the coda waveforms. From real or synthetic coda waveforms, the ‘experimental’ traveltime changeτ^expcan be measured as the valueτ maximizing the following cross-correlation:

CC(τ)= t+T/2

t−T/2 ψ0(t−τ)ψ1(t)dt, (1)

and the decorrelation DC^expcan be measured using:

DC^exp(t)=1−

t+T/2

t−T/2 ψ0(t)ψ1(t)dt t+T/2

t−T/2 ψ0²(t)dtt+T/2

t−T/2 ψ1²(t)dt

, (2)

whereψ0andψ1are the waveforms prior to and after the medium perturbation, respectively. DC^expandτ^expare considered within a time windowtwof duration T (typically a few periods).

To relate a measured phase shift (or traveltime variation) in the coda to a localized velocity perturbation (e.g. stress change), Pacheco & Snieder (2005) introduced the following acoustic coda- wave sensitivity kernel:

K(s,r,r,t)= t

0 I(s,r,t)I(r,r,t−t)dt

I(s,r,t) , (3)

wheresandr are the positions of the source and the receiver,r is the position of the local change andtis the centre of the time interval in the coda where the phase shift is evaluated.I(a,b,t) is the ensemble averaged intensity of the scattered wavefield froma to bat timet. The kernelKcan be interpreted as the volumetric density of time that the scattered waves spend at locationrin the medium.

Laroseet al.(2010) and Rossettoet al.(2011) showed that the same sensitivity kernel applies for modeling the coda-wave decorrelation induced by a local scattering (or structural) change (e.g.

fractures). The coda-waveform decorrelationDCcan then be related to the scattering cross-sectionσof the local structural change, while the relative traveltime variationτ in the coda can be related to the local relative velocity changes ^d_v^v as:

DC(r,t)= cσ

2 K(r,t), (4)

τ(r,t)= −

K(r,t)dv

v(r)d V. (5)

whereKis the sensitivity kernel at locationrandcthe velocity of the scattered waves.

Margerinet al. (2016) showed how each observable (either a phase shift or a decorrelation) generally requires a specific and more

(4)

936 A. Duran, T. Plan`es and A. Obermann

complex kernel than the one used in eq. (3). In the general case, the distribution of energy density is not sufficient; information about the directional dependence of energy propagation (i.e. the specific intensity) is also needed to describe the effect of the perturbation in the coda. The description of the general case scenario falls outside the scope of this paper and the reader is referred to the study of Margerinet al.(2016) for further details. In the case of a velocity- change perturbation, the general kernel (Margerinet al.2016) can be simplified to the kernel introduced by Pacheco & Snieder (2005) when the perturbation is located far-away from the sensors, for example, beyond the transport mean-free path, and when the diffusive regime applies—that is, when the energy radiates at the same rate in all directions. In the case of a scattering perturbation (or structural change), the general kernel can be simplified to the kernel of eq. (3) in the diffusive regime and at large distance, or when the scattering pattern produced by the structural change is isotropic. For acoustic waves, this is usually the case for a perturbation of much smaller size than the central wavelength. For elastic waves, theS-wave scattering pattern is generally anisotropic regardless of the size of the perturbation.

2.2 Elastic decorrelation sensitivity kernels

Sniederet al.(2019) proposed the construction of elastic kernels based on the separation of the displacement vectoru(r,t) at the perturbation location intoP-andS-wave components asu =uP

+uS. The intensity, proportional to the square of the displacement velocity ( ˙u), hence reads:

I ∝( ˙uP+u˙S)²=u˙²_P+u˙²_S+2 ˙uP·u˙S (6) u˙²_P+u˙²_Saccount for the energy contribution of each wave mode.

The interference term ˙uP·u˙Scan be positive or negative according to the specific (and random) composition of coda waves at a partic- ular position and time. Thus, by averaging over space and time, this term goes to zero.

Here, we build on the work of Sniederet al.(2019) and model the effect of a local isotropic scattering change on the coda-wave decorrelation. By expanding the intensities intoP- andS-wave contributions, the numerator of the kernel (eq.3) can be written as:

I^srI^r^r=

I_P^sr+I_S^sr

I^r_P^r+I_S^r^r

=I^sr_PI_P^r^r+I_S^srI_S^r^r

+I^sr_PI_S^r^r+I_S^srI_P^r^r, (7) where the following change of notation is used for simplification:

I(s,r,t)I(r,r,t−t)≡I^srI^r^r. The first two terms of the right- hand side do not involve mode conversion from the anomaly and were already introduced by Sniederet al.(2019) for modeling a weak velocity change. In our case, the scattering perturbation pro- duces mode conversions and the two additional cross-terms need to be considered.

