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Geophysical Journal International
, Volume 212 (2) – Feb 1, 2018

25 pages

/lp/ou_press/on-the-sensitivity-of-teleseismic-full-waveform-inversion-to-earth-OTda0HqNY5

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.
- ISSN
- 0956-540X
- eISSN
- 1365-246X
- D.O.I.
- 10.1093/gji/ggx480
- Publisher site
- See Article on Publisher Site

Summary Full-waveform inversion (FWI) is not yet a mature imaging technology for lithospheric imaging from teleseismic data. Therefore, its promise and pitfalls need to be assessed more accurately according to the specifications of teleseismic experiments. Three important issues are related to (1) the choice of the lithospheric parametrization for optimization and visualization, (2) the initial model and (3) the acquisition design, in particular in terms of receiver spread and sampling. These three issues are investigated with a realistic synthetic example inspired by the CIFALPS experiment in the Western Alps. Isotropic elastic FWI is implemented with an adjoint-state formalism and aims to update three parameter classes by minimization of a classical least-squares difference-based misfit function. Three different subsurface parametrizations, combining density (ρ) with P and S wave speeds (Vp and Vs) , P and S impedances (Ip and Is), or elastic moduli (λ and μ) are first discussed based on their radiation patterns before their assessment by FWI. We conclude that the (ρ, λ, μ) parametrization provides the FWI models that best correlate with the true ones after recombining a posteriori the (ρ, λ, μ) optimization parameters into Ip and Is. Owing to the low frequency content of teleseismic data, 1-D reference global models as PREM provide sufficiently accurate initial models for FWI after smoothing that is necessary to remove the imprint of the layering. Two kinds of station deployments are assessed: coarse areal geometry versus dense linear one. We unambiguously conclude that a coarse areal geometry should be favoured as it dramatically increases the penetration in depth of the imaging as well as the horizontal resolution. This results because the areal geometry significantly increases local wavenumber coverage, through a broader sampling of the scattering and dip angles, compared to a linear deployment. Inverse theory, Waveform inversion, Body waves, Computational seismology, Crustal imaging, Seismic tomography 1 INTRODUCTION Building high-resolution and quantitative images of the lithosphere from body waves is a key challenge in earthquake seismology. The methodological challenge results from the heterogeneous nature of the crust and the complex interaction of the wavefields with both these heterogeneities and the free surface. From the geodynamical viewpoint, building images of the lithosphere with the best possible resolution according to the available frequency band (namely, of the order of the wavelength) is crucial to correlate tectonic deformation in the crust with deeper mantellic process arising in the asthenosphere. Lithospheric images can be built either from regional earthquakes (earthquakes nucleated in the lithospheric target) or from distant earthquakes, located several thousands of kilometres away from the target and commonly referred to as teleseisms. One key advantage of the teleseismic configuration is to potentially provide a large catalogue of events that are suitable for lithospheric imaging by either traveltime tomography, receiver function analysis or waveform inversion techniques. The teleseismic wavefield can fairly be approximated by an incident planar wavefield incoming from outside of the lithospheric target with different incidence angles and backazimuths. This planar configuration is conducive to a rather uniform illumination of the lithospheric target but might be a limiting resolution factor due to the limited angular illumination proved by plane-wave sources as opposed to point sources. Teleseismic data sets are conventionally processed by either traveltime tomography or receiver function analysis. Traveltime tomography, when limited to the first-arrival P-wave, provides subsurface model with a resolution of the order of the first Fresnel zone width (Williamson 1991). In the teleseismic setting where the incident planar P-wave wavefields impinge the base of the lithospheric target with incidence angles ranging between 20° and 60°, the tomographic models tend to exhibit significant vertical smearing of lithospheric structures along a banana-shape sensitivity kernel (Marquering et al. 1999), hence attesting to the limited resolution of these approaches. This vertical resolution issue has prompted some authors to enrich teleseismic data sets with either regional (Bavalia et al. 2016) or global (Weidle & Widiyantoro 2005) data sets to perform P-wave traveltime tomography. Conversely, migration of receiver functions (Vinnik 1977; Langston 1979) can provide sharper images of lithospheric discontinuities (Ryberg & Weber 2000; Zhu & Kanamori 2000). However, stacking of P-wave receiver functions are inherently contaminated by secondary arrivals, that limit their use to first order PS converted waves generated by strong impedance contrast at major lithospheric discontinuities (Moho, 410 km and, 660 km boundaries). The current densification of permanent and temporary arrays in conjunction with the continuous growth of computing facilities permit the consideration of waveform inversion methods that are expected to provide quantitative images of the subsurface with a resolution close to the wavelength. Among teleseismic waveform inversion techniques, least-squares migration of scattered teleseismic body waves was first proposed by Bostock et al. (2001), Shragge et al. (2001), Rondenay et al. (2001) and Rondenay et al. (2005). In their approach, the forward problem is linearized with the single-scattering Born approximation, Green functions are computed by dynamic ray tracing and the migration is recast as a linear waveform inversion accordingly, where the misfit between the recorded and modelled scattered wavefields is minimized in a least-squares sense. Bostock et al. (2001) emphasized the key role of the free surface as a secondary source of down-going P and S waves. Compared to the migration of receiver functions, which are often focused on the forward-scattered converted P–S waves, migration of reflected waves back-scattered by lithospheric reflectors after a first reflection from the free surface produces sharp P- and S-wave velocity perturbation models of the lithosphere. A natural extension of the former linear waveform inversion is nonlinear full-waveform inversion (FWI), a technique also borrowed from exploration geophysics (Tarantola 1984; Mora 1987; Pratt 1999; Virieux & Operto 2009). The main difference with the above-mentioned linear waveform inversion is that the full wavefield residuals are minimized in FWI as opposed to the single-scattered wavefield residuals in scattering migration. This implies that the background subsurface model, in which seismic modelling is performed, is updated at each FWI iteration, while the background model is kept the same over iterations in scattering migration. A key advantage of the nonlinear inversion is that, as short-scale features are injected in the subsurface model over several inversion iterations, they can help to more fully account for converted waves and multiscattering during modelling. Conversely, a potential advantage of the linear inversion is that the explicit separation between the background wavefield and the scattered wavefield during a pre-processing stage and the use of ray tracing to select specific phases can be helpful to drive the inversion towards the best resolving information in the data (back-scattered P–S converted waves). Alternatively to FWI, Shang et al. (2012) proposed to image lithospheric reflectors by reverse time migration, while the background velocity models are built by reflection tomography from the free surface multiples (Burdick et al. 2014). This purely reflection approach discards the contribution of the incident wavefields and uncouples the update of the long wavelengths of the lithosphere by migration-based velocity analysis from the shorter ones by migration through an explicit scale separation as conventionally performed in multichannel seismic reflection processing. In its conventional form, FWI seeks to minimize in a least-squares sense the sample-to-sample misfit between the recorded and modelled seismograms. This implies that both traveltimes and amplitudes of all of the arrivals are involved in the misfit function. One question which arises in relation to the use of the amplitudes is whether secondary parameters such as density and attenuation can be reliably updated during FWI. While multi-parameter reconstruction for density and attenuation is an active field of research in controlled-source seismology (e.g. Hicks & Pratt 2001; Malinowski et al. 2011; Kamei & Pratt 2013; Kurzmann et al. 2013; Prieux et al. 2013; Groos et al. 2014), the path taken by the earthquake seismology community has been rather to modify the sensitivity kernels of the FWI to remove amplitudes effects and recast the waveform inversion as a finite-frequency traveltime inversion of the first arrival (Marquering et al. 1999) or selected wave packets (Maggi et al. 2009). This approach is generally referred to as adjoint tomography by the earthquake seismological community when seismic modelling is performed with full-wave numerical techniques such as the spectral element method (Komatitsch & Tromp 1999). Here, by adjoint is meant a mathematical technique that allows for the efficient computation of the gradient of a functional without forming the sensitivity matrix (see Tromp et al.2005; Fichtner et al.2006a,b; Plessix 2006 for a review). An alternative to the adjoint approach is the scattering-integral approach which relies on the explicit computation of the sensitivity matrix (Chen et al. 2007). Several applications of adjoint tomography at regional, continental and global scales have been presented by Tape et al. (2010), Fichtner et al. (2009, 2010, 2013), Zhu et al. (2012, 2015) and Bozdağ et al. (2016). During these last years, there have been a few attempts to apply FWI to teleseismic data for lithospheric imaging. The first attempts were performed in 2-D and 2.5-D using a frequency-domain implementation (Roecker et al. 2010; Pageot et al. 2013; Baker & Roecker 2014). A frequency domain formulation was used because 2-D seismic modelling can be performed efficiently with Gauss elimination techniques when a large number of sources are processed for a few discrete frequencies (Pratt 1999). However, Pageot et al. (2013) showed that a fine sampling of the frequencies was required during the inversion to balance the narrow and coarse scattering angle illumination provided by a few plane wave sources. This, along with the computational cost of large matrix factorization, lead them to the conclusion that a time-domain implementation was more suitable than the frequency-domain counterpart to perform 3-D teleseismic FWI. 3-D time-domain FWI of teleseismic data has been first investigated by Tong et al. (2014a), who assessed the ability of FWI to reconstruct reflector geometry and density, Vp and Vs models of simple volumetric structures such as cubic inclusion or slab through a careful analysis of sensitivity kernels of specific phases. Monteiller et al. (2015b) assessed the resolution power of FWI to image a simple crustal model with a sharp Moho from four teleseismic events. The reliability of FWI for teleseismic application was further demonstrated with an application to real data collected in the framework of the Pyrope experiment across the Pyrénées range (Wang et al. 2016). FWI of five events recorded by 29 stations provided tomographic P and S wave velocity (Vp and Vs) models of the Pyrennean lithospheric structure with an unprecedented resolution. Moreover, Wang et al. (2016) showed the consistency between the FWI Vp and Vs models and migrated images inferred from receiver functions. In parallel with this, Beller et al. (2017) applied FWI on nine teleseismic events collected during the CIFALPS experiment (Zhao et al. 2015) in the Western Alps. They use a FWI approach similar to Monteiller et al. (2015b) except that the density (ρ) was processed as an optimization parameter and jointly updated with Vp and Vs during the inversion. The FWI models show convincing images of the continental subduction and the Ivrea body as well as a low Vs zone in the lower lithosphere supporting the hypothesis of a slab detachment. The inversion was pushed up to a maximum frequency of 0.2 Hz. The ρ model shows structures that are consistent with those shown in the Vs model in the upper lithosphere. However, the maximum depth at which ρ can be imaged is smaller than for VS suggesting that ρ has been mostly reconstructed from free-surface multiples. Another striking result is the good resolution of the imaging in the cross-line direction, perpendicular to the main receiver line of the CIFALPS experiment, although a quite sparse distribution of stations in the cross-line direction. In particular, Beller et al. (2017) successfully recovered the arc shape of the Ivrea body, whose ρ model allows it to match the Bouguer anomaly. Despite this recent success, there is still a need to appraise the promises and pitfalls of 3-D teleseismic FWI. First, the plane-wave configuration raises the issue of the resolution with which lithospheric structures can be imaged from a few teleseismic events. Although Bostock et al. (2001) have emphasised the key role of the free surface multiples, the extent to which second-order back-scattered waves can be exploited by FWI when the full