Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES NANCY Unit´e de recherche INRIA Rennes, Iris[r]

32 Read more

and scattering in addition to intrinsic damping (Sheriff 1975). It is important that these latter effects are accounted **for** in order to obtain the true intrinsic attenuation.
Different techniques are used in laboratory and field experiments to study the attenuation of acoustic waves propagating through rocks. Toks¨oz & Johnston (1981) mainly focused on laboratory measurements of field samples. **Methods** generally used to measure attenuation in the laboratory may be classified into the following categories: (i) free vibration, (ii) forced vibration, (iii) wave propagation and (iv) observation of stress–strain curves (see Toks¨oz & Johnston 1981, **for** a review). Laboratory wave propagation techniques **for** estimation of sample attenuation within the lower ultrasonic frequency range are of particular interest since these techniques can be extended **for** use with data from field experiments. A migration/**inversion** method adapted to acquisition of multichannel seismic reflections was developed **for** 2-D and 3-D acoustic and 2-D elastic media (Jin et al. 1992; Lambar´e et al. 1992; Forgues 1996; Thierry et al. 1999a,b). Extensions of the method to the viscoacoustic and viscoelastic cases were developed by Ribodetti et al. (1995) and Ribodetti & Virieux (1998) to retrieve the attenuation factor in addition to velocities and density. The viscoacoustic method was finally adapted to laboratory experiments **for** the characterization of rock properties (Ribodetti et al. 2000).

Show more
20 Read more

METHOD
L. M´ ETIVIER ∗ , R. BROSSIER ∗ , J. VIRIEUX ∗ , AND S. OPERTO †
Abstract. Full **Waveform** **Inversion** (FWI) is a powerful method **for** reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing a distance between predicted and recorded data. The predicted data is computed as the solution of a wave propagation problem. Conventional numerical **methods** **for** the resolution of FWI problems are gradient-based **methods**, such as the preconditioned steepest-descent, or more recently the l-BFGS quasi-Newton algorithm. In this study, we investigate the interest of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction. The truncated Newton method allows one to better account **for** this operator. This method is based on the computation of the Newton descent direction by solving the corresponding linear system through an iterative procedure such as the conjugate gradient method. The large-scale nature of FWI problems requires however to carefully implement this method to avoid prohibitive computational costs. First, this requires to work in a matrix-free formalism, and the capability of computing efficiently Hessian-vector products. To this purpose, we propose general second-order adjoint state formulas. Second, special attention must be payed to define the stopping criterion **for** the inner linear iterations associated with the computation of the Newton descent direction. We propose several possibilities and establish a theoretical link between the Steihaug-Toint method, based on trust-regions, and the Eisenstat stopping criterion, designed **for** method globalized by linesearch. We investigate the application of the truncated Newton method to two test cases: the first is a standard test case in seismic imaging based on the Marmousi II model. The second one is inspired by a near- surface imaging problem **for** the reconstruction of high velocity structures. In the latter case, we demonstrate that the presence of large amplitude multi-scattered waves prevents standard **methods** from converging while the truncated Newton method provides more reliable results.

Show more
43 Read more

Denoising as the simplest inverse problem (Section 2.1) has contributed to enormous progress in developing sophisticated adap- tive and non-adaptive priors **for** complicated signal recovery from noisy signals (Milanfar 2012). Some recently proposed excellent denoising **methods** include nonlocal means filters (Milanfar 2012; Goyal et al. 2020) and block matching and 3D filtering (BM3D) (Dabov et al. 2007) and its variants (Goyal et al. 2020). These patch-based **methods** use both local and nonlocal redundancy of information in the input signal to preserve structures in the solution by yielding locally adaptive filters via similarity kernels. Specify- ing the kernel function in these **methods** is essentially equivalent to estimating a particular type of empirical prior from the input sig- nal (Milanfar 2012). This somehow contrasts with the traditional non-adaptive regularization **methods**, **for** which the prior is fixed and independent from the input signal (Tarantola 2005). Such an adaptive regularization has been applied to linear inverse problems in, e.g., Danielyan et al. (2011) and Venkatakrishnan et al. (2013). We refer the reader to Appendix A **for** a more detailed review of the BM3D method that will be used in this study.