Hence, four sensitivity (sub-)kernels are obtained: KP P ∝ I^sr_PI_P^r^r,K_{S S}∝I_S^srI_S^r^r,K_{P S}∝I_P^srI_S^r^r and K_{S P} ∝I_S^srI_P^r^r. Using general indicesMandNto represent eitherP- orS-wave modes, the four elastic sensitivity kernels are described generically as:

KM N(r,t)= t

0 IM(s,r,t)IN(r,r,t−t)dt

I(s,r,t) . (8)

The numerator of this equation describes multiple scattered waves arriving at location r as waves of type ‘M’, then being scattered/converted into to type ‘N’ by the perturbation, and finally

traveling to the receiverrand being recorded as an arbitrary wave- type (or combination or wave-types), depending on the chosen receiver. The numerical construction of these sensitivity kernels will be detailed in Section 3.

2.3 Modeling waveform decorrelation in the elastic case We generalize the formulation of eq. (4) to model the decorrelation induced by strong and localizedP- andS-wave scattering anomalies:

DC^num_P (r,t)= vPσP P

2 KP P(r,t)+ vPσP S

2 KP S(r,t). (9) DC^num_S (r,t)= vSσS S

2 KS S(r,t)+vSσS P

2 KS P(r,t). (10) We chose the labelDC^numfor ‘numerical decorrelation’ because this decorrelation modeling involves the construction of numerical kernels (see Section 3). Eq. (9) describes the change in waveform coherence due to a localizedP-wave scattering perturbation in the medium. The first term models the coda waves that are scattered fromPtoPwave by the perturbation. The second term describes mode conversion fromPtoSwave at the anomaly location.

An analogous decomposition applies forDC^num_S in eq. (10), which models waveform decorrelation due to a localizedS-wave scattering anomaly.

The elastic numerical decorrelation is then described as the sum ofP- andS-wave contributions:

DC^num(r,t)=DC^num_P (r,t)+DC_S^numS(r,t). (11)

3 I M P L E M E N T AT I O N O F N U M E R I C A L S E N S I T I V I T Y K E R N E L S

3.1 Heterogeneous velocity model and numerical solution of the wave equation

To compute the intensity propagators that form the building blocks of the sensitivity kernels, we perform numerical simulations of seismic waves over several realizations of heterogeneous 2-D acoustic and elastic media, without intrinsic attenuation. The scattering is weakly anisotropic, but the medium itself does not show any pref- erential direction. The medium size is 33.6 x 33.6 km², with a 20 m spatial pitch. Similar to Obermannet al. (2013b), we su- perimpose spatial velocity fluctuations ofδv_P(x, z)= 20 per cent and 5 per cent on a constant backgroundP-wave velocity v⁰_P = 6500 m s⁻¹(Fig.1a). The velocity fluctuations are themselves char- acterized by a spatial autocorrelation function of Von-Karman type (Frankel & Clayton1986; Holliger & Levander1992) with a correlation length ofa=300 m, which is of the order of one centralP wavelengthλP=325 m. This enhances the interaction between the waves and the heterogeneities of the medium. For elastic media, the totalS-wave velocity relates to the totalP-wave velocity, asvS= ^√^v^P₃ and hence undergoes the same respective velocity variations. The medium density is kept constant at 3750 kg m⁻².

Following Obermannet al.(2013b), we computed the scattering () and transport (^∗) mean free paths from the coherent and inco- herent part of the synthetic signals and found for the media with 20 per cent fluctuation:= 1200±40 m and^∗= 1200±80 m for the elastic and=700±10 m and^∗=740±10 m for the acoustic media. The acoustic and elastic model thus show negligible anisotropic scattering (≈^∗).

(5)

(a) (b)

Figure 1.(a) Heterogeneous velocity model following the Von-Karman autocorrelation function. The locationssandrcorrespond to the source and receiver, respectively.r_iare the anomaly locations. (b) Radiation patterns atr₁forPtoP,StoS,StoPandPtoSwave scattering. These radiation patterns are computed numerically by sending a plane wave over the velocity anomaly in an elastic homogeneous media.

To solve the wave equation, we use the 2-D spectral-element method developed by Komatitsch & Vilotte (1998) and imple- mented in the SPECFEM2D solver (version 6.1) by Trompet al.

(2008). For the acoustic simulations, we use an explosion as source mechanism, with a central frequency off0=20 Hz, and a relative bandwidth off/f0=0.6. For the elastic simulations, we also use a horizontal force in form of a Ricker wavelet at the receiver location (explained in Section 3.3). In both cases, we simulate an infinite medium with absorbing boundary conditions on all sides. We computed the equipartition ratio following Obermannet al.(2013b) and obtained a value of E_s/ E_p ≈ 3.1. This matches well with the theoretical value of 3, indicating effective absorbing boundaries.