wavefield is processed in one go, that is, without explicit separation between the incident and scattered wavefields, is not yet clear. A second issue is related to the choice of the best subsurface parametrization (the parameter set that fully describes the subsurface) and the optimization parameters (the subset of parameters that are updated during the inversion as opposed to the passive parameters that are kept fixed) for teleseismic FWI. This issue is far from being neutral as long as one seeks to update secondary parameters such as density, attenuation or anisotropic parameters (see Operto et al.2013, for a tutorial). The choice of the subsurface parametrization and the optimization parameters is mainly driven by the need to mitigate parameter cross-talk, while preserving a sufficient resolution in the imaging. The most common tools to define a suitable parametrization for FWI relies on the analysis of the so-called radiation patterns (Tarantola 1986; Forgues & Lambaré 1997; Gholami et al. 2013; Alkhalifah & Plessix 2014), the principal component analysis of the sensitivity matrix (Sieminski et al. 2009) or the eigenvector analysis of the Hessian operator (Plessix & Cao 2011). A third issue is related to the role of the initial model in teleseismic FWI. It is well acknowledged that this initial model should allow the prediction of traveltimes with an error that does not exceed half the period to prevent cycle skipping artefacts when a conventional least-squares difference-based misfit function is used (e.g. Virieux & Operto 2009). This condition is challenging to fulfil when a large number of wavelengths are propagated. This arises when seismic data lack low frequencies as in controlled-source seismology and the acquisition geometry involve long propagation distances. The low frequency content of teleseismic data should make FWI reasonably immune to cycle skipping. However, it is unclear if other factors than the kinematic accuracy of the initial model, for example the smoothness versus blocky character of the velocity model or site effects near the free surface, can drive the FWI towards a local minimum. A fourth obvious factor is related to the footprint of the station spread and sampling in teleseismic FWI. Clarifying this issue is crucial to design reliable surveys amenable to high resolution imaging methods such as FWI. Using too coarse stations network can generate aliasing artefacts of different nature (Shragge et al. 2001; Rondenay et al. 2005; Pageot et al. 2010). This has prompted the design of several teleseismic acquisitions in the form of dense profile of stations spaced 5 km apart at the expense of coarser areal deployments (Rondenay et al. (2001, Cascadia experiment), Iglesias et al. (2010, MASE experiment), Zhao et al. (2015, CIFALPS experiment). While such a dense linear set-up might be suitable to sample teleseismic events arriving with an azimuth aligned with the receiver line, it may not be suitable to record waves that are scattered off the receiver line or process events reaching the array with a significant obliquity. Among other parameters that may impact the reconstruction of lithospheric models by teleseismic FWI, we do not consider the effects of the heterogeneities that are located outside the lithospheric box. This issue has been recently investigated by Masson & Romanowicz (2017) for both traveltime and full-waveform tomography. For full-waveform tomography, they have shown that acceptable results can be obtained even when no a priori information other than a 1-D reference model is available outside the lithospheric target (Masson & Romanowicz 2017, their fig. 11). However, the FWI results can be significantly improved when a smooth traveltime tomography model is used rather than a 1-D reference model as a background model outside the lithospheric target (Masson & Romanowicz 2017, their fig. 14). We do not consider either the effects of the source wavelet estimation and noisy data. One issue with the source signature estimation is the potential trade-off with the subsurface parameter updates, in particular when these two optimization processes are performed in an alternating manner in the inversion iterations. This prompts us to estimate the source signature prior to the FWI and keep it fixed afterwards during the CIFALPS real data case study (Beller et al. 2017). This prior source estimation can be useful to mitigate the effects of inaccurate source location and moment tensor estimation, and also absorb the effects of remote large-scale heterogeneities located outside the lithospheric target as these heterogeneities will mainly generate a traveltime perturbation of the nearly planar incident teleseismic wavefields (Monteiller et al. 2015a). Since we aim to keep the conclusions of this study as general as possible, we do not add noise to the data during the synthetic experiments since the amount of noise as well as the ability of the acquisition to mitigate its effect by stacking is really case dependent. Also, for a real data application, noisy data can thoroughly be rejected or pre-processed as it was illustrated in Beller et al. (2017). The aim of this study is to gain new insights on the three above-mentioned issues through the application of 3-D time-domain FWI on a realistic synthetic case study inspired from the CIFALPS experiment. In the first part of this study, we review the main theoretical ingredients of the FWI. Then, we present the synthetic lithospheric model of the western Alps and describe the anatomy of the data computed in this model. We interpret the dominant arrivals recorded by the three geophone components and identify part of the signal that underwent aliasing. This is followed by a parametric analysis of the FWI. We first assess three different subsurface parametrizations: (ρ, λ, μ), (ρ, vp, vs) and (ρ, Ip, Is), where λ and μ are the Lamé parameters and Ip and Is the P and S impedances. Our numerical examples suggest that the (ρ, λ, μ) parametrization provides the best FWI results after recombining a posteriori the ρ, λ and μ optimization-parameter models into Ip and Is. Second, we assess the sensitivity of the FWI to the initial models. We show that a smooth tomographic model is the most suitable one for FWI, although 1-D reference models as PREM are enough to achieve acceptable FWI results. Third, we assess the sensitivity of the teleseismic FWI to two different kinds of acquisition geometry: coarse areal configuration versus dense linear acquisition. Our results strongly support that areal deployment should be first designed at the expense of a dense linear deployment for the application of teleseismic FWI. For such a configuration, areal deployments allows to increase the range of usable data, specifically, those with a high signal-to-noise ratio emanating from earthquakes not aligned with a linear array. Besides, 2-D arrays can sample a broader wavenumber spectrum than linear arrays, and hence improve the expected resolution of FWI images at the expense of data redundancy. This redundancy can nonetheless be achieved by including events occurring in a common tectonic area. 2 TELESEISMIC FULL-WAVEFORM INVERSION This section reviews some basic principles of FWI that will be useful to interpret the results of the numerical experiments carried out in this study. For sake of compactness, we review these principles with a matrix formalism suitable for the frequency-domain formulation of FWI (Pratt et al. 1998), although our teleseismic FWI is fully implemented in the time domain. 2.1 Theoretical background Let’s assume that a modelling engine such as those based upon hybrid methods is available to compute synthetic teleseismic seismograms (Monteiller et al. 2013; Tong et al. 2014b; Beller et al. 2017). The elastodynamic equations can be written in matrix form as \begin{equation} \mathbf {A}(\mathbf {m})\mathbf {u}=\mathbf {s}, \end{equation} (1)where the vectors s and u denote the seismic source and the synthetic wavefield, respectively, and the matrix A is the wave modelling operator, the coefficients of which depend on frequency (or time) and subsurface properties gathered in the vector m. The most natural parametrization of the subsurface involve the density ρ and elastic moduli c = cijkl since this parametrization directly results from the equation of motion and the Hooke’s constitutive law. FWI is a data fitting procedure that aims to find the subsurface parameters m* that minimize a certain measure $$\mathcal {C}$$ of the data misfit, in its most conventional form, the least-squares norm of the sample-to-sample difference between the observed and the modelled data \begin{equation} {{\bf m}}^* = \mathop{\rm arg \, min}\limits_{\mathbf {m}} \mathcal {C}(\mathbf {m}), \end{equation} (2)where $$\mathcal {C}(\mathbf {m})=\frac{1}{2} \Vert \boldsymbol{\Delta }\mathbf {d} \Vert _2^2$$ and $$\boldsymbol{\Delta }\mathbf {d}=\mathbf {R} \mathbf {u(m)} - \mathbf {d}$$ is the difference between the recorded data d and the modelled wavefield sampled at the receiver positions by the operator R. Because of the computational burden of the seismic modelling and the huge dimension of the model space, FWI is recast as an iterative local optimization problem that seeks the minimum of the misfit function in the vicinity of a starting model mk at iteration k. Zeroing the gradient of the misfit function after a second-order Taylor expansion of the misfit function (2) around the starting model provides the Newton descent direction. Multiplying this descent direction with a step length γk, that can be found by a line search procedure (Nocedal & Wright 2006), provides the model perturbation Δmk + 1 at iteration k \begin{equation} \Delta \mathbf {m}_{k+1} = -\gamma _k \ \mathbf {H}^{-1}(\mathbf {m}_k) \ \nabla _{\mathbf {m}} \mathcal {C} (\mathbf {m}_k) , \end{equation} (3)where H and $$\nabla _{\mathbf {m}} \mathcal {C}$$ are the Hessian (the second derivative of the misfit function) and the gradient of the misfit function with respect to m, respectively. 2.2 Gradient of the misfit function The gradient of the misfit function is given by \begin{equation} \nabla _{\mathbf {m}} \mathcal {C} = \left( \frac{\partial \mathbf {R \, u}(\mathbf {m})}{\partial \mathbf {m}}\right)^T \boldsymbol{\Delta }\mathbf {d} = \mathbf {J}^T \boldsymbol{\Delta }\mathbf {d}, \end{equation} (4)where J denotes the Fréchet derivatives matrix or sensitivity matrix (Pratt et al. 1998). Differentiation of eq. (1) with respect to a model parameter m gives \begin{equation} \mathbf {A}\frac{\partial \mathbf {u}}{\partial m} = - \frac{\partial \mathbf {A}}{\partial m}\mathbf {u}. \end{equation} (5)Analogy between eqs (1) and (5) shows that the partial derivative wavefield ∂u/∂m is the solution of the elastodynamic equations for a secondary virtual source − (∂A/∂m)u. The sparse operator ∂A/∂m defines the spatial and temporal supports of the virtual source at the position of m in space and at the arrival time of u at m in time, respectively as well as its radiation pattern. Therefore, the partial derivative wavefield can be interpreted as the wavefield scattered by a point diffractor located at m. The traveltime curve followed by this partial derivative wavefield sampled at the receiver positions (this sampling gives one column of the sensitivity matrix) describes a diffraction hyperbola (Fig. 1). The gradient of the misfit function is simply formed by the zero-lag correlation between the sampled partial derivative wavefield and the data residuals or in other words by the summation along the diffraction hyperbola of the product between the partial derivative wavefield and the data residuals. The aim of this correlation is to pick among all of the residuals those which result from a missing heterogeneity located at m. Figure 1. View largeDownload slide Sensitivity matrix and FWI imaging principle. The column j of the sensitivity matrix describes the synthetic wavefield that would be scattered by a diffractor located at mj and sampled at the stations positions. In this figure, we show the case of a P–S diffraction. The support of this scattered wavefield in the time-distance domain follows a diffraction hyperbola. The gradient of the misfit function is formed by a weighted stack of the data residuals along these trajectories. Conversely, the line i of the sensitivity matrix describes the positions in space along which a sample of the data residuals ∂ui is projected. These positions describe an isochronal ellipse as in scattering migration. Summation over all possible ellipses focuses by constructive interference the image of the scatterer with a resolution controlled by the source bandwidth and the angular illumination provided by the acquisition geometry (adapted from Rondenay et al.2005). Figure 1. View largeDownload slide Sensitivity matrix and FWI imaging principle. The column j of the sensitivity matrix describes the synthetic wavefield that would be scattered by a diffractor located at mj and sampled at the stations positions. In this figure, we show the case of a P–S diffraction. The support of this scattered wavefield in the time-distance domain follows a diffraction hyperbola. The gradient of the misfit function is formed by a weighted stack of the data residuals along these trajectories. Conversely, the line i of the sensitivity matrix describes the positions in space along which a sample of the data residuals ∂ui is projected. These positions describe an isochronal ellipse as in scattering migration. Summation over all possible ellipses focuses by constructive interference the image of the scatterer with a resolution controlled by the source bandwidth and the angular illumination provided by the acquisition geometry (adapted from Rondenay et al.2005). Although eqs (4) and (5) draw some clear connection