Show more
13 Read more

4.1 Introduction
The characterisation of the near surface (the first hundreds of meters) is essential **for** improving seismic imaging of both shallow and deeper exploration targets. Conven- tional seismic characterisation is done by analysing body waves. **For** example, first-arrival traveltime tomography is used to reconstruct the long wavelength velocity model ( Tail- landier et al. , 2009 ). Moreover, near-surface characterisation based on common-depth- point (CDP) reflection profiling requires ultra high-frequency seismic data acquisition (a few hundred Hz) ( Knapp and Steeples , 1986 ). In these imaging **methods**, surface waves are considered to be coherent noise that should be eliminated from the seismograms to enhance body waves. However, surface waves commonly represent more than half the seismic energy recorded in shot gathers and carry useful information. Surface waves are dispersive in heterogeneous media ( Thomson , 1950 ). Such property can be used to retrieve model parameters and characterise the near surface ( Nazarian and Stokoe II , 1984 ; Park et al. , 1999 ). Our objective is to use surface waves **for** reconstructing 2D high-resolution near-surface velocity models. We propose a surface-wave **inversion** approach based on a combination of the properties of two classical techniques: Surface Wave Analysis (SWA) and Full **Waveform** **Inversion** (FWI).

Show more
226 Read more

Abstract
Ground penetrating radar (GPR) is an efficient method **for** soil moisture mapping at the field scale, bridging the scale gap between small-scale invasive sensors and large-scale remote sensing instruments. Nevertheless, commonly-used GPR approaches **for** soil moisture characterization suffer from several limitations and the determination of the uncertain- ties in GPR soil moisture sensing has been poorly addressed. Herein, we used an advanced proximal GPR method based on full-**waveform** **inversion** of ultra-wideband radar data **for** mapping soil mois- ture and uncertainties in the soil moisture maps were evaluated by three different **methods**. First, GPR- derived soil moisture uncertainties were computed from the GPR data **inversion**, according to measure- ments and modeling errors and to the sensitivity of the electromagnetic model to soil moisture. Sec- ond, the reproducibility of the soil moisture map- ping was evaluated. Third, GPR-derived soil mois- ture was compared with ground-truth measurements (soil core sampling). The proposed GPR method ap- peared to be highly precise and accurate, with spa- tially averaged GPR **inversion** uncertainty of 0.0039 m 3 m −3 , a repetition uncertainty of 0.0169 m 3 m −3

Show more
18 Read more

Received 6 May 2004; revised 26 July 2004; accepted 17 August 2004; published 23 September 2004.
[ 1 ] Classical active seismic **methods** fail to sharply image
the earth’s deep crust. We present the first crustal-scale application of 2-D full **waveform** **inversion** based on dense ocean bottom seismic data to investigate the Eastern Nankai subduction system (Japan). This approach allows to quantify seismic velocities up to an unprecedented degree of resolution. Results reveal compressive tectonic features within both the subducting oceanic crust and the backstop. At depth, velocity anomalies along major faults and structural discontinuities bring evidence **for** the presence of fluids and weakened material and also **for** a possible co- seismic slip partitioning structure. I NDEX T ERMS : 0902 Exploration Geophysics: Computational **methods**, seismic; 3025 Marine Geology and Geophysics: Marine seismics (0935); 8010 Structural Geology: Fractures and faults; 8105 Tectonophysics: Continental margins and sedimentary basins (1212). Citation: Dessa, J.-X., S. Operto, S. Kodaira, A. Nakanishi, G. Pascal, J. Virieux, and Y. Kaneda (2004), Multiscale seismic imaging of the eastern Nankai trough by full **waveform** **inversion**, Geophys. Res. Lett., 31, L18606, doi:10.1029/2004GL020453.

Show more
content. Thus, tracers of different geophysical properties, which change (a) only electrical conductivity (e.g., salt [3]), and, (b) both electrical conductivity and permittivity (e.g., heat [4], ethanol [5]) are promising **for** GPR techniques.
In this regard, this abstract shows first a synthetic ethanol tracer test monitored by GPR FWI. As first step in the methodology, the synthetic tracer test is simulated and monitored by time-lapse crosshole GPR FWI, mimicking an experiment in typical aquifer conditions using a realistic aquifer model of the Krauthausen test site in Germany [6, 7]. Scenarios of different tracer types and magnitude of geophysical parameter changes are investigated. Thereby, different FWI starting models (SM) and two time-lapse FWI strategies are investigated to estimate the limitations of the techniques. The gained knowledge is used to perform real time-lapse GPR field measurements **for** several tracer tests. Field results, using **for** example heat as tracer (conducted at Krauthausen alluvial aquifer. site description in [6]), are preliminary interpreted in time-lapse by ray-based **inversion** and crosshole zero-offset (ZOP) attenuation analysis. In the next step, FWI will be applied on the GPR data with perspective whether it improves transport imaging resolution compared to other geophysical **methods**.