3.2 Computation of the intensity propagators

We recall that for the elastic case, we decompose the intensity propagator intoP- andS-wave components at the perturbation location (IPandIS). From the numerical simulations described above,IPis computed from the divergence of the displacementu(r,t) whileIS

is computed from the curl of the displacement:

I_P^num(s,r,t)=(λ+2μ) env²[∇ ·u(r,t)] (12) I_S^num(s,r,t)=μenv²[∇ ×u(r,t)] (13) Here, ‘env’ denotes the envelope of the recorded waveform computed with a Hilbert transform, andλandμare the Lam´e parameters of the medium.

Intensities are normalized by the energy of the sourceE0 and averaged over at least 20 numerical realizations of the heterogeneous media to reduce statistical fluctuations:

IM(s,r,t)=

I_M^num(s,r,t) E0

. (14)

Here,Mrepresents eitherPorSwaves.

3.3 Kernel construction

To construct the sensitivity kernels, the intensity propagators (nu- merators in eq. 8) need to be computed numerically at arbitrary pointsrin the medium (Kanu & Snieder2015a). Hence, two sets of simulations are needed, as depicted in Fig.2. In simulation set I, I(s,r,t) and I(s,r,t) are computed by placing the source at positionsand recording the displacement field at positionsr and r. In simulation set II,I(r,r,t−t) is computed by placing the source at positionrand recording the displacement field at positions sandr. Note that we use the following source–receiver reciprocity relation:I(r,r,t−t)=I(r,r,t−t).

For this relation to hold true, an important consistency consider- ation that was neglected by Kanu & Snieder (2015a) and pointed out by Sniederet al.(2019), necessitates that the source type used in Simulation II should be the ‘reciprocal’ of the receiver type used in Simulation I at locationr. For example, a pressure receiver atr in Simulation I requires the use of an explosive source atrin Simu- lation II. Likewise, a horizontal displacement receiver (x-direction) atrin Simulation I requires the use of a horizontal force source atrin Simulation II. The intensity propagators resulting from the two simulations are used to build the different subkernels according to eq. (8). In Fig. 3. we show snapshots of the different subkernels for infinite media, with two different degrees of heterogeneity corresponding to (a) 5 per cent and (b) 20 per cent of velocity fluctuations. For better visualization, the intensities are averaged over 20 realizations of numerical random media and computed at locations r_ithat are equally spread over the medium with interval distances of 240m (12 grid cells). The kernels are represented att= 2.6s, shortly after the directS-wave arrival time fromstor. Note that all of these subkernels correspond to the choice of an explosive source (at s) and a measurement of horizontal displacement (at receiver r).

In the case of a low level of heterogeneity (Fig.3a), the chosen receiver type at location r (horizontal displacement) affects the shape of the kernels. Equally, this can be understood as the effect of the radiation pattern of the ‘virtual’ (or reciprocal) source at r(simulation II, horizontal elastic force). For instance, the single

(6)

Figure 2. Sketch of the configuration used in each simulation in order to numerically compute the sensitivity kernel at an arbitrary pointr.

(a)

(b)

Figure 3. Elastic sensitivity kernels at 2.6 s for models with (a) 5 per cent and (b) 20 per cent of velocity fluctuations. The columns indicate the type of kernel.

The dashed line onKPPindicates the profile used to study the stability of the computations in Fig.4.

scattering ellipse ofKPPshows higher sensitivity values on its left- and right-hand sides, as compared to its top and bottom sides.

This is consistent with theP-wave radiation pattern of the virtual horizontal force source atr.K_SSshows an opposite effect, where the sensitivity values are larger above and below locationr, consistent with theS-wave radiation pattern of the virtual horizontal force source atr. Note that the radiation pattern generated by the explosive source at sis isotropic and does not contribute to the discussed sensitivity variations. The effect of the reciprocal source mechanism is also observed in the cross-term kernels. ForKPS, the pattern of S-wave radiation from the reciprocal source at r induces higher sensitivities above and below locationr. Additionally, the largeP- wave energy emitted by the explosive source ats, results in a large sensitivity around locations. To the contrary, forKSP, the pattern ofP-wave radiation from the reciprocal source atrinduces higher sensitivities left and right of locationr. Additionally, the low S-wave energy generated around the explosive source ats(only from local scattering) results in a smaller sensitivity around locations.

At later times, or in the presence of a higher level of heterogeneity (Fig.3b), Sniederet al.(2019) showed that the source imprint vanishes. In such cases, the sensitivity is more uniformly spread within the single scattering ellipse area. Note that in Fig. 3(a), because of the low velocity fluctuations (e.g. low heterogeneity level), theP-Psingle scattering ellipse shows a sharp boundary.

This boundary corresponds to a quasi-constant travel distance from the source to the receiver. In contrast, the high velocity fluctuations in Fig.3(b) result in a much more blurred single scattering ellipse boundary because of the larger possible variations in travel distances.

