# Main projects

Please see below some of our main research projects.

**Full waveform inversion and reverse-time migration**

Chauris, H., and Benjemaa, M., 2010, Seismic wave-equation demigration/migration, Geophysics, 75 (3), S111-S119, 2010.

Reverse-time migration is a well-known method based on single-scattering approximation and designed to obtain seismic images in the case of a complex sub-surface. It can however be a very time consuming task as the number of computations is directly proportional to the number of processed sources. In the context of velocity model building, iterative approaches require to derive a series of migrated sections for different velocity models. We propose to replace the summation over sources by a summation over depth-offsets or time delays defined in the sub-surface. For that, we provide a new relationship between two migrated sections obtained for two different velocity models: starting from one of the two images, we show how the second section can be correctly and efficiently obtained. In practice, for each time-delay, we compute a generalized source term by extending the concept of exploding reflector to non-zero offset. The final migrated section is obtained by solving the same wave equation in the perturbed model with the modified source term. We illustrate the methodology on 2D synthetic data sets, in particular when the initial and perturbed velocity models largely differ. |

*Left: velocity models (exact, initial, and 2 modified). Right: migrated section: initial, after demigration/migration and directly.*

Chauris, H., Noble, M., and Taillandier, C., 2008, What initial velocity model do we need for full waveform inversion?, Workshop 70th EAGE Conference and Technical Exhibition, Eur. Ass. of Geoscientists and Engineers (Rome)

In the context of velocity model building, we examine if the velocity models obtained after first-arrival traveltime tomography are accurate enough for subsequent full waveform inversion of reflected energy. For that purpose, we test the quality of the velocity model obtained by first-arrival traveltime tomography on the BP salt dome model. Several 1-D inversions are conducted in two different zones. In the simplest zone corresponding to smooth velocity models, the tomographic models are good enough for waveform inversion with realistic frequency contents. In the complex part going through a salt body, one need very low frequencies (starting at around 1 Hz) or a further refinement of the tomographic model. |

*1D full waveform inversion starting from the first-arrival travel time tomopgrahy result [Taillandier et al., 2009] for different frequency contents of the data. Dashed lines: exact model, dotted lines: initial model, and solid lines: inverted model.*

**Velocity estimation**

Taillandier, C., M. Noble, H. Chauris, and H. Calandra, First arrival travel time tomography based on the adjoint state methods, Geophysics,

**74**, (6), WCB57-WCB66, 2009.

Classical algorithms used for traveltime tomography are not necessarily well suited for handling very large seismic data sets or for taking advantage of current supercomputers. The classical approach of first-arrival traveltime tomography was revisited with the proposal of a simple gradient-based approach that avoids ray tracing and estimation of the Fréchet derivative matrix. The key point becomes the derivation of the gradient of the misfit function obtained by the adjointstate technique. The adjoint-state method is very attractive from a numerical point of view because the associated cost is equivalent to the solution of the forward-modeling problem, whatever the size of the input data and the number of unknown velocity parameters.An application on a 2Dsynthetic data set demonstrated the ability of the algorithm to image near-surface velocities with strong vertical and lateral variations and revealed the potential of the method. |

*(a): initial velocity model, (b): inverted velocity model after a few iterations, (c): final inverted velocity model, (d): exact velocity model.*

Chauris, H., and Lambaré, G., 2005, Seismic Velocity Analysis: in time or depth domain? IMA

Seismic velocity analysis is a crucial step needed to obtain consistent images of the sub-surface. Several new methods appeared in the last ten years, among them Slope Tomo-graphyand Differential Semblance Optimization. We discuss here the link between thesea prioridifferent methods. Slope Tomography is formulated in the prestackunmigrateddo-main and uses not only time information picked on seismic gathers, but also associatedslopes that better constrain the inversion scheme. On the other side, Differential Sem-blanceOptimization is formulated in the depth migrated domain where adjacent imagesare compared to obtain a final consistent image of the sub-surface. We analyze herethese two methods to show that they are in fact equivalent from a theoretical point of viewdespite the different formulation |

Chauris, H. and Noble, M., 2001, Two-dimensional velocity macro model estimation from seismic reflection by local Differential Semblance Optimization: applications to synthetic and real data sets, Geophysical Journal International,