Show more
I. I NTRODUCTION
F ULL **waveform** **inversion** (FWI) seeks to estimate consti- tutive parameters by nonlinear minimization of a distance between recorded and simulated wavefield measurements. This technology was originally developed in geophysical imaging [1], and has spread more recently into other fields of imaging sciences such as medical imaging [2] and oceanography [3]. This partial-differential equation (PDE)-constrained nonlinear inverse problem is classically solved with local reduced-space optimization **methods** [4]. In this linearized framework, a challenging source of non linearity is the so-called cycle skipping pathology which occurs when the initial model does not allow to match the data with a kinematic error smaller than half a period [5], [6]. Other sources of error are noise,

Show more
14 Read more

The short wavelengths provide the ine structure of the subsurface model, allowing to lo- calize relectors in depth. The short wavelengths can be obtained by migration techniques, assuming the background velocity is correct. There are two major categories of migration **methods**: ray-based **methods** [ Beylkin, 1985 , Bleistein, 1987 ], which are based on the high frequency assumption, and wave-equation based **methods** [ Baysal et al., 1983 , Whitmore et al., 1983 ]. [ Etgen et al., 2009 ] gives a comparison of diferent migration **methods**. The principle of migration is formulated by [ Claerbout, 1971 ], and it consists of propagating the source signal and the recorded data into to the medium and cross correlate these two wave- ields. The zero-lag cross correlation gives the locations of relectors. There exist other imaging conditions, such as deconvolution-based imaging condition [ Valenciano et al., 2003 ], source/receiver-normalized imaging condition [ Kaelin et al., 2006 ], extended imag- ing condition [ Sava and Fomel, 2006 ]. [ Chattopadhyay and McMechan, 2008 ] and [ Sava and Hill, 2009 ] give a summary of the imaging conditions. The classical correlation-based migration is qualitative, as it only provides a relectivity image. Alternately, quantitative migration [ Lambaré et al., 1992 , Jin et al., 1992 , Lameloise et al., 2015 , Symes, 2015 ] al- lows imaging the values of the physical parameters. Recent developments have shown that **for** migration-based velocity analysis, quantitative migration is preferable as it provides a more accurate migration image.

Show more
173 Read more

The main reason **for** the limitation in the applicability of FWI is related to what is usually referred to as cycle skipping, or phase ambiguity. In standard FWI, the oscillatory seismic data is matched in the least-squares sense where each observed sample is compared to the synthetic sample at the same position in time and/or in space. This choice is problematic: if the initial model predicts the signal with a shift larger than half a period, minimizing the least-squares distance between observed and calculated data amounts to match the observed data up to one or several phase shifts. This yields an incorrect estimation of the subsurface model which cannot be overcome through iterations: the optimization is locked into a local minimum. An illustration of this phenomenon, where the seismic data is considered schematically as a sinusoidal temporal signal, is presented in Figure 1. Overcoming this difficulty has been a recurrent objective since the introduction of FWI by Lailly (1983) and Tarantola (1984). Increasing the accuracy of the initial model through high resolution tomography **methods**, as well as designing hierarchical workflows focusing first on low frequency components of the data, early-arrivals, and/or short offsets, have been initial strategies proposed to challenge this issue (Kolb et al., 1986; Bunks et al., 1995; Pratt, 1999; Shipp and Singh, 2002; Sirgue and Pratt, 2004; Wang and Rao, 2009). They are still the ones implemented **for** real data applications to guarantee the success of FWI. This careful tuning is case-dependent, therefore, it reduces the flexibility of FWI, and requires an expert usage of FWI and pre-processing tools.