To construct the numerical kernels, ensemble-averaged intensities need to be used. Fig.4(left-hand panels) shows profiles of the numerical kernelsKPPat 2.6 s crossing the receiver location atz=21.5 km, as indicated in Fig.3, averaged over an increasing number of model realizations for (a) 5 per cent and (b) 20 per cent of velocity fluctuations. We see that the kernel stability already improves when averaged over five model realizations. In Fig. 4 (right-hand panels), we quantify the convergence of the kernels by integrating the sensitivity along the profile as

KPPdx. After an average across 20 model realizations, the fluctuations stabilize at about 5 percent, which we consider acceptable. This stabilization is similar for the kernelsKSS,KPSandKSP.

In Fig.5, we illustrate the lapse-time dependence of the numerical sensitivity computed at two exemplary locations:r₁, in the middle of the source–receiver axis, andr₂, close to the source (Fig.1). At early times (t<2 s, Fig.5, zoom),KPP has the largest contribution. At later times, the sensitivity toS-wave perturbations (KSS) becomes predominant. This is explained by the larger proportion of S-wave energy at equipartition (e.g. Henninoet al.2001). Snieder

(7)

(a)

(b)

Figure 4.The left-hand panels show a horizontal profile of the kernelKPP(dashed line in Fig.3) at timet=2.6 s averaged over an increasing number of velocity models for (a) 5 per cent and (b) 20 per cent of velocity fluctuations. The right-hand panels show the normalized integral along the profile for kernels KPP,KSS,KPSandKSPwith increasing model number. The numerical kernels stabilize after averaging over 20 realizations of random models. The horizontal line shows the mean of the integral values over models 20–60.

(a) (b)

Figure 5.Lapse-time dependence of the numerical elastic sensitivity kernels for media with 20 per cent velocity fluctuations, computed at the perturbation locationsr₁andr₂(Fig.1). The inset shows a zoom at early times when theP-wave energy dominates.

et al.(2019) also pointed out that the ratio of the sensitivity kernels stabilizes atKS/KP=9 in 2D, which is also the case for our simulations.

The largeS-wave energy contribution also plays a significant role for the cross-termsKSPandKPS. In equipartitioned conditions, as shown for the locationr₁ and equally valid for perturbations at locationr₃,K_SP=K_PS. At locationr₂, which is close to the source in s, the cross-term kernels are influenced by the intensity of ballistic Pwaves emitted by the nearby explosive source, thereforeKPS>

KSP.

4 VA L I D AT I O N O F T H E N U M E R I C A L A P P R O A C H

To validate the numerical construction of the sensitivity kernels, we compare measurements of decorrelation DC^exp with the modeled decorrelationDC^numin acoustic and elastic media. We additionally compare the kernel performance with the analytical solution of the radiative transfer,DC^RT(Shang & Gao1988; Sato1993; Paasschens 1997).

For the acoustic and elastic simulations, we successively intro- duce a scattering perturbation ofδvP/vP =7 per cent in one grid

(8)

Table 1. The coordinates of the source, receiver and of the three anomaly locations.

x-coordinate (km) y-coordinate (km)

Source 16.8 12.1

Receiver 16.8 21.5

r₁ 16.8 16.8

r₂ 16.8 14.4

r₃ 13.5 16.8

point (x=20m= ^λ₁₆^P) at three exemplary locationsr₁,r₂ andr₃ (Fig.1a). The coordinates of the source, receiver and of the three anomaly locations are gathered in Table1. Note that the distance between the source and the closest anomaly atr₂is about two mean- free paths.

These simulations use a background model with 20 per cent velocity fluctuations. The scattering cross-sections and radiation patterns of these perturbations are computed through an independent set of numerical simulations, performed in a homogeneous medium (Fig. 1b). A plane wave is sent towards the anomaly and the scattering pattern is measured by a set of sensors placed in a circular pattern around the anomaly. In the acoustic case, the small size of the perturbation compared to the wavelength induces isotropic scattering. This ensures the validity of the ‘simplified’ kernels constructed here (see Section 2), that is, kernels that do not take into account the angular dependence of the intensity flux. In the elastic case, the P-wave scattering is also isotropic but theS-wave scattering pat- tern is non-isotropic (Fig.1b). Strictly speaking, this means that the simplified elastic kernels developed here are only valid when the perturbation is located far away from the sensors (much beyond one transport mean-free path). Such configuration would ensure that the imprint of the scattering pattern of the perturbation does not affect the measurements.