**144**, 14-26

The quality of the migration/inversion in seismic reflection is directly related to the quality of the velocity macro model. We present here an extension of the differential semblance optimization method (DSO) for 2-D velocity field estimation. DSO evaluates via local measurements (horizontal derivatives) how that events in common-image gathers are. Its major advantage with respect to the usual cost functions used in reflection seismic inverse problems is that it is at least in the 1-D case unimodal and thus allows a local (gradient) optimization process. Extension of DSO to three dimensions in real cases involving a large number of inverted parameters thus appears much more feasible, because convergence might not require a random search process (global optimization). Our differential semblance function directly measures the quality of the common image gathers in the depth-migrated domain and does not involve de-migration. An example of inversion on a 2-D synthetic data set shows the ability of DSO to handle 2-D media with local optimization algorithms. The horizontal derivatives have to be carefully calculated for the inversion process. However, the computation of only a few commonimage gathers is sufficient for a stable inversion. As a Kirchhoff scheme is used for migration, this undersampling largely reduces the computational cost. Finally, we present an application to a real North Sea marine data set. We prove with this example that DSO can provide velocity models for typical 2-D acquisition that improve the quality of the final pre-stack depth images when compared to the quality of images migrated with a velocity model obtained by a classical NMO/DMO analysis. Whilst random noise is not a real difficulty for DSO, coherent noise, however, has to be carefully eliminated before or during inversion for the success of the velocity estimation. |

*Left: migrated section on North Sea data (DSO and classical approaches), Right: CIGs after velocity estimation by Differential Semblance Optimization.*

Chauris, H., Noble, M., Lambaré, G., and Podvin, P., 2002, Migration velocity analysis from locally coherent events in 2-D laterally heterogeneous media, Part II: applications on synthetic and real data, Geophysics,

**67**, 1213-1224

We demonstrate a method for estimating 2-D velocity models from synthetic and real seismic reflection data in the framework of migration velocity analysis(MVA).No assumption is required on the reflector geometry or on the unknown background velocity field, provided that the data only contain primary reflections/diffractions. In the prestack depth-migrated volume, locations where the reflectivity exhibits local coherency are automatically picked without interpretation in two panels:commonimage gathers (CIGs) and common offset gathers (COGs). They are characterized by both their positions and two slopes. The velocity is estimated by minimizing all slopes picked in the CIGs. We test the applicability of the method on a real data set, showing the possibility of an efficient inversion using (1) the migration of selected CIGs and COGs, (2) automatic picking on prior uncorrelated locally coherent events, (3) efficient computation of the gradient of the cost function via paraxial ray tracing from the picked events to the surface, and (4) a gradient-type optimization algorithm for convergence. |

*Left: migrated section on real data after 1D DSO, right: after 2D DSO.*

Chauris, H., Noble, M., Lambaré, G., and Podvin, P., 2002, Migration velocity analysis from locally coherent events in 2-D laterally heterogeneous media, Part I: theoretical aspects, Geophysics, **67**, 1202-1212

We present a new method based on migration velocity analysis (MVA) to estimate 2-D velocity models from seismic reflection data with no assumption on reflector geometry or the background velocity field. Classical approaches using picking on common image gathers (CIGs) must consider continuous events over the whole panel. This interpretive step may be difficult%GÃ¢Â€Â”%@ particularly for applications on real data sets. We propose to overcome the limiting factor by considering locally coherent events. A locally coherent event can be defined whenever the imaged reflectivity locally shows lateral coherency at some location in the image cube. In the prestack depth-migrated volume obtained for an a priori velocity model, locally coherent events are picked automatically, without interpretation, and are characterized by their positions and slopes (tangent to the event). Even a single locally coherent event has information on the unknown velocity model, carried by the value of the slope measured in the CIG. The velocity is estimated by minimizing these slopes. We first introduce the cost function and explain its physical meaning. The theoretical developments lead to two equivalent expressions of the cost function: one formulated in the depth-migrated domain on locally coherent events in CIGs and the other in the time domain.We thus establish direct links between different methods devoted to velocity estimation: migration velocity analysis using locally coherent events and slope tomography. We finally explain how to compute the gradient of the cost function using paraxial ray tracing to update the velocity model. Our method provides smooth, inverted velocity models consistent with Kirchhoff-type migration schemes and requires neither the introduction of interfaces nor the interpretation of continuous events. As for most automatic velocity analysis methods, careful preprocessing must be applied to remove coherent noise such as multiples. |

*Principle of Migration Velocity Analysis (left: common shot migration, right: common offset migration).*

**Curvelet and data processing**

Chauris, H., and T. Nguyen, Seismic demigration/migration in the curvelet domain, Geophysics, **73**, (2), S35-S46, 2008.

