A non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness and continuity constraints is proposed. The applied formalism explores curvelets as a directional frame that, by their sparsity on the image, and their invariance under the imaging operators, allows for a stable recovery of the amplitudes. Our method is based on the estimation of the normal operator in the form of an ’eigenvalue’ decomposition with curvelets as the ’eigenvectors’. Subsequently, we propose an inversion method that derives from estimation of the normal operator and is formulated as a convex optimization problem. Sparsity in the curvelet domain as well as continuity along the reflectors in the image domain are promoted as part of this optimization. Our method is tested with a reverse-time ’wave-equation’ migration code simulating the acoustic wave equation.

In this paper, we recover the amplitude of a seismic image by approximating the normal (demigrationmigration)operator. In this approximation, we make use of the property that curvelets remain invariant under the action of the normal operator. We propose a seismic amplitude recovery method that employs an eigenvalue like decomposition for the normal operator using curvelets as eigen-vectors. Subsequently, we propose an approximate non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness in the curvelet domain and spatial continuity constraints. Our method is tested with a reverse-time ’wave-equation’ migration code simulating the acoustic wave equation on the SEG-AA salt model.

In this paper, we present a nonlinear curvelet-based sparsity-promoting formulation for three problems in seismic processing and imaging namely, seismic data regularization from data with large percentages of traces missing; seismic amplitude recovery for subsalt images obtained by reverse-time migration and primary-multiple separation, given an inaccurate multiple prediction. We argue why these nonlinear formulations are beneficial.

Polymer components are used in many different applications, including in industries where critical applications put high requirements regarding quality assessment. Such applications might include medical or food where the presence of discontinuities might induce bacterial growth or other unpleasantries, thus certain manufacturers must be able to maintain a zero-tolerance towards damaged components. This leads to the need for efficient testing methods of nondestructive nature capable of testing large quantities of components in a production line environment. The authors have been tasked by Acoustic Agree AB and Trelleborg AB, a world-leading producer of polymer engineered solutions, to find a nondestructive testing method capable of detecting discontinuities in thin polymer components in a production line environment. Implementation in production line environments puts requirements on test cycle time and a goal is to complete a test cycle within 3-4 seconds. Due to restrictions regarding available equipment and expertise, the focus has been put on applying nonlinear acoustic methods for nondestructive testing instead of more conventional methods. These methods utilize the nonlinear distortion of acoustic waves which causes certain characteristics to appear, such as the generation of Higher Harmonics (HH), frequency modulation (NWMS), resonance frequency shift (NRUS), and amplitude recovery (IDAR). Visual testing was used to discern visibly damaged samples from seemingly undamaged ones. The only methods which showed the possibility of discerning damaged thin polymer components were NRUS and IDAR. Only the latter has the capability to maintain the prescribed test cycle time. Nonlinear acoustic methods seem to be capable of detecting discontinuities in thin polymer components within the given time frame. More work is required to properly investigate the performance of NWMS and IDAR. The configuration used in this work was mainly focused on IDAR, resulting in specific calibration for NWMS was neglected. The sample population was also too low to collect sufficient data to ensure statistical certainty regarding the performance for either method. / Polymerkomponenter används i många olika tillämpningar, i synnerhet i industrier där kritiska tillämpningar ställer höga krav på kvalitetsbedömning. Sådana tillämpningar kan finnas inom medicinska- eller livsmedelsindustrier där förekomsten av diskontinuiteter kan inducera bakterietillväxt eller andra obehagliga egenskaper, därför måste vissa tillverkare upprätta nolltolerans mot skadade komponenter. Detta leder till behovet av effektiva provningsmetoder av oförstörande natur som kan utvärdera stora mängder komponenter i en produktionslinjemiljö. Författarna har fått uppdrag av Acoustic Agree AB och Trelleborg AB, en världsledande producent av polymertekniska lösningar, för att finna en oförstörande provningsmetod som kan detektera diskontinuiteter i tunna polymerkomponenter i en produktionslinjemiljö. Implementering i produktionslinjemiljöer ställer krav på testcykeltid och ett mål är att kunna slutföra en testcykel inom 3-4 sekunder. Begränsningen av tillgänglig utrustning och expertis har lett till att författarna har fokuserat på att tillämpa olinjära akustiska metoder för oförstörande provning istället för mer konventionella metoder. Dessa metoder utnyttjar olinjär distortion av akustiska vågor vilket medför att vissa egenskaper uppträder, såsom generering av övertoner (HH), frekvensmodulering (NWMS), resonansfrekvensskift (NRUS) och amplitudåtergång (IDAR). Visuell provning användes för att skildra synligt skadade prover från till synes oskadade. De enda metoder som visade möjligheten att urskilja skadade tunna polymerkomponenter var NRUS och IDAR. Endast den sistnämnda har förmågan att utföra provning inom den föreskrivna provcykeltiden. Olinjära akustiska metoder verkar kunna detektera diskontinuiteter i tunna polymerkomponenter inom den givna tidsramen. Mer arbete krävs för att korrekt utvärdera prestandan av NWMS och IDAR. Konfigurationen som användes i detta arbete var huvudsakligen inriktad på IDAR, vilket resulterade i att specifik kalibrering för NWMS ej utförts. Provpopulationen var även för låg för att kunna samla in tillräckligt med data för att uppnå statistisk säkerhet angående metodernas prestanda.

In this paper, we recover the amplitude of a seismic image by approximating the normal (demigrationmigration) operator. In this approximation, we make use of the property that curvelets remain invariant under the action of the normal operator. We propose a seismic amplitude recovery method that employs an eigenvalue like decomposition for the normal operator using curvelets as eigen-vectors. Subsequently, we propose an approximate non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness in the curvelet domain and spatial continuity constraints. Our method is tested with a reverse-time ’wave-equation’ migration code simulating the acoustic wave equation on the SEG-AA salt model.