Show more
13 Read more

Key words: Born and Rytov formulations, diffraction tomography, finite difference **methods**, medium wavenumber illumination, seismic imaging, **waveform** **inversion**.
1 I N T R O D U C T I O N
Quantitative imaging using full wave equation has been achieved through the use of the adjoint formulation problem **for** seismic data in the last 20 yr. Both formulations in time domain (Lailly 1984; Tarantola 1984; Gauthier et al. 1986) and in frequency domain (Pratt et al. 1996; Pratt 1999; Ravaut et al. 2004) have been implemented and applied to various synthetic and real data examples with specific advantages on both sides. Easier seismic traces processing in time domain will allow progressive introduction of phases by increasing the time domain window in both observed and synthetic data (Kolb et al. 1986; Shipp & Singh 2002; Sheng 2004). Efficient ways of solving the forward problem in the frequency domain make the frequency formulation appealing (Stekl & Pratt 1998). Moreover, the progressive introduction of higher frequencies allows both to introduce and mitigate the non-linearity and recover shorter and shorter heterogeneities (Pratt 1999; Sirgue 2003). Furthermore, **for** wide-angle data acquisitions, this frequency approach efficiently takes benefit of the wavenumber redundancy by limiting the number of inverted frequencies (Pratt 1990; Sirgue & Pratt 2004). The attenuation may be introduced, which has been applied to real data examples (Hicks & Pratt 2001).

Show more
30 Read more

Key words: Born and Rytov formulations, diffraction tomography, finite difference **methods**, medium wavenumber illumination, seismic imaging, **waveform** **inversion**.
1 I N T R O D U C T I O N
Quantitative imaging using full wave equation has been achieved through the use of the adjoint formulation problem **for** seismic data in the last 20 yr. Both formulations in time domain (Lailly 1984; Tarantola 1984; Gauthier et al. 1986) and in frequency domain (Pratt et al. 1996; Pratt 1999; Ravaut et al. 2004) have been implemented and applied to various synthetic and real data examples with specific advantages on both sides. Easier seismic traces processing in time domain will allow progressive introduction of phases by increasing the time domain window in both observed and synthetic data (Kolb et al. 1986; Shipp & Singh 2002; Sheng 2004). Efficient ways of solving the forward problem in the frequency domain make the frequency formulation appealing (Stekl & Pratt 1998). Moreover, the progressive introduction of higher frequencies allows both to introduce and mitigate the non-linearity and recover shorter and shorter heterogeneities (Pratt 1999; Sirgue 2003). Furthermore, **for** wide-angle data acquisitions, this frequency approach efficiently takes benefit of the wavenumber redundancy by limiting the number of inverted frequencies (Pratt 1990; Sirgue & Pratt 2004). The attenuation may be introduced, which has been applied to real data examples (Hicks & Pratt 2001).

Show more
30 Read more

form inversions of time-lapse seismic data. The conventional approach **for** analysis using **waveform** tomography is to take the difference of the images obtained using baseline and sub- sequent time-lapse datasets that are inverted independently. By contrast, double-difference **waveform** **inversion** uses time- lapse seismic datasets to jointly invert **for** reservoir changes. We apply conventional and double difference **methods** to a field time-lapse walkaway VSP data set acquired in 2008 and 2009 **for** monitoring CO 2 injection at an enhanced oil recov-

1 Introduction
At the eld scale, evaluating the soil water content spatial variability is an important issue **for** many research and engineering applications [1]. **For** in- stance, in catchment hydrology, as the soil surface water content determines the partitioning of precipitation into run-o and inltration under specic weather conditions, disregarding the spatial variability of the soil water content can lead to erroneous predictions in eld run-o and, further, in discharge estimation of the whole catchment [2]. Usual soil water content measurement techniques at the eld scale are invasive **methods**, like gravimetric sampling or time domain reectometry (TDR). Although the TDR technology has been automated to some extent, the method remains problematic **for** mapping large areas due to the local measuring support of the TDR probe [3]. On the other hand, airborne and spaceborne remote sensing **methods** have been proven to be eective tools **for** estimating soil surface water content over larger areas, with either passive microwave radiometry or active radar instruments [4]. However, major limi- tations with current remote sensing techniques are the unknown within-pixel heterogeneity and the usually resulting poor agreement with calibrating and gravimetric sampling [59]. Hence, no absolute relation between the backscat- tered signals from synthetic aperture radar (SAR) and the soil water content exist, necessitating site-specic calibrations [10]. In particular, remote sensing radar systems are highly aected by soil roughness, due to the relatively high frequencies used in SAR systems, such that many studies have also addressed that problem [11]. Radar sensing is also aected by high apparent electrical conductivity values when not taken into account [12].

