The Dirac Equation in Geometric Quantization

The coadjoint orbit of the restricted Poincaré group corresponding to a mass m and spin 1/2 is described. The orbit is quantized using the geometric quantization. To include the discrete symmetries, one has to induce the irreducible representation of the restricted Poincaré group obtained by the quantization procedure to the full Poincaré group. The new representation is reducible and the reduction to an irreducible representation corresponds to the Dirac equation

Shot-gather time migration of planar reflectors without velocity model

Standard migration techniques require a velocity model. A new and fast prestack time migration method is presented that does not require a velocity model as an input. The only input is a shot gather, unlike other velocity-independent migrations that also require input of data in other gathers. The output of the presented migration is a time-migrated image and the migration velocity model. The method uses the first and second derivatives of the traveltimes with respect to the location of the receiver. These attributes are estimated by computing the gradient of the amplitude in a shot gather. The assumptions of the approach are a laterally slowly changing velocity and reflectors with small curvatures; the dip of the reflector can be arbitrary. The migration velocity corresponds to the root mean square (rms) velocity for laterally homogeneous media for near offsets. The migration expressions for 2D and 3D cases are derived from a simple geometrical construction considering the image of the source. The strengths and weaknesses of the methods are demonstrated on synthetic data. At last, the applicability of the method is discussed by interpreting the migration velocity in terms of the Taylor expansion of the traveltime around the zero offset

Migration without Velocity Model in Curvelet Domain

We present a formulation of velocity-less time migration in curvelet domain. In particular, we first decompose pre-stack gathers to curvelets, which unlike wavelets also contain directional information. Then, we use this directional information to perform migration that does not require any velocity model. This migration is performed directly with the curvelet coefficients. The resulting image is reconstructed by applying the inverse curvelet transform. The process is illustrated on synthetic data and compared to Kirchhoff migration

Simultaneous time imaging, velocity estimation and multiple suppression using local event slopes

We present and discuss the use of local event slopes and their associated attributes (referred to as XTP attributes here) as a way to estimate a time-imaging velocity field and to suppress organized noise including – but not restricted to - multiples. The 4 XTP attributes are: migration velocity, migrated spatial location, migrated zero offset time and stack domain dip. We derive the equations for XTP attributes from the double square root equation which illustrates the strong connection with Kirchhoff time migration. In this paper the XTP attributes are calculated in the shot and receiver domain. The advantages of shot/ receiver domain XTP noise suppression over similar efforts in the CMP and offset domain are discussed. In a companion presentation (Cooke et al, 2008), we discuss different methods of extracting these local event slopes