Geophysical tomography by viscoacoustic asymptotic ... - IEEE Xplore

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 5, OCTOBER 2001

Geophysical Tomography by Viscoacoustic Asymptotic Waveform Inversion of Ultrasonic Laboratory Data Alessandra Ribodetti, Henri-Pierre Valero, Stéphane Operto, Jean Virieux, and Dominique Gibert

Abstract—In this paper, we develop viscoacoustic asymptotic waveform inversion to estimate velocity and attenuation factor of a medium. Then, we present an application to laboratory ultrasonic data. Viscoacoustic asymptotic waveform inversion is performed using an optimization approach based on the iterative minimization of the mismatch between the seismic data and the computed response. To obtain a fast analytical imaging procedure, we include an asymptotic theory for attenuation in a linearized inverse scattering formulation. The forward modeling is solved by the Born approximation for a smooth and attenuative background medium. An asymptotic ray tracing method is used to calculate travel time, amplitude and attenuation between source, receiver, and scattering points. The inversion formula was specifically developed to account for the acquisition geometry designed in this study. Viscoacoustic asymptotic inversion is applied to ultrasonic data recorded during a physical-scaled laboratory experiment. This scaled experiment was used to test the reliability of our method when applied to a real dataset to estimate the attenuation factor . The results show that both the velocity and tomographic images allowed one to delineate the gross contour of the target. We obtained an excellent match between the observed data and the viscoacoustic Ray-Born synthetics. The match obtained with a viscoacoustic rheology was significantly better than for a purely acoustic one. The application presented in this paper suggests that the procedure that we designed (experimental setup, tomography) can be useful to estimate rock properties in the frame of the laboratory experiment. Index Terms—Attenution, ultrasonic data, viscoacoustic inversion.

I. INTRODUCTION

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N the last several years, the attenuative properties of the earth’s crust have become of increasing interest to seismologists. The interest ranges from simultaneous inversion of velocity and attenuation data to obtain improved earth models [1], to improve understanding of lithology, physical state, and the Manuscript received May 20, 1998; revised June 8, 2001. This work was supported in part by the European Commission and the Norwegian Research Council in the framework of the JOULE II Program (Project “Reservoir-Oriented Delineation Technology”). A. Ribodetti, S. Operto, and J. Virieux are with UMR Géosciences Azur, IRDCNRS-UNSA, 06235 Villefranche-sur-mer, France. H. P. Valero is with the Laboratoire de Géophysique Interne, 35042 Rennes, France, and also with Sonic Interpretation, Schlumberger K. K., Fuchinobe, Japan. D. Gibert is with the Laboratoire de Géophysique Interne, University of Rennes 1, 35042 Rennes, France. Publisher Item Identifier S 0018-9456(01)08098-6.

degree of saturation of subsurface rocks [2], and to the study of “bright spots” in hydrocarbon exploration [3]. Many workers have presented algorithms for the reconstruction of elastic parameters from seismic waveforms using different approximations and approaches [4]–[8]. However, the phenomenon of attenuation is much more complex than the elastic aspects of seismic wave propagation. Both in the field and in the laboratory, measurements are difficult to make. Some recent investigations concern the analysis of the full-waveform collected in field experiments for recovering attenuation properties (e.g., [9]–[18]). Laboratory measurements of attenuation in rock samples under varying pressures, temperatures, strain amplitudes, frequencies, and saturation conditions are presently being carried out [2]. We propose here an asymptotic method to estimate the compressional velocity and the attenuation factor using complete waveforms of data recorded during a physical-scaled laboratory ultrasonic experiment. This method is an extension of existing asymptotic inversion methods already developed for the reconstruction of elastic parameters [7] for acoustic parameters [8], [19], [20], for diffusive electromagnetic phenomena [21], and for viscoelastic parameters [22], [23]. Viscoacoustic asymptotic inversion is solved by iterative least-square minimization of the misfit between the observed and the computed traces using a quasi-Newtonian algorithm [7]. The forward problem is linearized with help from the Born approximation. High-frequency asymptotic ray theory is used to compute travel time, amplitude and attenuation in the medium for each source and receiver. Taking advantage of asymptotic theory, we developed a fast and flexible analytical imaging procedure which can account for various acquisition geometries including those with redundant or incomplete source-receiver coverage [7], [8], [19], [20]. To test our algorithm, we carried out a 2.5-D ultrasonic experiment in a modeling tank in Rennes. We simulated a fixed-offset experiment with a circular acquisition geometry providing a complete angular coverage of the target. The inversion formula was specifically developed to account for the acquisition geometry designed in this study. The target to be imaged is the section of a PVC cylinder set vertically in the water tank. The viscoacoustic tomography provided images of the target. We verified the both velocity and efficiency of the inversion by comparing the observed and the computed data. An excellent match between the two data sets was obtained after two iterations. The original motivation of this experiment was to test the reliability of the method when applied to real data to recover the attenuation factor