We run each simulation in 20 statistically different heterogeneous models, prior (ψ0(τ)) and after (ψ1(τ)) the introduction of the perturbation. We then compute the decorrelationDC^expbetween the waveforms following eq. (2). In Fig.6, we show the experimental, numerical and theoretical decorrelation for scattering perturbations in the acoustic case. We observe that perturbations located along the direct source–receiver path (r₁), or close to one of the sensors (r₂), have a larger impact on the decorrelation. At greater distances from the sensors, the sensitivity decreases and the perturbation causes smaller decorrelations (r₃). When the perturbation is placed close to the source (r₂, less than one transport mean-free path), the angular dependence of the energy scattered at the anomaly location must be considered. Since the sensitivity kernels used here do not consider such dependence, there is a mismatch between the measurements and the predicted values forr₂. For all other receiver positions, the experimental valuesDC^expare well predicted by the theoreticalDC^RTand numericalDC^numdecorrelations, validating the construction of the numerically computed sensitivity kernels.

In the elastic case, we first study the decorrelation in the presence of individualP- andS-wave scattering perturbations (Figs7a and b), before tackling the case of the decorrelation induced by simultaneousP- andS-wave scattering perturbation (Fig.7c).

At all except very early lapse times, the decorrelation induced by anS-wave scattering perturbation (Figs7a and b) is one order of magnitude larger than for aP-wave scattering perturbation (Fig.7a).

This observation is a direct consequence of the higherS-wave energy content of an equipartitioned elastic wavefield. Consequently, the decorrelation induced by a simultaneousP- andS-wave scattering perturbation is dominated by theS-wave sensitivity (Fig.7c). The

decrease in sensitivity in respect to the increasing distance to the sensors (r₃) is analogous to the acoustic case. The numerical kernels allow for accurately modeling the experimental decorrelations in all cases with exception of (r₂), which is in close proximity to the source.

As we use a 2-D model with absorbing boundaries (no surface waves), we also attempt to model the wavefield as a single effective mode of propagation dominated bySwaves. This approach was used in concrete (Laroseet al.2015; Zhanget al.2016). We use the analytic radiative transfer solution (Paasschens1997) with effective velocity and transport mean-free path, as determined from the simulations, to construct the kernelDC^bulk, which shows a good fit with the simulated elastic kernels (Fig. 7c). Note that this effective analytical approach is only valid for uniform infinite media.

However, the numerical elastic kernels validated here, could permit to model waveform decorrelation in more complex settings and ge- ometries (e.g. free surface, layered model, topography, non-uniform background scattering, etc.).

5 D I S C U S S I O N A N D C O N C L U S I O N We derived decorrelation sensitivity kernels in the acoustic and elastic regime using numerical simulations of wave propagation. We expanded on the numerical approach of Kanu & Snieder (2015a,b) that had only been applied to model velocity changes in acoustic media (P-wave propagation only). After addressing the effect of velocity changes to the elastic coda (Sniederet al.2019), in this work we model the effect of a scattering change to the elastic coda.

The numerical kernels that we built can account for various types of sources and receivers, as well as for the combined presence of scatteredPandSwaves, and their conversions. The elastic wavefield is decomposed intoP- andS-wave components at the perturbation location, in order to construct separatePtoP,StoS,PtoSand StoPscattering sensitivity kernels. In agreement with the higher content ofS-wave energy in an equipartitioned wavefield, we show that the sensitivity toS-wave perturbation is much higher than the P-wave sensitivity at all but very early lapse times in the coda. This is a consequence of the chosen high level of heterogeneity.

We show that the numerical kernels are accurate in modeling measurements of coda waveform decorrelation due to a strong and local scattering perturbation within the medium. In acoustic media, the measured decorrelation due toP-wave scattering anomalies at different locations, fits well with the numerical kernel prediction and the theoretical kernel based on the radiative-transfer solution.

In elastic media, the measured decorrelation induced byP-,S- and P+S-wave scattering anomalies at different locations, fits well with the predictions from the numerical kernels.

In the presented simulations, we did not include surface-wave propagation. However, they can easily be included in the proposed kernel-building approach by changing the upper boundary condition in the simulations (in 2-D or 3-D). Likewise, our 2-D demonstration can be naturally extended to 3-D simulations, provided access to appropriate computing resources. Such important computational cost originates from the numerous realizations of numerical media that are needed to stabilize the kernels. This forms one important disadvantage of this numerical approach.

Such 3D simulations, with the inclusion of surface-wave propagation, could form the next testing step before application to real data. This would also allow for the comparison of the present approach, with the “semi-analytical” approach of Obermannet al.

(2016a,2019).

(9)

(a) (b) (c)

Figure 6.Experimental, numerical and theoretical decorrelation for three different point anomalies at locationsr₁,r₂,r₃ (Fig.1) in the acoustic medium.

When the anomaly (r₂) is placed in the proximity of the source, the energy flux at the anomaly is non-isotropic. Thus, there is a mismatch between the predicted and experimental values.

(a)

(b)

(c)

Figure 7.Numerical and experimental decorrelations forP-,S- andP+S-wave scattering point anomalies at locationsr₁,r₂ andr₃ (Fig.1) in the elastic medium.