between FWI and diffraction tomography, the gradient of the misfit function can be more efficiently computed with the adjoint-state method at the cost of two forward simulations per source (Plessix 2006, for a review). Following Pratt et al. (1998), injecting the expression of the partial derivative wavefield (5) into (4) gives \begin{equation} \nabla _{m} \mathcal {C} = -\mathbf {u}^T \left(\frac{\partial \mathbf {A}}{\partial m} \right)^T \left(\mathbf {A}^{-1}\right)^T\boldsymbol{\Delta }\mathbf {d}^* = -\left( \frac{\partial \mathbf {A}}{\partial m} \mathbf {u} \right)^T {\bf u}^{\dagger *}, \end{equation} (6)where the adjoint wavefield u† satisfies \begin{equation} \mathbf {A}^{T} \ {\bf u}^{\dagger *}= \mathbf {R}^{T} \boldsymbol{\Delta }\mathbf {d}^*. \end{equation} (7)This expression shows that the gradient is obtained by a zero-lag correlation of the virtual source − (∂A/∂m)u with the adjoint wavefield u†, that is back-propagated in the subsurface using the residuals as a composite source term. Considering only one sample of the data residuals associated with a source s , a receiver r and a traveltime T, its energy will be smeared in the subsurface along positions x that satisfy the imaging condition Tsx + Trx = T. Here, Tsx and Trx denote the traveltimes of a wave connecting the source and the receiver to the scatterer x, respectively. These positions describe an isochronal surface, which is an ellipse in a 2-D homogeneous background medium (Fig. 1). All of these scattering concepts and imaging condition are typically met in migration techniques (such as ray+Kirchhoff or ray+Born migration) (McMechan & Fuis 1987; Miller et al. 1987; Lumley et al. 1994; Rondenay et al. 2005) and highlight the connections that can be drawn between FWI and generalized diffraction tomography (Devaney 1984; Wu & Toksöz 1987) as reviewed by Pratt et al. (1998). For a general elastic medium parametrized by the density ρ and the coefficients c = cijkl of the stiffness tensor, the gradient is simply given by: \begin{eqnarray} \nabla _\rho \mathcal {C} (\mathbf {x}) &=& \int _0^T \mathbf {u}^\dagger (\mathbf {x},t) \cdot \partial _t^2\mathbf {u}(\mathbf {x},t) dt, \nonumber\\ \nabla _{\mathbf {c}} \mathcal {C} (\mathbf {x}) &=& \int _0^T \boldsymbol{\epsilon }^\dagger (\mathbf {x},t)::\boldsymbol{\epsilon }(\mathbf {x},t) dt, \end{eqnarray} (8)where c denotes the stiffness tensor, u and u† the displacement and adjoint displacement wavefields, and $$\boldsymbol{\epsilon }$$ and $$\boldsymbol{\epsilon }^\dagger$$ the strain tensor and the adjoint strain tensor, respectively (Tromp et al. 2005). Other parametrizations may be considered and deduced from the former using the chain-rule of derivatives (Köhn et al. 2012), three of them are inspected in Section 2.4. 2.3 Resolution analysis We review here the main experimental factors that control the spatial resolution of the FWI. In the framework of diffraction tomography and FWI respectively, Wu & Toksöz (1987) and Sirgue & Pratt (2004) showed that the gradient of the misfit function with respect to one parameter has the form of a truncated inverse Fourier series in which the arguments of the basis functions are the wavenumber components locally injected in the targeted model at the position of the parameter. The truncation of this Fourier series, which results from the limited bandwidth of the source and the limited spread of the acquisition, limits the resolving power of FWI. In the high-frequency approximation (or local plane wave approximation), the local wavenumber vectors k injected at a given position in a 2-D subsurface model (Fig. 2a) can be inferred from the ray paths connecting the sources and the receivers to the scatterer leading to the following expression \begin{equation} \mathbf {k}\! = \mathbf {k}_s\!+\mathbf {k}_r=(k_x, k_y, k_z) = k (\sin \phi \cos \psi , \sin \phi \sin \psi , \cos \phi ) \end{equation} (9)where \begin{equation} k = \frac{2\omega }{c}\cos \left(\frac{\theta }{2}\right) \end{equation} (10)and c is the local wave speed, ω the angular frequency, ϕ, ψ and θ are the dip, azimuth and scattering angles, respectively (Miller et al. 1987; Wu & Toksöz 1987; Lambaré et al. 2003). Figure 2. View largeDownload slide Radiation patterns for elastic isotropic media. (a) Schematic illustration of the relationship between the wavenumber components mapped in the subsurface at a given position with the angular illumination provided by the acquisition and the reflectors of the background subsurface model (here, the free surface). In a teleseismic setting, the forward scattering generated during the upgoing propagation of the incident wavefield towards the surface will contribute to preferentially map the low-to-intermediate horizontal wavenumbers, while the backward scattering generated after the reflection of the incident wavefield from the free surface will contribute to preferentially map the high-to-intermediate vertical wavenumbers. Radiation patterns of elastic scatterers for the (b) (ρ, λ, μ), (c) (ρ, Vp, Vs) and (d) (ρ, Ip, Is) parametrizations . From left to right, each column presents the interaction of a P (1,4,7), Sv (2,5,8) and Sh-wave (3,6,9) incident plane wave with, from top to bottom, a ρ (1–3), λ (4–6) (resp. Vp and Ip) and μ (7–9) (resp. Vs and Is) scatterer. All the analytical radiation patterns (Wu & Aki 1985) are superimposed with the scattered wavefields that are computed numerically. Both patterns are scaled with respect to their respective maximum scattered energy. The anisotropic behaviour of the amplitudes of numerical patterns translates a slight deviation from Born theory (the heterogeneity is too wide comparatively to the propagated wavelength). Arrows indicate the direction of the incident plane wave (black for P-wave, red for Sv wave and red-black for Sh wave). Radiation-pattern curves are plotted in blue, green and purple for the P, Sv and Sh scattered modes, respectively. See the text for the interpretation. Figure 2. View largeDownload slide Radiation patterns for elastic isotropic media. (a) Schematic illustration of the relationship between the wavenumber components mapped in the subsurface at a given position with the angular illumination provided by the acquisition and the reflectors of the background subsurface model (here, the free surface). In a teleseismic setting, the forward scattering generated during the upgoing propagation of the incident wavefield towards the surface will contribute to preferentially map the low-to-intermediate horizontal wavenumbers, while the backward scattering generated after the reflection of the incident wavefield from the free surface will contribute to preferentially map the high-to-intermediate vertical wavenumbers. Radiation patterns of elastic scatterers for the (b) (ρ, λ, μ), (c) (ρ, Vp, Vs) and (d) (ρ, Ip, Is) parametrizations . From left to right, each column presents the interaction of a P (1,4,7), Sv (2,5,8) and Sh-wave (3,6,9) incident plane wave with, from top to bottom, a ρ (1–3), λ (4–6) (resp. Vp and Ip) and μ (7–9) (resp. Vs and Is) scatterer. All the analytical radiation patterns (Wu & Aki 1985) are superimposed with the scattered wavefields that are computed numerically. Both patterns are scaled with respect to their respective maximum scattered energy. The anisotropic behaviour of the amplitudes of numerical patterns translates a slight deviation from Born theory (the heterogeneity is too wide comparatively to the propagated wavelength). Arrows indicate the direction of the incident plane wave (black for P-wave, red for Sv wave and red-black for Sh wave). Radiation-pattern curves are plotted in blue, green and purple for the P, Sv and Sh scattered modes, respectively. See the text for the interpretation. The expression of k shows that the local wavelength and the angles ϕ, ψ and θ are the four parameters that control the spatial resolution of the FWI. In teleseismic applications when a few up-going wavefields can lead to a coarse and narrow sampling of θ, the use of a broad frequency band is useful to guarantee a proper sampling of the wavenumbers in the spatial directions spanned by the dip ϕ and azimuth ψ angles. Moreover, free-surface multiples may enrich the range of dip and short scattering angles sampled by the acquisition, hence increasing the vertical resolution of the imaging. Beyond the resolution limitations imposed by the limited frequency bandwidth of the source and the limited spread of the acquisition, the radiation pattern of the virtual sources further controls the amplitude versus θ variations of the partial derivative wavefields and hence potentially narrow the real range of wavenumbers mapped in the FWI gradient. In the next section, we analyse the radiation patterns of the virtual sources for the three subsurface parametrizations that are used in this study for a theoretical assessment of their impact on the FWI resolution and the trade-off between parameters. 2.4 Radiation patterns for different subsurface parametrization Inferences on the sensitivity of the wavefield to different parameter classes can be drawn from the radiation patterns of elastic waves scattered by small elastic heterogeneities (Wu & Aki 1985; Tarantola 1986). Such patterns give clear insights on the influence of a single parameter onto the data for different scattering regime (forwards versus backwards) (Fig. 2a). Moreover, the overlap of the radiation patterns of different parameters with θ indicates potential trade-off between these parameters. On the one hand, one may want to mitigate these trade-offs by choosing a parametrization that minimizes these overlaps. On the other hand, this strategy might narrow the influence of the parameters with θ, hence degrading the resolution with which parameters are reconstructed accordingly. Many different parametrizations can be considered for FWI. However, in the case of isotropic elasticity, the most common ones combine ρ with either wave speeds (Vp, Vs), impedances (Ip, Is) or elastic moduli (λ, μ). The radiation patterns of these three parametrizations are shown in Figs 2(b)–(d) for a 2-D homogeneous medium. The presentation of the radiation patterns is complemented with multi-parameter FWI gradients computed for an incident P wave with an incidence angle of 45o and one station (Fig. 3). The background medium is a homogeneous half space to which is added a small circular inclusion at 50 km depth to generate the data residuals. Figure 3. View largeDownload slide Anatomy of a teleseismic gradient for one source–receiver pair and different parametrizations. FWI gradients computed for (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) (ρ, Ip, Is) parametrizations. The source is a 45o incident plane-wave travelling through an homogeneous half-space with a small heterogeneity located on the centre of the figure (white circle). The gradient is computed for a three-component sensor located at the position of the red triangle. The isochronal curves of the scattered-waves generated through the interaction of the upgoing P-wave and downgoing P and S-wave with the small heterogeneity are superimposed on the gradients. See the text for interpretation. Figure 3. View largeDownload slide Anatomy of a teleseismic gradient for one source–receiver pair and different parametrizations. FWI gradients computed for (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) (ρ, Ip, Is) parametrizations. The source is a 45o incident plane-wave travelling through an homogeneous half-space with a small heterogeneity located on the centre of the figure (white circle). The gradient is computed for a three-component sensor located at the position of the red triangle. The isochronal curves of the scattered-waves generated through the interaction of the upgoing P-wave and downgoing P and S-wave with the small heterogeneity are superimposed on the gradients. See the text for interpretation. A first noticeable feature, that is shared by the three considered parametrizations, is that λ (resp. Vp and Ip) generates only P–P scattering with an isotropic radiation pattern (Figs 2b–d,(4–6), blue curve). Therefore, the radiation pattern of λ (resp. Vp and Ip) has no impact on the resolution with which this parameter is reconstructed: the local resolution is only controlled by the local P wavelength and the range of P–P scattering angles spanned by the acquisition. The FWI gradient with respect to λ (resp. Vp and Ip) highlights the dominant contribution of the forward-scattered P–P mode delineated by the half-ellipse isochronal curve with one focus at the station (Figs 3b,e and h, blue curve). This forward-scattering regime is associated with wide θ and will contribute to update the long wavelengths of λ (resp. Vp and Ip) (Fig. 2a). Superimposed on the forward-scattered P–P sensitivity kernel, we also see, with much smaller amplitudes, the migration isochrone associated with the doubly P–P back-scattered wave from the free-surface and the inclusion (Figs 3b, e and h, red curve). This back-scattering regime is associated with smaller θ and hence will contribute to update shorter wavelengths of λ (resp. Vp and Ip) (Fig. 2a). Since the forward-scattering regime is dominant, the FWI will tend to update the long wavelengths first before updating the shorter ones from the back-scattered waves once the residuals of the incident wave will have been reduced. This hierarchy between forward and backward scattering naturally honours a multi-scale approach which is useful to mitigate the nonlinearity of the FWI. In contrast, μ (resp. Vs and Is) gives rise to P–P, P–S, S–S and S–P scattering, with stronger diffracted S waves relative to the P counterparts (Figs 2b–d, 7–9). The P–S, S–S and S–P radiation patterns are identical for the μ, Vs and Is and the union of these three radiation patterns covers