0018–9456/01$10.00 © 2001 IEEE

RIBODETTI et al.: VISCOACOUSTIC ASYMPTOTIC WAVEFORM INVERSION

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recorded during a physical-scaled experiment which allows a good control of the experimental parameters (acquisition geometry, properties of the background medium). The future prospect of this work is the possibility to compare experimental measurements of the velocity and attenuation in rock samples [24]–[26] with those obtained at the scale of in-field seismic reflection experiments.

tifies fixed-offset inversion with a circular acquisition geometry to decouple the velocity and the perturbations. and are the asymptotic Green functions computed in the background medium for the ray paths connecting the source to the scatter point and from the scatter point to the receiver , respectively. In the context of asymptotic Ray theory for viscoacoustic medium, it takes the form

II. THEORETICAL BASIS OF 2.5-D VISCOACOUSTIC WAVEFORM INVERSION

(4)

To recover the spatially dependent parameters of the viscoacoustic medium (velocity and attenuation factor ), we propose an extension of the Ray Born approach, as described in [7], [8], [20]. The theory is based on a combination of the Ray theory to compute the asymptotic Green functions and the Born approximation to linearize the relation between the scattered wavefield and a model of small pertubations. in a modeling It is well known that the introduction of scheme is considered to be equivalent to treating velocity as a complex quantity [27]. The complex velocity is defined in [2] as

where amplitude; travel time; source term related to the dimension of propagation; attenuation integrated along the ray path. In 2.5-D [20] (5) and are the velocity and the factor in the background and are the velocity and factor permodel, and turbations, respectively. In compact form, we note the forward problem (2) as

(1) (6) In contrast to the algorithms of [7], [8], [20], the inversion scheme is implemented in the frequency domain in order to integrate easily the attenuation into the inversion. A frequency-domain inversion allowed us to set up, on the one hand, an analytical kernel for the Born approximation of asymptotic anelastic solutions used for the forward problem and on the other hand, an approximate analytical kernel for the linearized inversion. From the Born approximation, we obtain a solution of the linearized forward problem for viscoacoustic medium

where

.

Following the same approach proposed by [7] for the elastic inversion and by [8] and [20] for the acoustic inversion, we extend here the method to the viscoacoustic inversion. Velocity and perturbations are obtained by iterative least-squares minimization of the weighted misfit between the observed and the computed wavefields using a quasi-Newtonian algorithm. The solution of the quasi-Newtonian inversion is given by (7)

(2) is over the diffracting points where the integration domain and is the vector of components

(3) Note that the terms of complex vector depend only on the rheology of the background medium but not on the source-receiver configuration. This differs from other multiparameter inversion schemes (i.e., acoustic [8], [20] or elastic inversion [7]. Indeed, is the radiafor multiparameter inversion, the analogue of tion diagram which depends on the source-receiver aperture. This dependence allows one to decouple the different parameters with the help of data redundancy. Thus, we state that the viscoacoustic inversion can be viewed as a single complex parameter inversion rather than a multiparameter inversion. This jus-

and are the observed and the computed where scattered wavefields, respectively, and the superscript denotes since the the adjoint operator. At first iteration, background model is homogeneous and the perturbations are zero. is a weighting of the cost function (see Appendix A). In , is the Hessian. The Hessian (7), the term to be inverted, is an operator mapping the model space to itself and thus it represents a huge matrix whose dimension equals the square of the number of scatterers in the image. Due to its huge dimension, it is difficult to invert it numerically. The asymptotic (local) approximation of the Hessian for the acoustic case is computed in Appendix A. The final reconstruction formula is given by

(8)

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target zone is discretized with a uniform mesh of nodes. Mesh spacing is m. IV. PRE-PROCESSING OF DATA AND BACKGROUND MEDIUM ANALYSIS

Fig. 1. Water tank, data acquisition systems, and PVC cylinder are presented. The horizontal plane at Z 0:67 m depth beneath the water level containing source and receiver hydrophone and the horizontal section of the PVC cylinder is plotted.