The semi-analytical approach of Obermannet al.(2016a,2019) consists of building sensitivity kernels as a linear combination of body- and surface-wave sensitivity. The individual kernels are computed using analytical propagators. This approximation has proven to work well to describe the coda wave sensitivity in ’uniform media’, i.e. media with a uniform velocity distribution and a uniform heterogeneity distribution. However, this approach cannot describe the coda sensitivity of complex models, such as layered velocity structures, low velocity zones, topography, varying scattering strength, etc.

The present numerical approach has the advantage of taking into account the full complexity of the wave propagation and can apply to media with non-uniform scattering parameters and non-uniform background velocities. To compute realistic ensemble-average intensities in order to build the kernels, some prior knowledge of

’macro-properties’ of the model are needed, such as: macro-velocity and density model, heterogeneity level (i.e. scattering mean free path). The central frequency of the source is also needed.

From an imaging point of view, investigating the resolution differences at large versus short mean-free paths would be a key question

(10)

regarding potential applications. This would also allow for assessing in which situations the coda decorrelation could be used to discriminate P-wave scattering perturbations from S-wave scattering perturbations.

This however falls outside the scope of this study and remains to be investigated in the future.

A C K N O W L E D G E M E N T S

The research leading to these results has received funding in the form of a doctoral scholarship for AD from the Centro de Estudios Interdisciplinarios B´asicos y Aplicados en Complejidad (CeiBA), Bogot´a, Colombia. ThP acknowledges financial support from the Swiss National Fund and from the Swiss Federal Office of Energy.

The authors also want to thank the Leibniz-Rechenzentrum (LRZ) for access to and support in using the SUPERMUC system at the LRZ. The authors wish to thank Roel Snieder for fruitful discus- sions and C´eline Hadziioannou and an anonymous reviewer for their very constructive and detailed comments that helped to improve the manuscript.

R E F E R E N C E S

Brenguier, F., Campillo, M., Hadziioannou, C., Shapiro, N.M., Nadeau, R.M. & Larose, E., 2008. Postseismic relaxation along the San Andreas Fault at Parkfield from continuous seismological observations,Science, 321,1478–1481.

Brenguier, F., Rivet, D., Obermann, A., Nakata, N., Bou´e, P., Lecocq, T., Campillo, M. & Shapiro, N., 2016. 4-d noise-based seismology at volcanoes: ongoing efforts and perspectives,J. Volc. Geotherm. Res.,321, 182–195.

Donaldson, C., Caudron, C., Green, R.G., Thelen, W.A. & White, R.S., 2017. Relative seismic velocity variations correlate with deformation at k¯ılauea volcano,Sci. Adv.,3(6), e1700219, doi:10.1126/sciadv.1700219.

Frankel, A. & Clayton, R.W., 1986. Finite difference simulations of seismic scattering: implications for the propagation of short-period seismic waves in the crust and models of crustal heterogeneity,J. geophys. Res.,91, 6465–6489.

Froment, B., 2011.Utilisation du bruit sismique ambiant dans le suivi temporel de structures g´eologiques,PhD thesis, Universit´e de Grenoble.

Grˆet, A., Snieder, R., Aster, R.C. & Kyle, P.R., 2005. Monitoring rapid temporal change in a volcano with coda wave interferometry,Geophys.

Res. Lett.,32(6), 1–4.

Hennino, R., Tregoures, N., Shapiro, N.M., Margerin, L., Campillo, M., van Tiggelen, B.A. & Weaver, R.L., 2001. Observation of equipartition of seismic waves,Phys. Rev. Lett.,86(15), 3447–3450.

Hillers, G., Husen, S., Obermann, A., Plan`es, T., Campillo, M. & Larose, E., 2015. Noise-based monitoring and imaging of aseismic transient deformation induced by the 2006 Basel reservoir stimulation,Geophysics, 80(4), KS51–KS68.

Holliger, K. & Levander, A.R., 1992. A stochastic view of lower crustal fabric based on evidence from the Ivrea zone,Geophys. Res. Lett.,19(11), 1153–1156.

Kanu, C. & Snieder, R., 2015a. Numerical computation of the sensitivity kernel for monitoring weak changes with multiply scattered acoustic waves,Geophys. J. Int.,203(3), 1923–1936.

Kanu, C. & Snieder, R., 2015b. Time-lapse imaging of a localized weak change with multiply scattered waves using numerical-based sensitivity kernel,J. geophys. Res.: Solid Earth,120(8), 5595–5605.

Komatitsch, D. & Vilotte, J.P., 1998. The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures,Bull. seism. Soc. Am.,88(2), 368–392.

Larose, E., Plan`es, T., Rossetto, V. & Margerin, L., 2010. Locating a small change in a multiple scattering environment,Appl. Phys. Lett., 96(204101), 1–3.