a broad range of θ that is conducive to a broad-band reconstruction of these parameters if the imprint of all of these scattering modes is significant in the data. However, remembering that only incident P waves are considered in this study, the two lobes of the P–Sv radiation pattern pointing forward in the direction of the arrow with an angle of 45o in Figs 2(b)–(d) (7, green curves) show that the long wavelengths of μ (resp. Vs and Is) cannot be updated from the P–Sv mode since there is no radiation at very wide θ. This is supported by the gradient with respect to μ (resp. Vs and Is) in Figs 3c, f and i, dark green curve, which shows the more focused spatial domain spanned by the P–Sv forward scattering mode relative to the P–P one (Figs 3b, e and h, blue curve). Note also that the amplitudes of the P–Sv forward scattering mode are small compared to those of the doubly back-scattered modes in Figs 3(c), (f) and (i) due to the low values of the P–S transmission coefficient relative to those of the P–Sv reflection coefficient at the free surface (Pageot et al. 2013, their fig. 3). Interestingly, the only difference between the radiation patterns of μ, Vs and Is is that μ gives rise to two lobes of P–P scattering at wide and small θ (Fig. 2b,(7), blue curve), while Vs and Is give rise to P–P scattering at intermediate θ (Figs 2c and d,(7), blue curve). This difference manifests in Fig. 3(c), blue curve, by the clear imprint of the P–P forward-scattering mode in the μ gradient within a half-ellipse sensitivity kernel, unlike in the Vs and Is gradients. Owing that the long wavelengths of μ (resp. Vs and Is) cannot be updated from the P–Sv scattering mode as above-mentioned, the ability to update the long wavelengths of μ from the P–P mode might be an argument in favour of the (ρ, λ, μ) parametrization in particular if the inversion is started from a crude initial μ model. Potential trade-off between λ (resp. Vp and Ip) and μ (resp. Vs and Is) would result from the overlap between the P–P radiation patterns of μ (resp. Vs and Is) and λ (resp. Vp and Ip). These trade-offs might be more significant during the update of the long wavelengths of λ and μ because the P–P forward-scattering sensitivity kernel of these two parameters are very similar (Figs 3b and c). This comment highlights the fact that it is often challenging to update multiple parameter classes with a good resolution, while minimizing the trade-offs between them. The radiation patterns of ρ show significant differences for the three parametrizations, although ρ generates scattering for the P–P, P–S, S–S and S–P scattering modes in the three cases (Figs 2b–d,(1–3)). For (ρ, λ, μ), the radiation patterns are symmetric with respect to the incident plane wave front (Fig. 2b, (1–3)). In other words, forward and backward scattering are equally generated by ρ with this parametrization. Moreover, combining the contribution of the P and S scattered waves provides a sensitivity over the full range of scattering angles. Conversely, ρ scatters waves in a preferential direction with the two other parametrizations: backward for (ρ, Vp, Vs) (Fig. 2c,(1–3)) and forward for (ρ, Ip, Is) (Fig. 2d,(1–3)). This implies that (ρ, Ip, Is) is suitable to update the long to intermediate wavelengths of ρ, while (ρ, Vp, Vs) is more suitable to update the short to intermediate wavelengths. We might conclude that (ρ, λ, μ) might be the most suitable one to update ρ because the first-order P–P and P–S forward-scattering modes can yet generate models of good resolution, which can be further improved by the contribution of the doubly Sv–P and Sv–Sv back-scattered waves. Note also that the Sh–Sh radiation pattern is isotropic and hence can contribute to significantly increase the resolution of ρ. This theoretical analysis is supported by the gradients with respect to ρ (Figs 3a, d and g). For (ρ, λ, μ), we show well-balanced contributions of the forward- and backward-scattering regimes (Fig. 3a), while the broad P–P elliptical sensitivity kernel of the forward-scattering (delineated by the blue curve) is dominant in the (ρ, Ip, Is) parametrization (Fig. 3g). Conversely, the more spatially focused migration-like isochronal surfaces of the backward scattering are dominant for the (ρ, Vp, Vs) parametrization (Fig. 3d). Significant trade-offs are expected between λ (resp. Vp, Ip) and ρ and between μ (resp. Vs, Is) and ρ according to their respective radiation patterns. However, it is worth remembering that λ (resp. Vp, Ip) and μ (resp. Vs, Is) control both the kinematic and the dynamic attributes of seismic waves, while density mostly influences amplitudes. Heuristically, we may think that the long-wavelength reconstructions of λ (resp. Vp, Ip) and μ (resp. Vs, Is) will be primarily tied to the need to fit traveltimes and that amplitudes will play a second-order role in their update, hence limiting the imprint of these parameter trade-offs. Controlling the trade-off between the update of the small to intermediate wavelengths of ρ and λ (resp. Vp, Ip) or ρ and μ (resp. Vs, Is) from reflections at short to intermediate scattering angles is probably more challenging. A last comment concerns the presence of ghost isochronal surfaces in the gradients that have been focused off the position of the inclusion (Fig. 3). These isochrones result from the misinterpretation by the gradient of either (1) multiple-scattered waves or (2) different modes of converted waves generated by unmodelled heterogeneity. These artefacts can be partially cancelled out during the gradient computation by the destructive interferences of the contributions of several receivers and events. Moreover, the Gauss-Newton Hessian can further contribute to remove these artefacts by a deconvolution-like processing since the zero-lag correlation between the partial derivative wavefields will mimic the same undesired correlations as the zero-lag correlation between the partial derivative wavefields and the data residuals. Note that the artefacts related to multiple-scattered arrivals generated by the discontinuities of the background model should not be confused with those generated by multiple scattering from unmodelled heterogeneities. These multiple-scattered arrivals are present in the recorded data but are not modelled during the FWI gradient calculation, which relies on the single-scattering approximation. In this case, the artefacts generated by the recorded multiple-scattered arrivals can be removed from the FWI gradient by the second-order (nonlinear) term of the Hessian. 3 THE WESTERN ALPS LITHOSPHERIC MODEL 3.1 Model description Beller et al. (2017) applied FWI on nine teleseismic events recorded by the CIFALPS temporary array to image the lithospheric structure of the South-Western Alps (Fig. 4a). In the light of this study and for the purpose of the current study, we design a lithospheric model that is representative of the retrieved structure of the Alpine lithosphere. Figure 4. View largeDownload slide Synthetic example design. (a) The nine teleseismic events selected from the CIFALPS data set for FWI. (b) The 3-D Alps lithospheric model representative of a continental collision. Figure 4. View largeDownload slide Synthetic example design. (a) The nine teleseismic events selected from the CIFALPS data set for FWI. (b) The 3-D Alps lithospheric model representative of a continental collision. The synthetic model (Fig. 4b) represents the subduction of the European continental crust below the Adriatic plate in the Western Alps. This model is 400 km × 200 km long and 200 km deep. In its shallowest part, the model exhibits slow velocities/low density sedimentary basins, that are 10 km thick to mimic the presence of the SE-France and Po plain sedimentary basins (Fig. 4a, red). The upper European crust is indented by a piece of Adriatic lithospheric mantle (Fig. 4b, light blue) referred to as the Ivrea body mantle wedge. From the West to the East, the model displays two distinct lower crusts (Fig. 4a, green), the first dips towards the east up to a maximum depth of 100 km and characterizes the subduction of the European continental crust that under-thrusts the Ivrea body, whilst the other depicts the more conventional lower crust of the Adriatic plate. Beneath that, we designed a lithospheric mantle that gently varies from 100 to 125 km depth beneath the Ivrea body. The asthenospheric mantle located below also contains a piece of detached slab at the bottom end of the model (Fig. 4b). Overall velocities of the synthetic model are based on the IASP91 model (Kennett & Engdahl 1991) and densities are taken from the PEM-C model (Dziewonski et al. 1975). Seismic modelling in the lithospheric target is performed with a grid injection technique described in Beller et al. (2017): we first perform a global-earth simulation in the PREM model (Dziewonski & Anderson 1981) with the AxiSEM software (Nissen-Meyer et al. 2014) and store the 3-D teleseismic wavefield at the boundaries of the lithospheric target. Second, the teleseismic wavefield is propagated in the lithospheric target with the spectral element method (Komatitsch et al. 1998) through a grid injection technique (Monteiller et al. 2013). To avoid numerical artefacts during the grid injection, we force a smooth transition between our synthetic model and the outer PREM model with a cosine taper in a 20 km thick layer along the boundaries of the lithospheric target. For SEM modelling, the lithospheric target is discretised with an hexahedral mesh of 10 km wide elements. This leads to 2 000 000 degrees of freedom in the mesh. We perform SEM modelling in this lithospheric model using 9 computer nodes of 24 cores each. With this computational resources, one FWI gradient computations takes around 3 min of wall time. 3.2 Teleseismic data anatomy Since FWI is a data-fitting procedure, it is relevant to interpret the potential information content of the teleseismic wavefield and correlate ‘with one’s own eyes’ the geological features of the synthetic model with their seismic response. For that purpose, we show in Fig. 5 synthetic seismograms recorded by a dense profile of stations sampled 5 km apart for an event, the characteristics of which are given in Table 1. Figure 5. View largeDownload slide Synthetic data computed in the Alps lithospheric model. From left to right: radial, transverse and vertical components of particle velocities followed by radial and transverse receiver functions corresponding to the event 22 in Beller et al. (2017). E: early arrivals; L: late arrivals; R: reference arrivals; M: Moho reflection. Black ellipses indicate P-to-S wave conversions at upper-mantle reflectors (P410s and P660s), while the green ellipse highlights data aliasing of steep diffractions generated by the Ivrea body. Figure 5. View largeDownload slide Synthetic data computed in the Alps lithospheric model. From left to right: radial, transverse and vertical components of particle velocities followed by radial and transverse receiver functions corresponding to the event 22 in Beller et al. (2017). E: early arrivals; L: late arrivals; R: reference arrivals; M: Moho reflection. Black ellipses indicate P-to-S wave conversions at upper-mantle reflectors (P410s and P660s), while the green ellipse highlights data aliasing of steep diffractions generated by the Ivrea body. Table 1. Selected event for teleseismic waveform modelling indicating origin time, location and moment tensor solution from Global CMT project (Dziewonski et al. 1981; Ekström et al. 2012). Half duration (Hdur) and time shift (Tshift) are given in seconds. Id Event Lat (o) Dist (o) Hdur Depth Mw Lon (o) Baz (o) Tshift (km) CMT 22 2013-02-14 67.65 62.98 5.30 12 6.7 13:13:53.1 142.512 17.37 6.26 Id Event Lat (o) Dist (o) Hdur Depth Mw Lon (o) Baz (o) Tshift (km) CMT 22 2013-02-14 67.65 62.98 5.30 12 6.7 13:13:53.1 142.512 17.37 6.26 View Large The relatively shallow depth (12 km) of this event together with the dominant period of the source signature (≈5 s) prevents the identification of source-related multiples such as pP and sP phases, which are hidden within the coda of the direct P-wave arrival (their theoretical arrival times from the onset of the direct P arrival are 3.95 and 5.6 s, respectively). In contrast, the intermediate epicentral distance of the event (63o) allows the PcP phase to be well-separated from the incident P-wave within the displayed time window. This phase has a theoretical arrival time of 37.6 s, according to PREM and the TauP toolkit (Crotwell et al. 1999), and is characterized by a sub-planar move-out (Fig. 5c) . At first glance, owing to the low frequency content of the data, it is challenging to identify reflections from major lithospheric boundaries in the recorded data, which are dominated by the incident P wavefield and energetic diffraction from the Ivrea body. However, the footprint of several lithospheric structures, such as sedimentary basins and Ivrea body, is clearly visible in the onset time of the direct P arrival. In particular, the slow/buoyant sedimentary basins delay the traveltime of the direct P arrival and increase its amplitudes (Fig. 5, L), while their pinch-outs generate energetic diffractions with sharp dips (Fig. 5, Sed corners). In contrast, the fast Ivrea body advances the direct P arrival onset and decreases its amplitudes (Fig. 5, E). Moreover, it behaves as a strong secondary point source near the surface, that scatters waves downwards in the lithosphere. With respect to this comment, we interpret a late arrival with a hyperbolic moveout as the reflection from the Moho generated from this diffracting source (Fig. 5, M). The signature of the upper-mantle discontinuities corresponding to the top and bottom of the transition zone at 410 and 660 km depth are also visible at around 40 and 70 s after the first arrival (Fig. 5, P410s and P660s)). These discontinuities are not part of the lithospheric target and hence will not be imaged by FWI. However, their signature in the data will not impact upon the imaging since they will be injected in the modelled synthetic seismograms by the grid injection technique. The computation of receiver functions by frequency-domain deconvolution (Langston 1979) further helps us to analyse the contribution of late reflections from crustal heterogeneities by removing the imprint of the incident wavefield (Fig. 5d and e). The radial receiver function shows the geometry of the Moho of the European and Adriatic plate, with a pronounced dip for the former. On the transverse component, receiver functions are dominated by three diffraction hyperbola created by the pinch-out of the sedimentary basins and the Ivrea body. These arrivals are spatially aliased due to their significant dips. Later arrivals might indicate the location of the lithosphere–asthenosphere boundary (LAB). However, as is frequently the case in P-wave receiver function analysis, the signature of the LAB is hampered by the Moho free-surface multiples, which makes its identification challenging. 