=

where (9) is the angle between the segment and the -axis. The derivation of (8) is given in Appendix A. The derivation is given in Appendix B. of the Jacobian III. EXPERIMENTAL SETUP The ultrasonic experiment in a water tank was carried out to simulate geophysical tomography. The water tank ( m m m) was equipped with computer-based control and data acquisition systems (National Instruments) [28]. 5000 l of water was used as a constant velocity background medium. We used one hydrophone for the source and another for the receiver. The source and receiver configuration is shown in Figs. 1 and 2. Source and receiver are located on a horizontal plane and rotate together around a fixed vertical axis. The offset between source and receiver is kept constant during the acquisition (fixed-offset acquisition) with an angle of 15 between the source and the receiver radius. The source and receiver hym depth beneath the water level (see drophones are at Fig. 1). The initial angle was fixed to 35 . The angular step between two consecutive source positionings was 5 , which results in 72 seismograms per common-offset gather. The dominant frequency of our signals is 100 kHz, corresponding to a wavelength of 0.015 m in water. Waveforms are s. digitized with sampling interval of The object to be imaged consists of a PVC cylinder set vertically in the tank (see Fig. 1). The radius of the PVC of cylinder is 0.058 m and the height is 1.30 m. The width the cylinder is 0.0015 m (see Fig. 2). Note that the axis of the cylinder does not coincide with the axis of the source-receiver is system. The center of the cylinder in the plane m m. Because of the acquisition (source and receiver lie in a plane) and target geometry (shape of the common cylinder is invariable along ), our experiment is 2.5-D. Thus, the target is a cross section of the cylinder located in the horizontal plane defined by the source and receiver positions (see Fig. 2). On the bottom of Fig. 2, we present a zoom of the target zone which will be considered during the inversion. The

In this section, we describe the acquisition steps, the pre-processing of the data and the background medium properties. To separate the scattered wavefield from the total wavefield, the acquisition is performed in two steps [29]. First, we put the object inside the water tank, scanning the target with the source and receiver around it, and measuring the total wavefield. After removing the object, we repeat the same scanning procedure in order to obtain the incident wavefield. The difference between these two data sets is the wavefield scattered by the object. This dual-experiment method also helps to minimize interferences from the experimental setup such as walls of the tank and the hydrophone itself. is displayed in Fig. 3(a). The The diffracted wavefield arrival labeled Fig. 3(a) is the reflection from the external edge of the cylinder. The arrival labeled Fig. 3(b) is the reflection from the second cylinder surface. The arrival labeled Fig. 3(c) is a multiple from inside the cylinder. The arrival labeled Fig. 3(d) is a reflection from the bottom of the water tank plus a reflection from the water level. Arrival labeled Fig. 3(e) is the free surface multiple of the scattered wavefield. Note that the arrivals Fig. 3(d) and (e) were not fully eliminated by the dual procedure [see Fig. 3(d)] because we did not take the change of water level into account when the cylinder with its prop was inserted in the water. In our experiment, the background model is composed of water. In the context of an ultrasonic experiment for which very short wavelengths are considered, an accurate estimation of water velocity is important. Indeed, the localization of the structures in the image is sensitive to the accuracy of the background model (i.e., accuracy of the water velocity estimation in our case). We used a velocity of 1520 m/s and a factor of 210 000 in the water corresponding to a temperature of 34 C and a frequency of 100 kHz [2], [30]. V. INVERSION RESULTS We performed our inversion between 25 kHz and 130 kHz. Two iterations were computed to obtain the final velocity and images. The recovered velocity and images are shown in Fig. 4 at iteration 1 and in Fig. 5 at iteration 2. The estimated center of the recovered object is at (0.09 m; 0.19 m) that corresponds to the center of the cylinder. The shape and dimensions of the recovered object are difficult to assess accurately because the source signature and the multiples contaminate the images. Nevertheless, the gross contours of the cylinder both on the velocity and the images are clearly identifiable in Fig. 4. Note that the effect of the iterations is to update the amplitude of the perturbations while the shape and the localization of the object do not vary over iterations. To verify the efficiency of the iterative inversion, we compared the observed data and the Ray-Born synthetics and displayed the residuals between the two sets of seismograms [see