Larose, E., Obermann, A., Digulescu, A., Plan`es, T., Chaix, J.F., Mazerolle, F. & Moreau, G., 2015. Locating and characterizing a crack in concrete with LOCADIFF: a four-point bending test,J. acoust. Soc. Am,138(1), 232–241.

Lesage, P., Reyes-Dávila, G. & Arámbula-Mendoza, R., 2014. Large tec- tonic earthquakes induce sharp temporary decreases in seismic velocity in Volcán de Colima, Mexico,J. geophys. Res.: Solid Earth,119(5), 4360–4376.

Maeda, T., Sato, H. & Nishimura, T., 2008. Synthesis of coda wave envelopes in randomly inhomogeneous elastic media in a half-space: single scattering model including Rayleigh waves,Geophys. J. Int.,172(1), 130–

154.

Margerin, L., Campillo, M., Shapiro, N.M. & van Tiggelen, B., 1999. Res- idence time of diffuse waves in the crust and the physical interpretation of coda Q. Application to seismograms recorded in Mexico,Geophys. J.

Int.,138,343–352.

Margerin, L., Plan`es, T., Mayor, J. & Calvet, M., 2016. Sensitivity kernels for coda-wave interferometry and scattering tomography: theory and numerical evaluation in two-dimensional anisotropically scattering media, Geophys. J. Int.,204(1), 650–666.

Margerin, L., Bajaras, A. & Campillo, M., 2019. A scalar radiative transfer model including the coupling between surface and body waves,Geophys.

J. Int.,219(2), 1092–1108.

Minato, S., Tsuji, T., Ohmi, S. & Matsuoka, T., 2012. Monitoring seismic velocity change caused by the 2011 Tohoku-oki earthquake using ambient noise records,Geophys. Res. Lett.,39(9).1-6

Obermann, A. & Hillers, G., 2019. Seismic time-lapse interferometry across scales,Adv. Geophys., Adv. Seismol.,60(3), 65–143.

Obermann, A., Plan`es, T., Larose, E. & Campillo, M., 2013a. Imaging pre- and co-eruptive structural and mechanical changes on a volcano with ambient seismic noise,J. geophys. Res.,118(12), 6285–6294.

Obermann, A., Plan`es, T., Larose, E., Sens-Sch¨onfelder, C. & Campillo, M., 2013b. Depth sensitivity of seismic coda waves to velocity perturbations in an elastic heterogeneous medium,Geophys. J. Int.,194(1), 372–382.

Obermann, A., Froment, B., Campillo, M., Larose, E., Plan`es, T., Valette, B., Chen, J.H. & Liu, Q.Y., 2014. Seismic noise correlations to image structural and mechanical changes associated with the Mw 7.9 2008 Wenchuan earthquake,J. geophys. Res.,119(4), 3155–3168.

Obermann, A., Kraft, T., Larose, E. & Wiemer, S., 2015. Potential of ambient seismic noise techniques to monitor the St. Gallen geothermal site (Switzerland),J. geophys. Res.,120(6), 4301–4316.

Obermann, A., Lupi, M., Mordret, A., Jakobsd´ottir, S.S. & Miller, S.A., 2016a. 3D-ambient noise Rayleigh wave tomography of Snæfellsj¨okull volcano, Iceland,J. Volc. Geotherm. Res.,317,42–52.

Obermann, A., Plan`es, T., Hadziioannou, C. & Campillo, M., 2016b. Lapse- time-dependent coda-wave depth sensitivity to local velocity perturbations in 3-D heterogeneous elastic media,Geophys. J. Int.,207(1), 59–66.

Obermann, A., Plan`es, T., Larose, E. & Campillo, M., 2019. 4-D Imaging of subsurface changes with coda waves: Numerical studies of 3-D combined sensitivity kernels and applications to the Mw 7.9, 2008 Wenchuan earthquake,Pure appl. Geophys.,176(3), 1–12.

Olivier, G., Brenguier, F., Campillo, M., Roux, P., Shapiro, N. & Lynch, R., 2015. Investigation of coseismic and postseismic processes using in situ measurements of seismic velocity variations in an underground mine, Geophys. Res. Lett.,42(21), 9261–9269.

Olivier, G., Brenguier, F., de Wit, T. & Lynch, R., 2017. Monitoring the stability of tailings dam walls with ambient seismic noise,Leading Edge, 36(4), 350a1–350a6.

Paasschens, J.C.J., 1997. Solution of the time-dependent Boltzmann equation,Phys. Rev. E,56(1), doi: 10.1103/PhysRevE.56.1135.

Pacheco, C. & Snieder, R., 2005. Time-lapse travel time change of multiply scattered acoustic waves,J. acoust. Soc. Am.,118,1300–1310.

Peng, Z. & Ben-Zion, Y., 2006. Temporal changes of shallow seismic velocity around the Karadere-D¨uzce branch of the north Anatolian fault and strong ground motion,Pure appl. Geophys.,163(2–3), 567–600.