Curvelets can represent local plane waves. They efficiently decompose seismic images and possibly imaging operators. We study how curvelets are distorted after demigration followed by migration in a different velocity model.We show that for small local velocity perturbations, the demigration/ migration is reduced to a simple morphing of the initial curvelet. The derivation of the expected curvature of the curvelets shows that it is easier to sparsify the demigration/migration operator than the migration operator.An application on a 2D synthetic data set, generated in a smooth heterogeneous velocity model and with a complex reflectivity, demonstrates the usefulness of curvelets to predict what a migrated image would become in a locally different velocity model without the need for remigrating the full input data set. Curvelets are thus well suited to study the sensitivity of a prestack depthmigrated image with respect to the heterogeneous velocity model used for migration. |

*Common Image Gathers in the initial model (a), in the perturbed model (b) and after prediction in the curvelet domain (c). The (b) and (c) sections should be similar..*Sun, B., Ma, J., Chauris, H., and Huizhu, Y., 2009, Solving the wave equation in the curvelet domain: a multi-scale and multi-directional approach, Journal of Seismic Exploration,

**18**(4), 385-399

Seismic imaging is a key step in seismic exploration to retrieve the Earth properties from seismic measurements at the surface. One needs to properly model the response of the Earth by solving the wave equation. We present how curvelets can be used in that respect. Curvelets can be seen from the geophysical point of view as the representation of local plane waves. The unknown pressure, solution of the wave equation, is decomposed in the curvelet domain. We derive the new associated equation for the curvelet coefficients and show how to solve it. In this paper, we focus on a simple homogeneous model to illustrate the feasibility of the curvelet-based method. This is a first step towards the modeling in more complex models. In particular, we express the derivative of the wave field in the curvelet domain. The simulation results show that our algorithm can give a multi-scale and multi-directional view of the wave propagation. A potential application is to model the wave motion in some specific directions. We also discuss the current limitations of this approach, in particular the extension to more complex models. |

*Seismic wave propagation combined with a decomposition in the curvelet domain.*

Chauris, H., and Nguyen,T., 2008, Seismic velocity estimation in the curvelet domain, 70th EAGE Conference and Technical Exhibition, Eur. Ass. of Geoscientists and Engineers (Rome)

Curvelets can be seen from the geophysical point of view as the representation of local plane waves. They are known to efficiently decompose any seismic gathers and possibly imaging operators. We study here how curvelets can be useful for velocity estimation. In that context, we first show that the Differential Semblance Optimization technique has a very simple expression in the curvelet domain. We then derive the gradient of the cost function, still in the curvelet domain. An application on a 2-D synthetic data set, generated in a smooth heterogeneous model and with a complex reflectivity, demonstrates the usefulness of curvelets to derive how the velocity model can be improved to better focalize energy in the sub-surface after migration. |

*Initial model (top), after inversion (single iteration) in the curvelet domain (middle) and exact model (bottom).*

Chauris, H., and Nguyen,T., 2007, Towards interactive seismic imaging with curvelets, Workshop WO8 Curvelets, contourlets, seislets, ... in seismic data processing - where are we and where are we going?, 69th EAGE Conference and Technical Exhibition, Eur. Ass. of Geoscientists and Engineers (London)

We present a new methodology towards interactive seismic imaging. A first seismic image is obtained with a prestack depth migration code run in a reference velocity model. The objective is to efficiently derive what the migrated section would be in a perturbed velocity model. Instead of migrating the full input data in the second velocity model, the reference migrated section is first decomposed in the curvelet domain. This provides coefficients associated to the representation of local plane waves and described by two positions, a direction and a central frequency. Perturbing the velocity model consists of de-migrating the local plane waves by ray tracing and re-migrating them in the second model. In practice, this simply leads to the shift of coefficients in the curvelet domain. Only the coefficients affected by the velocity perturbations have to be considered. The final section is then obtained by applying the inverse curvelet transform. As an initial example, this methodology is used on a 2-D zero-offset section migrated in a heterogeneous model. It allows for better understanding the sensitivity of a seismic image with respect to the velocity model used for migration. |