4 PARAMETRIC ANALYSIS OF TELESEISMIC FWI 4.1 Experimental setup For all the experiments involved in the parametric study, we perform FWI for the same nine events as those used for the CIFALPS experiment (Fig. 4a). We refer the reader to Beller et al. (2017, their table 1) for the precise location of the nine events used for FWI. We performed a multi-scale FWI (Bunks et al. 1995) by successive inversion of data sets of increasingly high-frequency content. We have generated three data sets by low-pass filtering the raw data with cut-off periods of 20, 10 and 5 s, respectively. For each frequency band, the optimization is stopped when: (1) the misfit reaches 1 per cent, (2) 47 iterations or the dedicated wall-time is reached and (3) when no suitable step-length is found according to Wolfe criteria (Nocedal & Wright 2006). We do not add noise to the data sets during the FWI experiments to minimise the role of the regularization and hence maximise the footprint of the experimental factors we want to assess (subsurface parametrization, initial model, acquisition geometry). Unlike the real CIFALPS data application, the temporal source signature is assumed to be known and hence is not estimated during FWI. We do not apply any time-dependent preconditioning (namely, time windowing or amplitude balancing) on the data, but we normalize each observed and computed common-event gathers by the maximal amplitude of the vertical component of the observed data to equally balance their contribution in the gradient of the misfit function. We use the l-BFGS optimization algorithm (Nocedal 1980), a quasi-Newton approach that recursively estimates the action of the inverse Hessian on the gradient, without other additional preconditioning on the gradient. The optimization parameters are the logarithm of the physical parameters to further balance the parameters in depth. A mild regularization is applied by smoothing the gradient with a Gaussian function with a correlation length of 2.4 km in all three Cartesian directions. In the following numerical examples, each experimental configuration deviates from the one we consider as the reference one in only one respect: subsurface parametrization, starting model or acquisition geometry. In the reference case, FWI is applied using: a (ρ, Vp, Vs) parametrization, a starting model that consists in a smoothed version of the true model (this model will be referred to as the Tomographic model), a 2-D regular grid of stations with a receiver-sampling interval of 5 km in the two horizontal Cartesian directions, as shown in the second line of Table 2. Table 2. Statistics of the FWI applications. Setup: experimental setup in terms of initial model (m0), optimization parameters (param) and acquisition geometry (acqui). Misfit function (5 s): initial/final data misfit function and data misfit function reduction in per cent for the last multi-scale step (cut-off period: 5 s). Misfit reduction: data misfit reduction in per cent for each frequency band (20 s, 10 s and 5 s). Model misfit: initial/final correlation-based model misfit and misfit reduction in per cent. Iterations: number of FWI iteration for each frequency band (20, 10, 5 s). Setup Misfit function (5 s) Misfit reduction ( per cent) Model misfit Iterations mo param acqui fo fend per cent 20 s 10 s 5 s mo-mtrue m-mtrue per cent 20 s 10 s 5 s Tomo (ρ, λ, μ) N05 7.57 × 103 4.00 × 102 5.28 1.88 7.87 12.86 3.89 × 107 2.73 × 107 70.33 44 47 45 Tomo (ρ, Vp, Vs) N05 7.57 × 103 8.95 × 102 11.82 1.51 29.38 23.31 3.89 × 107 2.51 × 107 64.76 42 14 36 Tomo (ρ, Ip, Is) N05 7.57 × 103 1.17 × 103 15.46 3.32 8.22 26.29 3.89 × 107 3.22 × 107 82.98 44 42 19 Smooth (ρ, Vp, Vs) N05 1.64 × 104 1.07 × 103 6.41 1.72 19.4 18.84 4.59 × 107 2.82 × 107 61.37 34 41 31 PREM (ρ, Vp, Vs) N05 1.61 × 104 1.01 × 103 6.27 1.56 19.04 21.58 4.66 × 107 3.17 × 107 68 35 11 40 Gradient (ρ, Vp, Vs) N05 5.84 × 104 1.73 × 103 2.96 1.58 15.08 18.44 1.33 × 108 5.76 × 107 43.15 35 39 37 Tomo (ρ, Vp, Vs) N10 2.04 × 103 2.39 × 102 11.72 1.7 18.12 16.04 3.89 × 107 2.90 × 107 74.7 36 44 29 Tomo (ρ, Vp, Vs) N25 4.00 × 102 1.68 × 102 42 1.16 14.97 75.34 3.89 × 107 2.92 × 107 75.24 38 14 2 Tomo (ρ, Vp, Vs) N50 9.13 × 101 3.79 × 101 41.51 2.16 12.14 48.97 3.89 × 107 4.33 × 107 111.49 35 13 4 Tomo (ρ, Vp, Vs) L05 3.21 × 102 2.58 × 102 67.07 2.36 24.16 100 3.89 × 107 4.87 × 107 125.19 44 11 0 Tomo (ρ, Vp, Vs) L10 1.64 × 102 1.10 × 102 80.37 2.35 7.8 93.22 3.89 × 107 4.80 × 107 123.39 44 42 2 Tomo (ρ, Vp, Vs) CIF 6.43 × 102 1.37 × 102 21.31 7.78 11.34 20.79 3.89 × 107 5.11 × 107 131.36 30 35 44 Setup Misfit function (5 s) Misfit reduction ( per cent) Model misfit Iterations mo param acqui fo fend per cent 20 s 10 s 5 s mo-mtrue m-mtrue per cent 20 s 10 s 5 s Tomo (ρ, λ, μ) N05 7.57 × 103 4.00 × 102 5.28 1.88 7.87 12.86 3.89 × 107 2.73 × 107 70.33 44 47 45 Tomo (ρ, Vp, Vs) N05 7.57 × 103 8.95 × 102 11.82 1.51 29.38 23.31 3.89 × 107 2.51 × 107 64.76 42 14 36 Tomo (ρ, Ip, Is) N05 7.57 × 103 1.17 × 103 15.46 3.32 8.22 26.29 3.89 × 107 3.22 × 107 82.98 44 42 19 Smooth (ρ, Vp, Vs) N05 1.64 × 104 1.07 × 103 6.41 1.72 19.4 18.84 4.59 × 107 2.82 × 107 61.37 34 41 31 PREM (ρ, Vp, Vs) N05 1.61 × 104 1.01 × 103 6.27 1.56 19.04 21.58 4.66 × 107 3.17 × 107 68 35 11 40 Gradient (ρ, Vp, Vs) N05 5.84 × 104 1.73 × 103 2.96 1.58 15.08 18.44 1.33 × 108 5.76 × 107 43.15 35 39 37 Tomo (ρ, Vp, Vs) N10 2.04 × 103 2.39 × 102 11.72 1.7 18.12 16.04 3.89 × 107 2.90 × 107 74.7 36 44 29 Tomo (ρ, Vp, Vs) N25 4.00 × 102 1.68 × 102 42 1.16 14.97 75.34 3.89 × 107 2.92 × 107 75.24 38 14 2 Tomo (ρ, Vp, Vs) N50 9.13 × 101 3.79 × 101 41.51 2.16 12.14 48.97 3.89 × 107 4.33 × 107 111.49 35 13 4 Tomo (ρ, Vp, Vs) L05 3.21 × 102 2.58 × 102 67.07 2.36 24.16 100 3.89 × 107 4.87 × 107 125.19 44 11 0 Tomo (ρ, Vp, Vs) L10 1.64 × 102 1.10 × 102 80.37 2.35 7.8 93.22 3.89 × 107 4.80 × 107 123.39 44 42 2 Tomo (ρ, Vp, Vs) CIF 6.43 × 102 1.37 × 102 21.31 7.78 11.34 20.79 3.89 × 107 5.11 × 107 131.36 30 35 44 View Large For a quantitative insight on the relevance of the FWI models, we computed a correlation-based misfit between a vertical section of the FWI models and the Alps lithospheric model, this vertical section being aligned with the CIFALPS profile (Beller et al. 2017, their fig. 3). Even though such a norm may not be the most suitable one (in the sense that it cannot discriminate the footprint of resolution and amplitude errors), it complements in a more quantitative way the qualitative visual assessment of the subsurface models. The statistics of the results of the parametric analysis are outlined in Table 2. 4.2 Which earth parametrization for teleseismic FWI? We first provide some insights on the most suitable subsurface parametrization for teleseismic FWI. The conclusions we draw in this section, do depend on the acquisition geometry considered for FWI applications. While they are arguably valuable for teleseismic application where few incident plane waves impinges the lithospheric target from beneath, these conclusions cannot be generalized for other kind of seismic acquisitions such as reflection ones for which the reader is referred to Tarantola (1986). We perform FWI for three different elastic parametrizations, (ρ, λ, μ), (ρ, Vp, Vs) and (ρ, Ip, Is), and we jointly update the three parameter classes involved in the chosen parametrization. For each of the three inversions, we show the parameters that have been updated by the inversion, referred to as optimization parameters, as well as the parameters of the two other parametrizations, these parameters being built by the a posteriori nonlinear recombination of the optimization parameters. These parameters will be referred to as visualization parameters. Three (ρ, λ, μ), (ρ, Vp, Vs) and (ρ, Ip, Is) models are shown in Figs 6(a)–(i), 7(a)–(i) and 8(a)–(i), respectively. For each of these figures, two multi-parameter models (visualization parameters) have been built a posteriori by the nonlinear recombination of the optimization parameters, while the third one directly shows the optimization parameters. On each figure (Figs 6a–i, 7a–i and 8a–i), the optimization parameters are (ρ, λ, μ), (ρ, Vp, Vs), (ρ, Ip, Is), from top to bottom. The presentation of the subsurface models are complemented by Fig. 9, whose histograms show the level of correlation between the reconstructed models generated by each FWI and the true subsurface models. Figure 6. View largeDownload slide Subsurface parametrization assessment. From left to right are shown the ρ, λ and μ models that are built a posteriori by the nonlinear recombination of the optimization parameters. These optimization parameters are (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) (ρ, Ip, Is). The first row directly shows the optimization parameters (ρ, λ, μ) accordingly. The last row (j–l) shows the true (ρ, λ, μ)) models for comparison. Figure 6. View largeDownload slide Subsurface parametrization assessment. From left to right are shown the ρ, λ and μ models that are built a posteriori by the nonlinear recombination of the optimization parameters. These optimization parameters are (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) (ρ, Ip, Is). The first row directly shows the optimization parameters (ρ, λ, μ) accordingly. The last row (j–l) shows the true (ρ, λ, μ)) models for comparison. Figure 7. View largeDownload slide Subsurface parametrization assessment. From left to right are shown the ρ, Vp and Vs models that are built a posteriori by nonlinear recombination of the optimization parameters. These optimization parameters are (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) (ρ, Ip, Is). The second row directly shows the optimization parameters (ρ, Vp, Vs) accordingly. The last row (j–l) shows the true (ρ, Vp, Vs) models for comparison. Figure 7. View largeDownload slide Subsurface parametrization assessment. From left to right are shown the ρ, Vp and Vs models that are built a posteriori by nonlinear recombination of the optimization parameters. These optimization parameters are (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) (ρ, Ip, Is). The second row directly shows the optimization parameters (ρ, Vp, Vs) accordingly. The last row (j–l) shows the true (ρ, Vp, Vs) models for comparison. Figure 8. View largeDownload slide Subsurface parametrization assessment. From left to right are shown the ρ, Ip and Is models that are built a posteriori by nonlinear recombination of the optimization parameters. These optimization parameters are (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) ρ, Ip, Is). The third row directly shows the optimization parameters (ρ, Ip, Is) accordingly. The last row (j–l) shows the true (ρ, Ip, Is) models for comparison. Figure 8. View largeDownload slide Subsurface parametrization assessment. From left to right are shown the ρ, Ip and Is models that are built a posteriori by nonlinear recombination of the optimization parameters. These optimization parameters are (a–c) (ρ, λ, μ), (d–f) (ρ, Vp, Vs) and (g–i) ρ, Ip, Is). The third row directly shows the optimization parameters (ρ, Ip, Is) accordingly. The last row (j–l) shows the true (ρ, Ip, Is) models for comparison. Figure 9. View largeDownload slide FWI model quality. The colours label the subsurface parametrization used for FWI or in other words the FWI optimization parameters. Initial refers to the initial models for each parameter class. The horizontal axis labels the visualization parameter that has been built a posteriori by nonlinear recombination of the optimization parameters. The vertical axis labels the correlation-based misfit function between the visualization and the true models. The three arrows point the quality of the optimization parameters μ, Vs and Is (see the text for explanations). Figure 9. View largeDownload slide FWI model quality. The colours label the subsurface parametrization used for FWI or in other words the FWI optimization parameters. Initial refers to the initial models for each parameter class. The horizontal axis labels the visualization parameter that has been built a posteriori by nonlinear recombination of the optimization parameters. The vertical axis labels the correlation-based misfit function between the visualization and the true models. The three arrows point the quality of the optimization parameters μ, Vs and Is (see the text for explanations). As a guideline for the visual assessment of the subsurface models, we focus on the reconstruction of the sedimentary basins, the Ivrea body and the subducting slab. According to these 3 criteria, we conclude that the best subsurface models for the kinematic