Perton, M., S´anchez-Sesma, F., Rodr´ıguez-Castellanos, A., Campillo, M.

& Weaver, R., 2009. Two perspectives on equipartition in diffuse elastic fields in three dimensions,J. acoust. Soc. Am.,126(3), 1125–1130.

(11)

Planès, T., 2013.Imagerie de hangements locaux en régime de diffusion multiple,PhD thesis, Université de Grenoble.

Plan`es, T., Larose, E., Margerin, L., Rossetto, V. & Sens-Sch¨onfelder, C., 2014. Decorrelation and phase-shift of coda waves induced by local changes: multiple scattering approach and numerical validation,Waves Random Complex Media,2(24), 1–27.

Plan`es, T., Larose, E., Rossetto, V. & Margerin, L., 2015. Imaging multiple local changes in heterogeneous media with diffuse waves,J. acoust. Soc.

Am.,2(137), 660–667.

Poupinet, G., Ellsworth, W.L. & Frechet, J., 1984. Monitoring velocity variations in the crust using earthquake doublets: an application to the Calaveras fault, California, J. geophys. Res., 89(B7), 5719–5731.

Poupinet, G., Ratdomopurbo, A. & Coutant, O., 1996. On the use of earthquake multiplets to study fractures and the temporal evolution of an active volcano,Ann. Geophys.,39,253–264.

Rossetto, V., Margerin, L., Plan`es, T. & Larose, E., 2011. Locating a weak change using diffuse waves: Theoretical approach and inversion proce- dure,J. appl. Phys.,109(034903), 1–11.

Roux, P. & Ben-Zion, Y., 2014. Monitoring fault zone environments with correlations of earthquake waveforms,Geophys. J. Int.,196(2), 1073–

1081.

Salvermoser, J., Hadziioannou, C. & St¨ahler, S., 2015. Structural monitoring of a highway bridge using passive noise recordings from street traffic,J.

acoust. Soc. Am.,138(6), 3864–3872.

S´anchez-Pastor, P., Obermann, A. & Schimmel, M., 2018. Detecting and locating precursory signals during the 2011 El Hierro, Canary Islands, submarine eruption,Geophys. Res. Lett.,45(10), 288–297.

S´anchez-Pastor, P., Obermann, A., Schimmel, M., Weemstra, C., Verdel, A. & Jousset, P., 2019. Short-and long-term variations in the reykjanes geothermal reservoir from seismic noise interferometry,Geophys. Res.

Lett.,46(11), 5788–5798.

Sato, H., 1993. Energy transportation in one- and two-dimensional scattering media: analytic solutions of the multiple isotropic scattering model, Geophys. J. Int.,112,141–146.

Schaff, D.P. & Beroza, G.C., 2004. Coseismic and postseismic velocity changes measured by repeating earthquakes, J. geophys. Res., 109(B10302), 1–18.

Sens-Sch¨onfelder, C. & Wegler, U., 2006. Passive image interferometry and seasonal variations of seismic velocities at Merapi Volcano, Indonesia, Geophys. Res. Lett.,33(21), L21302, doi:10.1029/2006GL027797.

Shang, T. & Gao, L., 1988. Transportation theory of multiple scattering and its application to seismic coda waves of impulsive source,Sci. Sin.,31, 1503–1514.

Snieder, R., Duran, A. & Obermann, A., 2019.Locating Velocity Changes in Elastic Media with Coda Wave Interferometry,pp. 188–217(chap. 6), Cambridge University Press, Cambridge, UK.

Taira, T., Nayak, A., Brenguier, F. & Manga, M., 2018. Monitoring reservoir response to earthquakes and fluid extraction, Salton Sea geothermal field, California,Sci. Adv.,4(1), doi:10.1126/sciadv.1701536.

Takagi, R., Okada, T., Nakahara, H., Umino, N. & Hasegawa, A., 2012.

Coseismic velocity change in and around the focal region of the 2008 Iwate-Miyagi Nairiku earthquake,J. geophys. Res,117(B06315), 1–19.

Tromp, J., Komatitsch, D. & Liu, Q.Y., 2008. Spectral-element and adjoint methods in seismology,Comm. Comp. Phys.,3,1–32.

Ugalde, A.,et al., 2013. Passive seismic monitoring of an experimental co2 geological storage site in hontom´ın (northern spain),Seismol. Res. Lett., 84(1), 75–84.

Wu, C., Peng, Z. & Ben-Zion, Y., 2009. Non-linearity and temporal changes of fault zone site response associated with strong ground motion, Geophys. J. Int.,176(1), 265–278.

Zhang, Y., Plan`es, T., Larose, E., Obermann, A., Rospars, C. & Moreau, G., 2016. Diffuse ultrasound monitoring of stress and damage development on a 15-ton concrete beam,J. acoust. Soc. Am.,139(4), 1691–1701.