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Fig. 2. Plot of the view of the section of the water tank, in the horizontal plane (X; Y ) perpendicular to the Z -axis of the PVC cylinder, at Z = 0:67 m depth beneath the water level, and containing the source-receiver system. On the top-right, a close-up of the data acquisition geometry during experiment. On the bottom, the horizontal section of the cylinder is displayed in the target region.

Fig. 3(b)–(d)]. A good fit between observed and predicted seismograms was obtained. This fit was significantly improved until the second iteration validating the efficiency of the iterative procedure. An inversion without attenuation (acoustic inversion) was also performed for comparison with the viscoacoustic inversion. In Fig. 6, trace 22 is superimposed for the observed data, the acoustic Ray-Born synthetics [20] and the viscoacoustic Ray-Born synthetics [using (2)]. A better match is obtained for the viscoacoustic seismograms than for the acoustic one. Indeed, the better fit obtained when accounting for attenuation

cannot prove that the viscoacoustic model is more rheologically significant than the acoustic one since the viscoacoustic rheology is parameterized by one more parameter than the acoustic one. Nevertheless, the excellent match between the observed and computed seismograms together with the result of the theoretical study of the Hessian operator which showed that the velocity and parameters are decoupled in the inversion make us confident in the validity of the viscoacoustic model. For a better understanding of the convergence process, the misfit function is quantitatively analyzed in the frequency domain where we are able to control its variation at low as well

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Fig. 3. (a) Scattered wavefield; 72 traces are recorded with fixed-offset geometry. (b) Ray-Born viscoacoustic synthetic scattered wavefield at iteration 2. (c) Residuals. This section contains mainly the multiples and the reflection from the free surface indicating a good match between observed and computed scattered wavefield. (d) Comparison between trace 22 extracted from the observed (dotted blue line) and the Ray–Born viscoacoustic synthetics (continuous red line). On the left, the computed trace was obtained at iteration 1 and on the right, at iteration 2. Note the improvement of the fit resulting from the iterative procedure.

as high frequencies during the iterations. In Fig. 7, for each selected frequency bandwidth, the -misfit is normal-

ized by the -misfit function obtained by summing over all frequencies (25–130 kHz). We calculated the variance of


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Fig. 4. (a) Recovered velocity section at iteration 1. (b) Recovered section at iteration 1. The shape of the cylinder is clearly identified on both sections although they are contaminated by the source signature and multiples.

Fig. 5. (a) Recovered velocity section at iteration 2. (b) Recovered at iteration 2.

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the error reduction and we represent a percentage defined as . Fig. 7 shows the misfit behavior as a function of the frequency bandwidth. The total bandwidth (25–130 kHz) was split into eight bands of 12 kHz each. At the first iteration, we obtain a normalized misfit close to 8% for the first bandwidth from 25 kHz to 37 kHz. The misfit reaches 28% for the second frequency interval and then it jumps nearly 30% for frequencies between 51 and 63 kHz. For this latter interval, we observe the greatest misfit value. This may be related to a resonance frequency (as corresponds to the cylinder radius). For the following band (64–76 kHz), we observe a sudden decrease of the misfit which is of the order of 20%. For the latter bands, the misfit keeps on decreasing sharply to reach values less than 1%.

Fig. 6. Trace 22 from the observed (green line), from the viscoacoustic-synthetics (blue line), and from the acoustic-synthetics (red line). Arrows indicate the improvement of the fit (phase and amplitude) between observed and synthetics when attenuation is taken into account.

VI. CONCLUSION In this paper, we have presented 2.5D viscoacoustic asymptotic iterative inversion and an application to fixed-offset ultrasonic experimental data. We have shown, with help from an analysis of the Hessian operator, that for the acquisition geometry designed in this study, the velocity and the attenuation factor were decoupled in the asymptotic inversion. perturbations Both tomographic images of velocity and allow us to clearly identify the object. Moreover, we obtained a very good fit between observed and viscoacoustic predicted synthetics which made us confident in the viscoacoustic model. The fit obtained in the viscoacoustic case was significantly better than the one obtained with an acoustic rheology.

