We introduce two dimensional shot-profile migration data reconstruction (SPDR2). SPDR2 reconstructs data using leastsquares shot-profile migration with a constant migration velocity model. The velocity model is chosen both for efficiency and so that minimal assumptions are made about earth structure. At the core of least-squares migration are forward (de-migration) and adjoint (migration) operators, the former mapping from model space to data space, and the latter mapping from data space to model space. SPDR2 uses least-squares migration to find an optimal model which, in turn, is mapped to data space using de-migration, providing a reconstructed shot gather. We apply SPDR2 to real data from the Gulf of Mexico. In particular, we use SPDR2 to extrapolate near offset geophones.
In seismic data reconstruction, algorithms tend to fall into one of two categories, being rooted in either signal processing or the wave equation. Examples of the former include Spitz (1991), G¨ul¨unay (2003), Liu and Sacchi (2004), Hennenfent and Herrmann (2006), and Naghizadeh and Sacchi (2007), while examples of the later include Stolt (2002), Chiu and Stolt (2002), Trad (2003), Ram´ırez et al. (2006), and Ram´ırez and Weglein (2009). SPDR2 belongs to the family of wave equation based methods for data reconstruction. It differs from previous efforts in its parameterization of model space, being based on shot-profile migration (e.g. Biondi, 2003) and de-migration operators. Additionally, it relies on data fitting methods such as those used in Trad (2003), rather than direct inversion and asymptotic approximation which are used in, for example, Stolt (2002). A challenge in data reconstruction is alias. In particular, when aliased energy is present and interferes with signal, their separation becomes challenging (but, not impossible). A recent example of data reconstruction is Naghizadeh and Sacchi (2007). They use the non-aliased part of data to aid in the reconstruction of the aliased part of data. An alternative approach is to transform data via some operator that maps from data space to some model space, and such that in that model space, the corresponding representation of signal and alias are separable. This is a common approach in many signal processing methods, and is also the approach that we take in SPDR2. In particular, the SPDR2 model space is the sum of constant velocity shot-profile migrated gathers (i.e. a sum of common shot image gathers). This means that the SPDR2 model space is a representation of the earth’s reflectors parameterized by pseudo-depth (i.e. depth under the assumption of a constant migration velocity model) and lateral position. We will show that under the assumption of limited dips in the earth’s reflectors, the SPDR2 model space allows for the suppression of alias while preserving signal, thus allowing for the reconstruction of aliased data. We begin with a description of shot-profile migration and demigration built from the Born approximation to the acoustic wave-field and constant velocity Green’s functions. We apply shot-profile migration to an analytic example in order to illustrate its mapping of signal and alias from data space (shot gathers) to model space.