parameters, namely, Vp, λ and Ip and Vs, μ and Is, are the Ip and Is (visualization) models inferred from the (ρ, λ, μ) optimization models (Fig. 8b and c). This qualitative assessment is supported by the metric outlined in Fig. 9, that confirms that these Ip and Is models show the best correlation with the true ones. The ρ model inferred from the (ρ, λ, μ) parametrization seems slightly better resolved than the one inferred from the (ρ, Vp, Vs) parametrization, at the expense of the noise level. In contrast, the ρ model inferred from the (ρ, Ip, Is) parametrization is much smoother, that is consistent with the former analysis of the radiation pattern. This visual assessment is supported by Fig. 9 in the sense that the ρ model inferred from the (ρ, Vp, Vs) parametrization shows the best correlation with the true model, while the one inferred from the (ρ, λ, μ) parametrization shows the poorest one, which might result from the noise impacting this model. However, this noise did not prevent a reliable recombination of the Ip and Is models from the (ρ, λ, μ) optimization parameters. One may wonder why the (ρ, λ, μ) parametrization provided the best subsurface models after recombination of Ip and Is. If we limit our comparative analysis to the optimization parameters, μ is the one which shows the best correlation with the true model among the kinematic parameter more closely related to shear waves (Vs, Is, μ), (Fig. 9, arrows). This might result from the key contribution of the forward-scattered P–P mode during the long-wavelength reconstruction of μ (Fig. 3c). A second reason may be related to the high-wavenumber content of the ρ model obtained with the (ρ, λ, μ) parametrization (Fig. 3a). When Is is inferred from ρ and μ by multiplication $$(I_s=\sqrt{\rho \mu })$$, it is likely that the high wavenumber content of ρ has complemented the small-to-intermediate wavenumber content of μ to focus a broadband Is model. This improved resolution of the visualization Ip and Is models relative to the λ and μ optimization models is quite visible in the reconstruction of the Ivrea body and subducting slab (compare Figs 8b and c with 6b and c). A similar line of though would explain why the sedimentary basins in the visualization Ip and Is models inferred from the (ρ, λ, μ) optimization parameters (Figs 8b and c) are better focused than those of the Ip and Is optimization models (Figs 8h and i). The lower high-wavenumber content of the optimization μ model, in particular at great depths (Fig. 6c, subducting slab), relative to the optimization Is and Vs models might result from the dominant imprint of the P–P forward scattering in the μ gradient relative to that of the back-scattering components. This might have dominantly driven the deep update of μ towards a low-to-intermediate wavenumber reconstruction at the expense of the high wavenumber components. Again, multiplication of μ with ρ might have contributed to build a posteriori an Is model, whose quality is higher than that of the optimization Is and Vs model. The same comment could apply to the Ip reconstruction from λ and μ. Another issue concerns potential cross-talk between ρ and the two kinematic parameters. The multiplication of ρ with a kinematic parameter to build the impedance might help to remove cross-talk as suggested by Prieux et al. (2013). A common practice in FWI studies to appraise the quality of output models consists in comparing FWI models vertical profiles to their respecting counterparts in the initial and targeted models. Such a quality assessment is presented in Fig. 10 for impedance models of the three above-mentioned parametrization. The localization of interest is x = 100 km. FWI clearly succeed in recovering the absolute values of Ip and Is models as well as those of the ρ model. Biases, appearing as vertical undulations, are attributed to limited-bandwidth effects resulting from the limited bandwidth of signals and the limited angular illumination or by trade-off effects with the wave speed. This is particularly true for the density that is characterized by a more selective radiation pattern than the S wave speed). Note that, whatever the choice of parametrization used during optimization, Is and Ip models (obtained directly by inversion or after recombination of the other optimization parameters) fairly well reproduce the amplitudes of the targeted Ip and Is structures aside from the previously mentioned filtering effects. Therefore, the P and S impedances might be robust parameters for structural interpretations. Figure 10. View largeDownload slide FWI vertical logs. From left to right are presented vertical profiles (at x = 100 km) of ρ, Ip and Is visualization parameters generated from FWI models obtained from the optimization of Lamé parameters (lam), seismic velocities (vel) and seismic impedances (imp). Their counterparts in the initial (ini) and targeted models (true) are also given for comparison. Figure 10. View largeDownload slide FWI vertical logs. From left to right are presented vertical profiles (at x = 100 km) of ρ, Ip and Is visualization parameters generated from FWI models obtained from the optimization of Lamé parameters (lam), seismic velocities (vel) and seismic impedances (imp). Their counterparts in the initial (ini) and targeted models (true) are also given for comparison. 4.3 Which initial model for teleseismic FWI ? FWI is a nonlinear and ill-posed inverse problem that is generally tackled with local optimization techniques. In this framework, a requirement for the optimization algorithm to converge towards a reliable subsurface model is that the initial model should be accurate enough to position the starting point within the attraction basin of the global minimum. Forwardly, this issue is closely related to the cycle skipping phenomenon that occurs in FWI whenever a seismic wiggle obtained from an inaccurate predictive model is shifted by more than half the period with respect to the corresponding observed wiggle (Virieux & Operto 2009). If cycle-skipping occurs, data residuals will be back-projected onto a wrong isochrone surface hence conducting FWI towards a local minimum. A conventional way of mitigating cycle skipping effects is to progressively increase the number of propagated wavelengths during seismic modelling, since the relative time error is inversely proportional to this number (Pratt 2008). A first data-driven strategy to achieve this goal relies on a frequency continuation approach where data sets of increasing high-frequency content are hierarchically processed (Bunks et al. 1995). This frequency continuation can be complemented by a time continuation that can be designed by gently increasing the selected time window over iterations. Both data continuation strategies contribute to implement a multi-scale imaging proceeding from the low wavenumber updates to the higher ones. The frequency continuation approach requires that the recorded data have a sufficiently low frequency content, which is not always the case in exploration seismology although the design of broadband technology has become an active field of research (e.g. Baeten et al. 2013). To overcome this issue, this community has focused its efforts on the upstream building of reliable initial model by traveltime and slope tomography and migration-based velocity analysis, the design of elaborated misfit functions with better convexity properties or by expanding the search space (e.g. Symes 2008; Lambaré 2008; van Leeuwen & Herrmann 2013; Métivier et al. 2016). In the case of teleseismic data, the cycle skipping issue should be prevented by the availability of long period (low frequency) data. Furthermore, reference 1-D global earth model such as PREM, IASP91 or AK135 are generally accurate enough to predict the arrivals of global seismic phases with a confidence interval in between one and two seconds therefore preventing the cycle skipping issue for representative teleseismic periods (20 s to 1 s). To verify the previous statement, we analyse the sensitivity of the FWI to four different starting models. A first initial model (Figs 11d–f) is built by Gaussian smoothing of the true Alps lithospheric model with a correlation length of 50 km in the three spatial directions. Despite being smooth, this model still captures some of the 3-D complexity of the target and can be considered as analogous to an initial model built by first-arrival traveltime tomography. In absence of such a knowledge about the lithospheric velocity and density structure, another option is to resort to available 1-D reference earth models. We investigate here two possible initial models based on the PREM reference model: the layered original one and a smoothed version obtained by Gaussian filtering with a 10 km correlation length (Figs 11g–i) and (Figs 11j–l). A last starting model is a simple vertical velocity gradient (Figs 11m–o). Figure 11. View largeDownload slide Starting models for FWI. From left to right ρ, vp and vs models for: (a–c) the input model used to generate synthetic data, (d–f) tomographic model computed by smoothing of the input model, (g–i) smoothed version of PREM model, (j–l) PREM model and (m–o) vertical gradient. Figure 11. View largeDownload slide Starting models for FWI. From left to right ρ, vp and vs models for: (a–c) the input model used to generate synthetic data, (d–f) tomographic model computed by smoothing of the input model, (g–i) smoothed version of PREM model, (j–l) PREM model and (m–o) vertical gradient. The FWI results for the different initial models are shown in Fig. 12. As expected, the best FWI model has been inferred from the initial tomography-like model (Figs 12d–f and Table 2). Using the smooth PREM model as an initial model leads to acceptable FWI results, that may be improved by a more aggressive regularization (Figs 12g–i and Table 2). The FWI Vp model is very similar to the one inferred from the tomography-like model, while the ρ and Vs models are slightly noisier. Overall, this result confirms that smoothed reference earth models are accurate enough for teleseismic FWI. When the original PREM model is used as initial model, the footprint of the sharp crustal layering of the initial model overlap the smoother structures updated by FWI (Figs 12j–l). Indeed, this high-wavenumber footprint cannot be removed or updated by FWI because its spectral content is beyond the resolution limits of FWI. Moreover, this prior high-wavenumber content in the initial model, which can be mispositioned in depth, can drive the inversion towards a local minimum by generating, during seismic modelling, transmitted and reflected waves that are badly positioned in time. Figure 12. View largeDownload slide Assessment of initial models. From left to right, ρ, Vp and Vs models. (a–c) True models. (d–o) FWI models inferred from (d–f) the smooth tomography-like model, (g–i) the smoothed version of the PREM model, (j–l) the original PREM model and (m–o) the vertical gradient model. Figure 12. View largeDownload slide Assessment of initial models. From left to right, ρ, Vp and Vs models. (a–c) True models. (d–o) FWI models inferred from (d–f) the smooth tomography-like model, (g–i) the smoothed version of the PREM model, (j–l) the original PREM model and (m–o) the vertical gradient model. Interestingly the vertical gradient model is accurate enough to produce acceptable velocity models (Figs 12m–o and Table 2). However, the ρ model shows saturated amplitudes suggesting a higher sensitivity of the inversion to this parameter. A possible reason is that, among all of the tested initial models, the vertical gradient model is the one which does not generate any wave partitioning except at the free surface, hence leading to significant amplitude residuals. In this case, ρ has been over-saturated, that is to say more important than for the other density models, to remove these amplitude residuals during the early iterations. This over-saturation of ρ might also have favoured some leakage with the velocity updates. This leakage from the velocities to the ρ updates is suggested by the less contrasted reconstruction of the Vp parameter in Fig. 12(n). In summary, a good initial model for teleseismic FWI should be smooth enough to avoid involving wavenumber components in the initial models that cannot be easily updated by FWI or which are beyond its resolution limit. On the other hand, our numerical experiments suggest that the initial model should be accurate enough to predict well enough amplitude phenomena in particular if second-order parameter such as density is jointly updated with the wave speeds. If this condition is not satisfied, the update of the density might become exaggeratedly sensitive to these amplitude residuals and might prevent the update of the high wavenumber components of the wave speeds in the upper part of the lithospheric model. 