Fig. 7. Percentage of the misfit function is displayed for each selected frequency bandwidth.

Future work will concern some improvements of the experimental setup in order to verify more rigorously the estimation of the absolute amplitude of the parameters and, then, to improve more efficiently the signal-to-noise ratio. This will require a more accurate control of the source-receiver positions together

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with a better a priori knowledge of the rheological properties of the experimental material (unknown in this study). The experimental setup and the processing developed in this study can be used as a tool for rock properties estimation [24]–[26]. Future work will also consist of post-processing of the images to remove the source signature from the image and thus, to recover the detailed shape of the object as well as the absolute estimation of the velocity and perturbations. APPENDIX A DERIVATION OF (7) The basis of the algorithm developed by [7], [8], and [20], is to choose a weight function such that a local approximation of the Hessian is diagonal and thus can be easily inverted. is given by The local approximation of the Hessian [20, p. 41, eq. (3.15)]

(A.1) for the viscoacoustic case

Fig. 8. Illustration of the geometrical relation between d and d when the source position is perturbed by dS .

We choose to

such that the kernel of the integral in (A.5) equals . We obtain for (A.6)

With this value of , the Hessian term is a function and thus its matrix is the identity matrix. Given that the Hessian matrix is the identity matrix, the percan be obtained by formulating the gradient tubations

(A.2) with (A.7)

(A.3) The angle is the angle between the -axis and the segment (see Fig. 8). is the angle step between two adjacent sources (Fig. 8). We would like to mention that the wavenumber (see Fig. 8). Note that the summation in (A.1) is performed over only which allows a full parametrization of the source and receiver positions. We mention, at this step, that the function in a two-dimensional (2-D) cartesian coordinate system is given by (A.4) denotes a domain of the 2-D space. where We apply a change of variables in the integrals of (A.1) from to to have the same variable of integration as in (A.4). We assume that the Jacobian of this change of variable never vanishes for any scatterer and any angle . Indeed, this assumption can be easily checked given the simple acquisition geometry and the fact that the background model is homogeneous. After applying the change of variable, we have

Inserting (A.6) in (A.7)

(A.8) .

where

, based on The derivation of the Jacobian simple geometrical calculations, is presented in Appendix B. APPENDIX B DERIVATION OF THE JACOBIAN In this Appendix, we derive the Jacobian to . variable from Given that

of the change of

(B.1) the components of

(see Fig. 8) are decomposed as (B.2)

(A.5)

where and

represent source and receiver, respectively.


We define the angles and as the angles made by and with the -axis, respectively, (see Fig. 8). is the length and . and are the distances of the segments between the source and the receiver and the scatterer , respectively, (see Fig. 8). For the source we have (B.3) We consider now a little perturbation of angles , , and the , and we define associated small perturbation of angles , angle as indicated in Fig. 8. We have the two relations (B.4) and combining these two relations, we have

(B.5) Similarly, we have for the receiver (B.6) is the angle between the segments and . where as well as the As the norms of and are known angles and (by geometrical construction), all the terms in (B.1) can be easily computed. ACKNOWLEDGMENT The authors thank G. Lambaré of Ecole des Mines, Paris, for helpful discussions. REFERENCES [1] M. Randall, “Attenuative dispersion and frequency shifts of the earth’s free oscillations,” Phys. Earth Planet. Int., vol. 2, pp. 1–4, 1976. [2] M. Toksöz and D. Johnston, “Seismic wave attenuation,” Soc. Expl. Geophys., 1981. [3] R. Sheriff, “Factors affecting seismic amplitudes,” Geophys. Prosp., vol. 23, pp. 125–138, 1975. [4] N. Bleistein, “On the imaging of reflectors in the earth,” Geophysics, vol. 52, pp. 931–942, 1987. [5] W. Beydoun and M. Mendes, “Elastic Ray-Born l2 migration/inversion,” Geophys. J. Int., vol. 97, pp. 151–160, 1989. [6] G. Beylkin and R. Burridge, “Linearized inverse scattering problems in acoustics and elasticity,” Wave Motion, vol. 12, pp. 15–52, 1990. [7] S. Jin, R. Madariaga, J. Virieux, and G. Lambaré, “Two-dimensional asymptotic iterative elastic inversion,” Geophys. J. Int., vol. 108, pp. 575–588, 1992. [8] G. Lambaré, J. Virieux, S. Jin, and R. Madariaga, “Iterative asymptotic inversion in the acoustic approximation,” Geophysics, vol. 57, pp. 1138–1154, 1992. [9] M. Dietrich and M. Bouchon, “Measurements of attenuation from vertical seismic profiles by iterative modeling,” Geophysics, vol. 50, pp. 931–949, 1985. [10] A. Tarantola, “Theoretical background for the inversion of seismic waveforms including elasticity and attenuation,” Pageoph, vol. 128, pp. 365–399, 1988. [11] M. Brzostowski and G. A. McMechan, “3-D tomographic imaging of near-surface seismic velocity and attenuation,” Geophysics, vol. 57, pp. 396–403, 1992.

