Tiago A. Coimbra

Offset continuation (OCO) ray tracing using OCO trajectories

Offset continuation (OCO) is a seismic configuration transform designed to simulate a seismic section as if obtained with a certain source-receiver offset using the data measured with another offset. Since OCO is dependent on the velocity model used in the process, comparison of the simulated section to an acquired section allows for the extraction of velocity information. An algorithm for such a horizon-oriented velocity analysis is based on so-called OCO rays. These OCO rays describe the output point of an OCO as a function of the Root Mean Square (RMS) velocity. The intersection point of an OCO ray with the picked traveltime curve in the acquired data corresponding to the output half-offset defines the RMS velocity at that position. We theoretically relate the OCO rays to the kinematic properties of OCO image waves that describe the continuous transformation of the common-offset reflection event from one offset to another. By applying the method of characteristics to the OCO image-wave equation, we obtain a raytracing-like procedure that allows to construct OCO trajectories describing the position of the OCO output point under varying offset. The endpoints of these OCO trajectories for a single input point and different values of the RMS velocity form then the OCO rays. A numerical example demonstrates that the developed ray-tracing procedure leads to reliable OCO rays, which in turn provide high-quality RMS velocities. The proposed procedure can be carried out fully automatically, while conventional velocity analysis needs human intervention. Moreover, since velocities are extracted using offset sections, more redundancy is available or, alternatively, OCO velocities can be studied as a function of offset.

Common-reflection-point time migration

Since the early days of seismic processing, time migration has proven to be a valuable tool for a number of imaging purposes. Main motivations for its widespread use include robustness with respect to velocity errors, as well as fast turnaround and low computation costs. In areas of complex geology, in which it has well-known limitations, time migration can still be of value by providing first images and also attributes, which can be of much help in further, more comprehensive depth migration. Time migration is a very close process to common-midpoint (CMP) stacking and, more recently, to zero-offset commonreflection- surface (CRS) stacking. In fact, Kirchhoff time migration operators can be readily formulated in terms of CRS parameters. In the nineties, several studies have shown advantages in the use of common-reflection-point (CRP) traveltimes to replace conventional CMP traveltimes for a number of stacking and migration purposes. In this paper, we follow that trend and introduce a Kirchhoff-type prestack time migration and velocity analysis algorithm, referred to as CRP time migration. The algorithm is based on a CRP operator together with optimal apertures, both computed with the help of CRS parameters. A field-data example indicates the potential of the proposed technique.

Migration Velocity Analysis Using Residual Diffraction Moveout in the Pre-stack Depth Domain

In this paper we present a methodology for migration velocity improvement and diffraction localisation based on a moveout analysis of over- or undermigrated diffraction events in the depth domain. The method does not depend on any requirement apart from a fairly arbitrary initial velocity model as input. We demonstrate that the method can be applied in both the pre- or post-stack domains without any restrictions. For each iteration, the method provides an update to the velocity model and consequently to the diffraction locations. The algorithm is based on the focusing of remigration velocity rays from uncollapsed migrated diffraction curves. These velocity rays are constructed from a ray-tracing like approach applied to the image-wave equation for velocity continuation. After choosing and picking the diffraction events in the pre-stack migrated domain, the method has a very low computational cost, and the diffraction points are located automatically. We demonstrate the feasibility of our method using a synthetic nonzero-offset data example.

Migration velocity analysis using residual diffraction moveout: a real-data example

Unfocused seismic diffraction events carry direct information about errors in the migration-velocity model. The residual-diffraction-moveout (RDM) migration-velocity-analysis (MVA) method is a recent technique that extracts this information by means of adjusting ellipses or hyperbolas to uncollapsed migrated diffractions. In this paper, we apply this method, which has been tested so far only on synthetic data, to a real data set from the Viking Graben. After application of a plane-wave-destruction (PWD) filter to attenuate the reflected energy, the diffractions in the real data become interpretable and can be used for the RDM method. Our analysis demonstrates that the reflections need not be completely removed for this purpose. Beyond the need to identify and select diffraction events in post-stack migrated sections in the depth domain, the method has a very low computational cost and processing time. To reach an acceptable velocity model of comparable quality as one obtained with common-midpoint (CMP) processing, only two iterations were necessary.

A Computer Library for Ray Tracing in Analytical Media

Ray tracing technique is an important tool not only for forward but also for inverse problems in Geophysics, which most of the seismic processing steps depends on. However, implementing ray tracing codes can be very time consuming. This article presents a computer library to trace rays in 2.5D media composed by stack of layers. The velocity profile inside each layer is such that the eikonal equation can be analitically solved. Therefore, the ray tracing within such profile is made fast and accurately. The great advantage of an analytical ray tracing library is the numerical precision of the quantities computed and the fast execution of the implemented codes. Although ray tracing programs already exist for a long time, for example the seis package by Cervený, with a numerical approach to compute the ray. Regardless of the fact that numerical methods can solve more general problems, the analytical ones could be part of a more sofisticated simulation process, where the ray tracing time is completely relevant. We demonstrate the feasibility of our codes using numerical examples.

Stretch-free generalized normal moveout correction: Stretch-free GNMO correction

The effective application of normal moveout correction processes mainly depends on four factors: the chosen traveltime approximation, the stretching associated with the given traveltime, crossing events and phase changes, the last two being inherent to the seismic data. In this context, we conduct a quantitative analysis on stretching considering a general traveltime expression depending on half-offset and midpoint coordinates. Through this analysis, we propose a mathematically proven procedure to eliminate stretching, which can be applied to any traveltime approximation. The proposed method is applied to synthetic and real data sets, considering different traveltime approximations and achieved complete elimination of stretching.

A new interpretation on fuzzy differential equations: Linguistic concepts evolving over time

Fuzzy differential equations have been explored in many different ways, including theoretical aspects and in applications. The interpretation that is given to the fuzziness is a uncertainty due to fluctuations accumulated in the measurement process or due to partial information. This has much to do with a probabilistic interpretation. We propose to interpret the fuzziness as Zadeh first stated: membership to a non-sharp concept. As example we solve a fuzzy initial value problem whose independent variable is the time and the unknown variable is regarded as a single concept (the word “tall”) whose compatibility (membership) function varies over time. In summary, the main novelty is that the fuzzy differential equation models a concept evolving over time.

A CRS-based Heterogeneity Attribute

An attribute describing heterogeneity is proposed based on the diffracted-wave contribution. In areas of complex geology the amount of diffracted energy will increase relative the specular reflections. By use of the modified CRS technique, a reliable measure of the diffractions can be obtained. Proper scaling with the output from a conventional CRS stack gives a normalized measure of the complexity of the geology. The new attribute has possible use both in seismic texture analysis and as a weight factor employed to combine CRS based reflection- and diffraction-enhanced stacks. The potential of the heterogeneity attribute is demonstrated using seismic data from offshore Brazil.