4.4 Sensitivity to acquisition design 4.4.1 Linear versus areal acquisitions High-resolution seismic imaging techniques such as receiver function migration or waveform tomography require dense array of seismic sensors to adequately sample the teleseismic wavefield as well as the imaging kernel. On the one hand, seismic experiments have been designed for high-resolution receiver function analysis or scattering migration with dense profiles of stations spaced 5 to 10 km apart. Cascadia, Hi-Climb, MASE, CIFALPS or Pyrope experiments are examples of such dense linear acquisitions. On the other hand, areal geometries designed with a coarse grid of stations (30 to 70 km interval) have been carried out to perform tomography at a larger scale. USArray, Iberarray or the incoming AlpArray are examples of such experiments, where the receiver grid is either fixed or moving. To the best of our knowledge, no synthetic experiment has been performed to assess which acquisition geometry is the most suitable one for teleseismic FWI at the lithospherical scale. To fill this gap, we performed FWI for seven station layouts, that have been designed based upon the two above-mentioned acquisition trends. Four areal acquisitions consist of 2-D receiver grids with a station-sampling interval of 5, 10, 25 and 50 km, leading to pools of 2001, 525, 105 and 21 stations, respectively (Figs 13d–o). Two acquisitions consist of one line of stations spaced 5 and 10 km apart (Figs 13p–u), leading to pools of 69 and 35 stations, respectively (Figs 13p–u). Finally, for sake of comparison with the real data case study presented in Beller et al. (2017), we also assess the CIFALPS acquisition, that combines a dense profile of stations spaced 5 km apart with a sparse irregular spread of permanent stations (Figs 13v–x). Figure 13. View largeDownload slide Sensitivity of FWI to the station layout. From left to right, ρ, Vp and Vs models. (a–c) True (ρ, Vp, Vs) models used to generate the recorded data; FWI models inferred from 2-D grids of station with inter-station spacings of 5 km (d–f) , 10 km (g–i) , 25 km (j–l) and 50 km (m–o). FWI models inferred from linear acquisition with inter-station spacings of 10 km (p–r) and 5 km (s–u). (v–x) FWI models inferred from the CIFALPS acquisition as used in Beller et al. (2017). Figure 13. View largeDownload slide Sensitivity of FWI to the station layout. From left to right, ρ, Vp and Vs models. (a–c) True (ρ, Vp, Vs) models used to generate the recorded data; FWI models inferred from 2-D grids of station with inter-station spacings of 5 km (d–f) , 10 km (g–i) , 25 km (j–l) and 50 km (m–o). FWI models inferred from linear acquisition with inter-station spacings of 10 km (p–r) and 5 km (s–u). (v–x) FWI models inferred from the CIFALPS acquisition as used in Beller et al. (2017). The FWI models for each experiment are shown in Fig. 13. As expected, the denser areal geometry gives the best resolved models for each of the three parameter classes. The coarsening from 5 to 10 km of the receiver grid does not significantly degrade the reconstruction of the synthetic model neither in terms of spatial resolution nor in terms of noise (Figs 13d–i). Furthermore, the 25 km acquisition still provides reliable reconstruction of the synthetic model even though its resolution is clearly more limited. Here, this poorer resolution results because the FWI stopped after two iterations during the processing of the last frequency band (see Table 2). For a 50 km acquisition, despite a quite low resolution, the main structural heterogeneities of the synthetic model are still recognizable. However, a strong acquisition footprint in the form of energetic velocity anomalies at the station positions every 50 km is quite visible in the P-wave velocity model (Fig. 13n). Overall, we show a progressive degradation of the spatial resolution as the receiver sampling becomes coarser. This results because the range of scattering and dip angles sampled by the source-receiver layout becomes narrower and sparser as the station interval increases. The narrowing of the scattering angle illumination will limit the resolution of the reconstruction perpendicularly to the dip, while the coarsening of the scattering-angle illumination can generate wraparound of the structures if narrow-frequency bands are processed. The scattering-angle illumination as a function of the dip angle (noted ϕ in the current study, eq. 9), is illustrated in Rondenay et al. (2005, their fig. 3) for a linear teleseismic acquisition and two different station samplings. Compared to areal geometries, linear acquisitions drastically affect the ability of FWI in recovering well-resolved lithospheric models. Although six of the nine earthquakes have backazimuths approximately aligned with the receiver line (Fig. 4a), the FWI models show a degraded resolution of the Vp and Vs models as well as incomplete reconstruction of density at great depths. This results because the linear deployment illuminates more incompletely (in terms of bandwidth and sampling) the scattering and the two dip angles than an areal geometry, even for the image points located in the vertical plane of the receiver line. As an illustrative example, any non-cylindrical structure cross-cutting the vertical plane of the profile such as the Ivrea body will generate out-of-plane scattering which cannot be recorded without an areal geometry. The limited number of off-line stations available in the real CIFALPS acquisition helps increasing the spatial resolution at the expense of the signal-to-noise ratio, which might have been degraded due to spatial aliasing (Figs 13v–x). It is worth remembering that, for sake of consistency between the different experiments, we did not apply significant regularization during FWI. When dealing with such sparse acquisition, more aggressive smoothing constraints along the spatial directions that are not sampled well by the acquisition geometry would improve the results as suggested by the results of Beller et al. (2017) on the CIFALPS real data. 4.4.2 Acquisition footprint and aliasing The noise in the recovered FWI model might be due to the conjunction of two factors. First, FWI models are contaminated by acquisition footprint in the form of high-amplitude perturbations at the station positions, this acquisition footprint being particularly obvious for the 50 km station grid. These localized perturbations result from the high amplitudes of the adjoint wavefields at the station positions and the arrival time of the incident planar wavefields at the station. Therefore, the product of these two wavefields generate some bright spots at the station positions if no efficient correction for these amplitude effects are applied. The Hessian, in particular its diagonal coefficients, is expected to correct for them. However, the poor approximation of the Hessian provided by l-BFGS during the first FWI iterations might have contributed to leave a significant imprint of this amplitude singularities in the FWI models. These high-amplitude perturbations can behave as strong diffractors that propagate noise in the FWI models over iterations. Second, aliasing effects may contaminate the recovered image at high-frequencies therefore adding noise in the reconstructed image. In the framework of Kirchhoff migration, Lumley et al. (1994) defined three kinds of aliasing: data, image and operator aliasing. Data aliasing occurs when the recorded wavefield is not properly sampled in one or more dimension according to the Nyquist theorem. Spatial aliasing can occur when the apparent wavelength of the incoming wavefield at the surface is not sampled adequately by the receiver grid. Even though the 5 s-period incident P wavefield satisfies the sampling criterion, sharp small-scale structures near the surface can behave as energetic diffractors that scatter energy along steep diffraction hyperbola in the time–distance domain. In this study, the edges of the Ivrea body and the pinch-out of the sedimentary basins generate such energetic diffractions that undergo aliasing even for a 5 km station-sampling interval (Fig. 5, green ellipse). An obvious means to mitigate data aliasing is to limit the inversion to low-frequency data at the expense of the spatial resolution of the imaging. Another strategy would consist in re-sampling the acquisition by interpolation of seismic traces (Wilson & Guitton 2007). Image aliasing occurs when the mesh, that parametrizes the subsurface target, is too coarse to properly sample the output of the imaging. In FWI, the maximum resolution is half a wavelength and the subsurface should be meshed accordingly, that is with at least four grid points per wavelengths in a finite-difference framework. In our case, we design the hexahedral mesh with at least five degrees of freedom per wavelength according to the above-mentioned requirement. Operator aliasing occurs when the integral operator that transforms the data in model updates is not properly sampled or, in other words, when the dip angle of the wavenumber vectors, eq. (9), are undersampled, leading to wraparound in space. This form of aliasing in the model space can occur even if the data are not aliased. In the framework of inverse scattering techniques, Rondenay et al. (2005, their Fig. 4) illustrated how the sampling of the stations impacts upon the sampling of the weighting function applied to the scattered data during their weighted stack along diffraction hyperbolas. This migration operator becomes increasingly smooth with depth, which means that operator aliasing is mostly generated at shallow depths. Undersampling this weighting function leads to wraparound artefacts in the form of high wavenumber ‘speckle’ in the image. Considering that the FWI gradient and the migration kernel obey the same imaging principle, it is likely that this kinds of aliasing has affected our FWI results. Other sources of noise can result from the poor fold of teleseismic acquisitions. The subsurface imaging in FWI relies on the Huygen’s principle: any continuous structure in the subsurface is imaged by the interference of the isochrone surfaces generated by back-projection of data residual samples. These interferences are constructive at the locus of the missing subsurface structures and destructive off this locus. If the stations sampling is too coarse, the destructive interferences off the locus of the missing heterogeneities is not efficient enough to cancel out the undesired contributions of the isochrones. Beyond the receiver sampling, the source distribution in teleseismic imaging can potentially generate aliasing noise and degrade spatial resolution. In a parametric study of 2.5-D frequency domain teleseismic FWI, Pageot et al. (2013) have shown the footprint of the source distribution and sampling on the quality of the imaging. Narrowing the range of incidence angles sampled by the events contribute to narrow the wavenumber coverage accordingly and hence, degrade the spatial resolution of the imaging. However, the source sampling for a fixed range of incidence angles has little effects on the imaging quality provided that the temporal frequency dimension is finely sampled. This results because a planar wavefield leads to a uniform spatial sampling of the subsurface at the expense of the angular illumination. More serious aliasing artefacts occurred when a 2-D lithospheric target is illuminated from only one side, leading to significant gaps in the dip angle coverage. In 3-D applications, this aliasing may be more prominent since two dip angles need to be sampled. 4.4.3 Resolution analysis We assess now more precisely the footprint of the receiver sampling on the spatial resolution of the FWI. We performed two spike tests, the aim of which is to image a grid of small spherical inclusions spaced 20 km apart. These inclusions are superimposed on the smooth tomographic lithospheric model, which is used as a starting model for FWI. The diameter of the anomalies is 20 km and the amplitude of the perturbations are ±100 m s−1 and ±100 kg m−3 for wave speeds and ρ, respectively, with an alternating sign from one inclusion to the next. We perform this resolution analysis for two station deployments: the first consists of a grid of stations sampled 25 km apart (Fig. 15), while the second one involves a line of stations spaced 5 km apart (Fig. 16). The number of stations involved in the two acquisitions is roughly equivalent. In Figs 15 and 16, we extract a vertical section from the FWI models along the receiver line of the linear acquisition and several horizontal slices at increasing depths. To assist the interpretation of these FWI results, we show the local wavenumber coverage, eqs (9) and (10), that is provided by the nine events and the two station layouts for a homogeneous background model, at four diffractor points located at (x[km], y[km], z[km]) = (0,0,10) (Fig. 14a), (0,0,50) (Fig. 14b),(0,50,10) (Fig. 14c) and (0,50,50) (Fig. 14d). Note that the (2π) factor in the expression of the wavenumber modulus, eq. (10), has not been considered to generate Fig. 14 for a more direct access to the sampled wavelength. The wavenumber coverage was computed considering the forward-scattered P–P and P–S modes and all the mode conversions of the doubly back-scattered waves from the free surface and the scatterers discretizing the homogeneous model. The P and S wave speeds in the homogeneous medium are 7 and 4 km s−1, respectively. These figures give some idea of the bandwidth of the wavenumber spectrum illuminated by the acquisitions as well as the sampling of this bandwidth, keeping in mind that the weighting performed by the radiation patterns and the reflection coefficients at the free-surface are not taken into account. As a guideline, a circle of radius 0.04 km−1 is superimposed on each wavenumber spectrum to assess the sampling of the dip angles for a fixed wavenumber modulus. This angular sampling is directly controlled by the station spacing. Any undersampling of these dip angles will manifest in the space domain by spatial wraparound. A wavenumber of 0.04 km−1 roughly corresponds to the upper bound of the spectrum of a Gaussian inclusion of correlation length 10 km. Therefore, a full reconstruction of the inclusions shown in Figs 15 and 16 would be achieved if the area covered by the red circle was filled up in Fig. 14. Figure 14. View largeDownload slide Wavenumber illumination provided by the linear (top panels) and areal (bottom panels) acquisitions for a scatterer located at (x[km], y[km], z[km]) = (a) (0,0,10), (b) (0,0,50), (c) (0,50,10) and (d) (0,50,50). The first two scatterers are aligned with the station profile of the linear acquisition, while the last two ones are out