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[12] W. S. D. Wilcock, S. C. Solomon, G. M. Purdy, and D. R. Toomey, “The seismic attenuation structure of a fast-spreading mid-ocean ridges,” Science, vol. 258, pp. 1470–1474, 1992. [13] J. Blanch, J. O. Robertsson, and W. W. Symes, “Modeling of a constant : Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique,” Geophysics, vol. 60, pp. 176–184, 1995. [14] Q. Liao and G. A. McMechan, “Multifrequency viscoacoustic modeling and inversion,” Geophysics, vol. 61, pp. 1371–1378, 1996. [15] D. Vasco, J. Peterson, and E. Majer, “A simultaneous inversion of seismic traveltimes and amplitudes for velocity and attenuation,” Geophysics, vol. 61, pp. 1738–1757, 1996. [16] M. Matheney and R. Nowack, “Seismic attribute inversion for velocity and attenuation structure using data from the glimpce lake superior experiment,” J. Geophys. Res., vol. 102, pp. 9949–9960, 1997. [17] A. Ribodetti and J. Virieux, “Asymptotic theory for imaging the attenuation factor ,” Geophysics, vol. 63, pp. 1767–1778, 1998. [18] E. Causse, R. Mittet, and B. Ursin, “Preconditioning of full-waveform inversion in viscoacoustic media,” Geophysics, vol. 64, pp. 130–145, 1999. [19] E. Forgues, “Inversion linearisée multiparamètres via la théorie des rais,” Ph.D. dissertation, Inst. Français du Pétrole—Inst. Phys. Globe du Paris, France, 1996. [20] P. Thierry, “Migration/inversion 3-D en profondeur à amplitude préservée: application aux données de sismique réflexion avant sommation,” Ph.D. dissertation, Univ. Paris VII, Paris, France, 1997. [21] J. Virieux, C. Flores-Luna, and D. Gibert, “Asymptotic theory for diffusive electromagnetic imaging,” Geophys. J. Int., vol. 119, pp. 857–868, 1994. [22] A. Ribodetti, J. Virieux, and S. Durand, “Asymptotic theory for viscoacoustic seismic imaging,” in 65th Annu. Int. Meeting Soc. Expl. Geophys., 1995, pp. 631–634. [23] A. Ribodetti, “Imagerie sismique haute résolution pour les milieux dissipatifs,” Ph.D. dissertation, Univ. Nice, France, 1998. [24] G. Saracco, A. Ribodetti, S. Turquety, and F. Conil, “Ultrasonic seismic imaging of lava samples by viscoacoustic asymptotic waveform inversion,” in Proc. Calibration and Developments, vol. 1, 2000, IMTC/00, pp. 380–385. [25] G. Saracco, S. Turquety, A. A. Ribodetti, and F. Conil, “Estimation of rheological parameters of lava samples by viscoaoustic asymptotic waveform inversion,” in Proc. EGS, vol. 74, 2000, pp. 81–81. [26] S. Turquety, “Development and Calibration of Acoustic Tomography Experiment to Identify the Velocity and Attenuation of Rock Samples: Application to Lava Samples,” Univ. Rennes 1, France, DEA Memo., 2000. [27] H. Chang and G. McMechan, “Numerical simulation of multiparameter seismic scattering,” Bull. Seis. Soc. Amer., vol. 86, pp. 1820–1829, 1996. [28] H. P. Valero, “Endoscopie sismique,” Ph.D. dissertation, Inst. Phys. Globe du Paris, Paris, France, 1997. [29] T. Lo, M. N. Toksöz, S. Xu, and R. S. Wu, “Ultrasonic laboratory tests of geophysical tomographic reconstruction,” Geophysics, vol. 53, pp. 947–956, 1988. [30] K. Fujii and R. Masui, “Accurate measurements of the sound velocity in pure water by combining a coherent phase-detection technique and a variable path-length interferometer,” J. Acoust. Soc. Amer., vol. 93, pp. 273–282, 1993.