Nguyen, T., and Chauris, H., 2010, The Uniform Discrete Curvelet Transform, accepted for publication to IEEE Transactions on Signal Processing

An implementation of the discrete curvelet transform is proposed in this work. The transform is based on and has the same order of complexity as the Fast Fourier Transform (FFT). The discrete curvelet functions are defined by a parameterized family of smooth windowed functions that satisfies two conditions: (i)$\bm{2}\pi$ periodic, (ii) their squares form a partition of unity. The transform is named the Uniform Discrete Curvelet Transform (UDCT) because the centers of the curvelet functions at each resolution are positioned on a uniform lattice. The forward and inverse transform form a tight frame, in the sense that they are the exact transpose of each other. Generalization to $M$ dimensional version of the UDCT is also presented. The novel discrete transform has several advantages over existing transforms, such as lower redundancy ratio, hierarchical data structure and ease of implementation. |

*3D curvelet with the Uniform Discrete Curvelet Transform, in the spatial (a) and frequency (b) domains.*

Donno, D., Chauris, H., and Noble, M., Curvelet-based multiple prediction, accepted for publication to Geophysics

The suppression of multiples is a crucial task when processing seismic re%GÃ¯Â¬Â‚%@ection data. We investigate how curvelets could be used for surface-related multiple prediction. From a geophysical point of view, a curvelet can be seen as the representation of a local plane wave, and is particularly well suited for seismic data decomposition. For the prediction of multiples in the curvelet domain, we propose to first decompose the input data into curvelet coefficients. These coefficients are then convolved together to predict the coefficients associated to multiples, and the final result is obtained by applying the inverse curvelet transform. The curvelet transform offers two advantages. The directional characteristic of curvelets allows to exploit Snel's law at the sea surface. Moreover, the possible aliasing in the predicted multiple can be better managed by using the curvelet multi-scale property to weight the prediction according to the low- frequency part of the data. 2D synthetic and field data examples show that some artifacts and aliasing effects can be indeed reduced in the multiple prediction with the use of curvelets. |

*Multiple prediction on a real data, with the classical SRME method (a and c) and with in the curvelet domain (b and d).*

Ma, J., Plonka, G., and Chauris, H., 2010, A new sparse representation of seismic data using adaptive easy-path wavelet transform, accepted for publication to Geoscience and Remote Sensing Letters

Sparse representation of seismic data is a crucial step for seismic forward modeling and seismic processing such as multiple separation, migration, imaging, and sparsity-promoting data recovery. In this paper, a new locally adaptive wavelet transform, called easy-path wavelet transform (EPWT), is applied for the sparse representation of seismic data. The EPWT is an adaptive geometric wavelet transform that works along a series of special pathways through the input data and exploits the local correlations of the data. The transform consists of two steps: reorganizing the data following the pathways according to the data values, and then applying a one-dimensional wavelet transform along the pathways. This leads to a very sparse wavelet representation. In comparison to conventional wavelets, the EPWT concentrates most energy of signals at smooth scales and needs less significant wavelet coefficients to represent signals. Numerical experiments show that the new method is really superior over the conventional wavelets and curvelets in terms of sparse representation and compression of seismic data. |

*(a) Original section, (b) gathers after selection of 1024 coefficients in the curvelet domain (b), wavelet domain (c) and Easy-Path Wavelet domain (d).*

Chauris, H., Karoui, I., Garreau, P., Wackernagel, H., Craneguy, P., Bertino, L., 2009, The circlet transform: a new tool for ocean eddy tracking, accepted for publication to Computers and Geosciences

We present a novel method for detecting circles on digital images. This transform is called the circlet transform and can be seen as an extension of classical 1D wavelets to 2D: each basic element is a circle convolved by a 1D oscillating function. In comparison with other circle-detector methods, mainly the Hough transform, the \emph{circlet} transform takes into account the finite frequency aspect of the data: a circular shape is not restricted to a circle but has a certain width. The transform operates directly on image gradient and does not need further binary segmentation. The implementation is efficient as it consists of a few Fast Fourier Transforms. The circlet transform is coupled with a soft-thresholding process and applied to a series of real images from different fields: ophthalmology, astronomy and oceanography. The results show the effectiveness of the method to deal with real images with blurry edges. |

*Sea Surface Temperature (top: original, bottom: gradient), with the associated main detected circlet elements.*