Show more
27 Read more

Imaging remote objects in the deep Earth, such as, subducting slabs, mantle plumes, or large low shear velocity provinces and ultra low velocity zones is key **for** understanding Earth’s structure and the geodynamical processes involved as it cools. In order to image these structures, we developed a strategy **for** performing regional-scale full-**waveform** inversions at arbitrary location inside the Earth [1]. Our approach is to confine wave propagation computations inside the region to be imaged. This local wavefield modeling is used in combination with wavefield extrapolation techniques in order to obtain synthetic seismograms at the surface of the Earth [2]. This allows us to evaluate a misfit functional and sensitivity kernels can then be computed locally using the adjoint state method [3]. The Green’s functions needed **for** extrapolating the wavefield are computed once **for** all in a 3D reference Earth model using the spectral element software Specfem/3DGLOBE. We will present benchmark tests demonstrating that the proposed method allows us to image 3D localized structures - this without having to model wave propagation in the entire Earth at each iteration, which is prohibitively costly, thus improving the feasibility of accurate imaging of regional structures anywhere in the Earth using numerical **methods**. We will show that our method permits to account **for** additional data in regional inversions, that is to account **for** distant earthquakes that are located outside the region of the study - preliminary results **for** the tomography of the north American continent will be presented.

Show more
Motivated by the recent growth of high performance computing HPC, we will try to tackle the high non-linearity of the problem to minimize, using global optimization methods which are eas[r]

Darse, B.P. 48, 06235 Villefranche sur Mer CEDEX, France E-mail: ludovic.metivier@ujf-grenoble.fr
Abstract. Full **Waveform** **Inversion** (FWI) is a promising seismic imaging method. It aims at computing quantitative estimates of the subsurface parameters (bulk wave velocity, shear wave velocity, rock density) from local measurements of the seismic wavefield. Based on a particular wave propagation engine **for** wavefield estimation, it consists in minimizing iteratively the distance between the predicted wavefield at the receivers and the recorded data. This amounts to solving a strongly nonlinear large scale inverse problem. This minimization is generally performed using gradient-based **methods**. We investigate the possibility of applying the truncated Newton (TrN) method to this problem. This is done through the development of general second-order adjoint state formulas that yield an efficient algorithm to compute Hessian- vector products, and the design of an adaptive stopping criterion **for** the inner conjugate gradient (CG) iterations. Numerical results demonstrate the interest of using the TrN method when multi-scattered waves dominate the recorded data.

Show more
Keywords: Domain decomposition method, Schwarz **waveform** relaxation algorithm, multilevel preconditioning, nonlinear Schr¨ odinger equation, dynamics, stationary states
1. Introduction
This paper is devoted to the derivation of a multilevel Schwarz **Waveform** Relaxation (SWR) method **for** computing both in real- and imaginary-time the solution to the NonLinear Schr¨ odinger Equation (NLSE) [4, 5, 6, 10, 11]. Domain decomposition SWR **methods** **for** solving wave equations have a long history from the classical SWR method with overlapping zones to optimized version without overlap (see e.g. [7, 9, 12, 15, 16, 17, 18, 19, 22, 8] as well as http://www.ddm.org, **for** a complete review and references about this method). Basically in SWR **methods**, the transmission conditions at the subdomain interfaces are derived from the solution to the corresponding wave equation, usually using Dirichlet boundary conditions (Classical SWR), Robin boundary conditions, transparent or high-order Absorbing Boundary Conditions (ABCs) including Dirichlet-to-Neumann (DtN) transmitting conditions (Optimized SWR), or Perfectly Matched Layers [1, 9, 21]. We also refer to [1, 2, 20, 23] **for** some reviews on truncation techniques **for** quantum wave equations in infinite domains. SWR **methods** can be a priori applied to any type of wave equation [13, 14, 15].

Show more
17 Read more

with inverses given respectively by · ⊗
A α and · ⊗ B α.
Example 1 A basic example is A = C(V) and B = C 0 (T ∗ V) where V is a closed
smooth manifold ([ 21 , 8 ], see also [ 13 ] **for** a description of the Dirac element in terms of groupoids). This **duality** allows to recover that the usual quantification and principal symbol maps are mutually inverse isomorphisms in K -theory:

40 Read more