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Alessandra Ribodetti received the M.Sc. degree in applied mathematics from Università degli Studi di Genova, Italy, in 1993, and the Ph.D. degree in applied geophysics from the Universitée de Nice-Sophia Antipolis, France, in the framework of the European JOULE Program “3-D Asympotic Seismic Imaging” in 1998. From 1998 to 2000, she was a Postdoctoral Fellow at the Ecole des Mines de Paris in the framework of the European Nonnuclear Energy Program: “Estimation of AVO Attributes After Preserved Amplitude Migration of Seismic Data.” In 2000, she became a Research Geophysicist at the Institut de la Recherche pour le Developpement, Villefranche-sur-mer, France. Her current research interests include theoretical seismology, seismic tomography, wave propagation in attenuating media, inverse theory, exploration seismology, and seismic data processing. Dr. Ribodetti is a member of AGU and EAGE. In 1995, she received an award for “the most outstanding contribution” by the Honors and Awards Committee of the Society of Exploration Geophysics, 65th Annual SEG Meeting, Houston, TX.

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Henri-Pierre Valero received the Ph.D. degree in geophysics from the Institut de Physique du Globe de Paris, France, in 1998, where he developed seismic endoscopy to obtain azimuthal images around the borehole.This subject required the development of an experimental probe with acquisition hardware. The algorithms and software related to the proposed method were also implemented. From 1995 to 1996, he was with the National Spacial Center at CNES, Toulouse, France, in the Geodesic Department under the direction of A.Cazenave. The study concerned the computation of a high-resolution geoid using ERS1 (orbit 35 days). In March 1999, he joined Schlumberger K.K. as an Engineer and now works in the Sonic Interpretation Department, Fuchinobe, Japan. His current interest is in acoustic wave propagation in boreholes and the application of signal processing (wavelet analysis, deconvolution methods) to the data. Dr. Valero is a member of SEG and SPWLA.

Stéphane Operto received the engineering degree from Ecole Nationale Supéerieure des Arts et Méetiers, France, in 1990, and the Ph.D. degree in marine geophysics from the University of Paris VI, France, 1995. In 1996, he was a Postdoctoral Fellow at the University of Texas at Austin and in 1997 he was a Research Geophysicist at the Ecole des Mines de Paris. In 1998, he joined CNRS, Villefranche-sur-mer, France, as a Research Geophysicist. His current interests are in seismic data modeling, tomography, and seismic exploration of continental margins.

Jean Virieux received the B.S. degree in 1972, the M.S. degree in physics in 1976 from the Ecole Normale Superiéure de Paris, Paris, France, and the Thése d’Etat from the University of Pierre and Marie Curie, France, in 1979. Currently, he is an Associate Professor at Institut de la Recherche pour le Developpement, Villefranche-sur-mer, France. His main interest is the modeling of wave propagation using asymptotic as well as numerical tehcniques. As a consequence, recovering the medium structure is a primary importance in his work. Extensions to nonlinear processes as well as to attenuation or diffusion are subjects in which he is interested. Dr. Vilrieux is a member of AGU, SEG, and SSA.

Dominique Gibert received the Ph.D. degree in geophysics and the Habilitation Diriger des Recherches from the Institut de Physique du Globe du Paris, France, mainly in the field of geomagnetism and geodynamics. Currently, he is Professor of Geophysics at the University of Rennes 1, Rennes, France. From 1984 to 1988, he was a Research Scientist with Ifremer, where he was in charge of the satellite altimetry program. From 1989 to 1991, he initiated the geophysical laboratory of the Ecole Nationale Suprieure des Mines, Paris, where, together with geophysicist P. Podvin, he developed several research programs in the field of geophysical imaging. His current research interests concern geophysical imaging and geomagnetism.