of plane. In (a–d), the left panels show the (kx, kz) spectrum, while the right panels show the (kx, ky) spectrum. The red circle is superimposed as a guideline to assess the sampling of the dip angle along this circle. Figure 14. View largeDownload slide Wavenumber illumination provided by the linear (top panels) and areal (bottom panels) acquisitions for a scatterer located at (x[km], y[km], z[km]) = (a) (0,0,10), (b) (0,0,50), (c) (0,50,10) and (d) (0,50,50). The first two scatterers are aligned with the station profile of the linear acquisition, while the last two ones are out of plane. In (a–d), the left panels show the (kx, kz) spectrum, while the right panels show the (kx, ky) spectrum. The red circle is superimposed as a guideline to assess the sampling of the dip angle along this circle. Figure 15. View largeDownload slide Resolution test for the areal acquisition. (a) Vertical section in the middle of the receiver grid. (b–f) Depth slices at 10, 50, 90, 130 and 170 km depth. From left to right, reconstructed ρ, Vp and Vs inclusions. The dash ellipse in (a) (density perturbations) highlights the inclusion distortion with alternated positive and negative vertical perturbations resulting from limited bandwidth effects (see the text for details). Figure 15. View largeDownload slide Resolution test for the areal acquisition. (a) Vertical section in the middle of the receiver grid. (b–f) Depth slices at 10, 50, 90, 130 and 170 km depth. From left to right, reconstructed ρ, Vp and Vs inclusions. The dash ellipse in (a) (density perturbations) highlights the inclusion distortion with alternated positive and negative vertical perturbations resulting from limited bandwidth effects (see the text for details). Figure 16. View largeDownload slide Resolution test for the linear acquisition. (a) Vertical section along the receiver line. (b–f) Depth slices at 10, 50, 90, 130 and 170 km depth. From left to right, reconstructed ρ, Vp and Vs inclusions. Figure 16. View largeDownload slide Resolution test for the linear acquisition. (a) Vertical section along the receiver line. (b–f) Depth slices at 10, 50, 90, 130 and 170 km depth. From left to right, reconstructed ρ, Vp and Vs inclusions. For the areal geometry, FWI succeeds in reconstructing the three parameters at shallow depths (Fig. 15). The resolution and the amplitudes of the Vp updates degrade with depth faster than those of Vs and ρ. This might result because all of the scattering modes contribute to the update of Vs and ρ unlike Vp hence leading to a higher fold during the Vs and ρ updates. Moreover, the P–S reflection coefficient at the free surface is higher than the P–P one (Pageot et al. 2013, their fig. 3), making the doubly P–S–S reflections from the lithospheric reflectors more energetic than the P–P–P counterparts. Increasing the fold is quite important when the imaging mostly rely on the contribution of second-order back-scattered waves which is likely the case at these depths. Moreover, the signature of the forward scattered wavefield diffracted from the deep inclusions might have been healed when arriving at the surface, hence contributing to degrade the Vp resolution at depth. We also show alternating positive and negative perturbations in the ρ vertical section at large depths (Fig. 15a, dash ellipse). This distortion of the inclusion shape highlights limited bandwidth effects, which result because ρ is mainly updated from back-scattered waves according to its radiation pattern. The range of scattering angles sampled by the acquisition decreases with depth hence narrowing the vertical wavenumber bandwidth accordingly. More surprisingly, the horizontal resolution is not impacted significantly by the limited backazimuthal coverage provided by the nine events (Figs 15b–f). In other words, the reconstruction of the inclusions in the horizontal planes shows only a moderate smearing along the y direction, which becomes more obvious below 130 km depth. This moderate smearing in the y direction is consistent with the elliptic shape of the (kx, ky) coverage provided by the areal acquisition with a major axis along the kx axis (Figs 14a–d-4). This good horizontal wavenumber coverage can be explained by the fact that spherical inclusions behave as point diffractors that scatter waves in all the directions. These scattered waves are recorded by the areal geometry along all of the horizontal directions hence providing a significant coverage of the horizontal components of the wavenumber vectors. Noise is also shown at shallow depths (Figs 15a and b) and decrease with depth. This noise has a higher frequency content on the Vs and ρ models than on the Vp model because S waves have a leading role in the Vs and ρ updates, unlike in the Vp one. This noise can result from the acquisition footprint from the receiver side and spatial aliasing due to dip angle undersampling. The decrease of the noise with depth is consistent with the fact that (kx, kz) bandwidth becomes wider towards low wavenumbers with depth as highlighted in Figs 14(a)–(d)-2. This contributes to make the (kx, kz) coverage more uniform with depth, and hence less prone to aliasing, in the area delineated by the red circle (Fig. 14a–d-2). For the line geometry, the reconstruction of the three parameters has been significantly degraded hence confirming the previous FWI results on the Alps model (Fig. 16). In the vertical section, only the shallow Vp, Vs and ρ inclusions are reconstructed at 10 km depth, while the inversion failed to reconstruct them at greater depths. Comparing these results with those obtained with the areal geometry highlights the key contribution of off-line stations to reconstruct the inclusions at great depths in the vertical plane defined by the central receiver line. Again, this results because the spherical inclusions behave as point diffractor that scatter waves in all spatial directions. Therefore, the wavefield scattered by the in-line inclusions can be recorded by the full station layout leading to a broad scattering and dip angle illumination and a high fold. The broader and more-uniform (kx, kz) coverage generated by the areal acquisition relative to the linear one supports this statement (Figs 14a–d (1 versus 2)). Note however that it is not guaranteed that more cylindrical geological structures would emphasise the same add-value of the areal geometry relative to the linear one. A second possible factor that explains the rapidly decreasing sensitivity in depth of the imaging is that the surface waves that are generated by conversion of the incident P wave onto Rayleigh waves near the surface might have a significant contribution in both the vertical and horizontal resolution. The wavelength of the Rayleigh wave at 0.1 Hz (30 km) is consistent with the depth resolution shown in Fig. 16(a). Concerning horizontal resolution, the depth slices at 10 km depth show quite noisy reconstructions of the ρ and Vs inclusions, while only the inclusions along the receiver line are reconstructed in the Vp slices. Compared to the results obtained with the areal geometry, the smearing along the cross-line (y) direction is more significant, which reflects a deficit of wavenumber coverage along this dimension, while the significant amount of noise in the ρ and Vs models is likely due to aliasing. The poor illumination of the ky components by the linear acquisition is well illustrated by the narrow (kx, ky) coverage in Figs 14(a)–(d-3). The failure of the VP reconstruction off the receiver line might result from the small amplitudes of the P–P waves that are back-scattered towards the receiver line (back-scattered waves are those which dominantly sample the horizontal components of the wavenumber vectors), while Vs and ρ reconstructions might have benefited from the various off-line back-scattering modes generated by the inclusions and the free surface. As for the vertical resolution, a second factor which can explain the contrasted horizontal resolution in the shallow updates of Vp compared to Vs and ρ is related to the contribution of Rayleigh waves, which are more sensitive to Vs perturbations. We remind that our synthetics tests were performed for noise-free synthetic data thus limiting our acquisition assessment to ideal cases where only the scattering, dip and azimuth angle illumination provided by the source-receiver geometry controls the resolution of the FWI images. In this setting, we have shown that, compared to a linear acquisition, an areal acquisition will contribute to improve the resolution of the imaging by sampling a broader wavenumber spectrum of the lithospheric target. For a real data application, noise will indeed impact upon this resolution. The impact of this noise can be mitigated by involving more redundant information in the imaging process either by injecting more close events in the inversion or by decreasing the station interval. The issue with the second option is that, for a given number of stations, a refinement of the station interval along a preferential direction (linear-type acquisition) will be performed at the expense of the angular illumination and hence the imaging resolution (Fig. 14). Moreover, depending on the geodynamical context (that will guide the orientation of the receiver line along the main structural dips) and the distribution of the teleseismic events, there might be a limited number of events reasonably aligned in the azimuth of the receiver line, hence further degrading the useful angular illumination. This is the reason why we would favour the first option, that is combining a coarse areal acquisition to cover a broad angular bandwidth with a large number of events to guarantee a redundant sampling of this bandwidth. In practice, moderate to strong earthquakes tend to be very localized on the Earth since they mostly occur along plate boundaries. Therefore, it is reasonable to assume that large catalogues of close events can be constituted to build redundant teleseismic data sets for FWI. In summary, combining coarse areal station deployments with large catalogues of redundant events is from our viewpoint the best strategy to conciliate the resolution and the signal-to-noise specifications. 5 CONCLUSIONS We have analysed several theoretical and experimental factors that have a significant influence on teleseismic FWI for lithospheric imaging. We have designed a realistic lithospheric model of the Western Alps and used nine events collected during the CIFALPS experiment to assess teleseismic FWI with a realistic experimental set-up. The first open question that has been addressed deals with the best subsurface parametrization for teleseismic FWI in the elastic isotropic approximation. We have concluded that (ρ, λ, μ) were the most suitable optimization parameters based on the analysis of the radiation patterns. Recombining a posteriori these three parameters into P and S impedances provide the lithospheric reconstruction that correlates best with the true model. During this recombination, sharp density reconstruction through the multiplication with the Lamé parameters, contribute to better focus some key structures such as shallow sedimentary basin, the Ivrea body and the subducting slab by injecting high wavenumber components. However, (ρ, Vs, Vs) or (ρ, Ip, Is) optimization parameters also provide reliable and close results without recombination. The conclusions drawn from this synthetic experiment will need to be validated against real data applications where the presence of noise might damage the reconstruction of the density and hence the quality of the Ip and Is built by recombination of the (ρ, λ, μ) optimization parameters. A second important conclusion is that 1-D global reference models such as PREM provide reliable initial models for teleseismic FWI. Such reference models need to be smoothed before being used for FWI in order to the imprint of the layering. The short wavelengths associated with this layering cannot be updated by FWI as they are be beyond the resolution limits of FWI and may drive the inversion to a local minimum if their geometries is inaccurate. The low frequency content of the teleseismic data makes the inversion almost immune to cycle skipping. However, too strong amplitude residuals generated by too simple initial models, such as vertical gradient models, can make the inversion to overupdate the density parameter at the expense of the wave-speed updates. Designing reliable acquisition for teleseismic FWI is a key issue since the best trade-off before station sampling and layout spread should be found for a given pool of stations. We have tested two main acquisition trends corresponding to areal geometries designed with coarse regular grids of stations and dense linear acquisitions. Our results clearly indicate that areal geometry should be designed to improve both the penetration in depth of the imaging and the horizontal resolution. The main reason is that, compared to a linear acquisition, an areal geometry will tremendously increase the fold and the horizontal and vertical wavenumber coverages with which a diffractor point will be imaged. This statement applies both to diffractor points located in the vertical section defined by the linear acquisition and out of this vertical section. Acknowledgements This study was partially funded by the SEISCOPE consortium (http://seiscope2.osug.fr), sponsored by CGG, CHEVRON, EXXON-MOBIL, JGI, SHELL, SINOPEC, STATOIL, TOTAL and WOODSIDE. This study was granted access to the HPC resources of SIGAMM infrastructure (http://crimson.oca.eu), hosted by Observatoire de la Côte d’Azur and which is supported by the Provence-Alpes Côte d’Azur region, and the HPC resources of CINES/IDRIS under the allocation 046091 made by GENCI. 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