Partha S. Routh

Derivation of forward and adjoint operators for least-squares shot-profile split-step migration

The forward and adjoint operators for shot-profile least-squares migration are derived. The forward operator is demigration, and the adjoint operator is migration. The demigration operator is derived from the Born approximation. The process begins with a Greens function that allows for a laterally varying migration velocity model using the split-step approximation. Next, the earth is divided into horizontal layers, and within each layer the migration velocity model is made to be constant with respect to depth. For a given layer, (1) the source-side wavefield is propagated down to its top using the background wavefield. This gives a background wavefield incident at the layers upper boundary. (2) The layers contribution to the scattered wavefield is computed using the Born approximation to the scattered wavefield and the background wavefield. (3) Next, its scattered wavefield is propagated back up to the measurement surface using, again, the background wavefield. The measured wavefield is approximated by the sum of scattered wavefields from each layer. In the derivation of the measured wavefield, the shot-profile migration geometry is used. For each shot, the resulting wavefield modeling operator takes the form of a Fredholm integral equation of the first kind, and this is used to write down its adjoint, the shot-profile migration operator. This forward/adjoint pair is used for shot-profile least-squares migration. Shot-profile least-squares migration is illustrated with two synthetic examples. The first uses data collected over a four-layer acoustic model, and the second uses data from the Sigsbee 2a model.

Encoded Simultaneous Source Full-Wavefield Inversion For Spectrally Shaped Marine Streamer Data

In this paper, we apply encoded simultaneous source fullwavefield inversion (FWI) to marine streamer data. FWI of large scale 3D data is a challenging problem, especially constraining the inversion using the high frequencies available in exploration seismic data. Two methodologies that make high-frequency FWI feasible for field data are: (a) applying encoded simultaneous source full-wavefield inversion (ESSFWI) and (b) shaping the data to provide a preferential weighting to the low-frequency components of the data. These two methods in combination provide us with the computational efficiencies needed for large 3D runs. To date, most encoded simultaneous source methods have been applied to fixed-receiver data; i.e., each receiver records data from all shots in the survey. We developed an approach that enables us to apply ESSFWI to marine streamer data that are non-fixed spread. The approach uses a normalized cross-correlation objective function with multiple realizations of the encoded data at each iteration of the nonlinear FWI. The method can be applied to 2D/3D data with any survey geometry. Here we demonstrate the methodology and discuss its details with synthetic examples. Although not presented here our initial investigations on 3D field streamer data look encouraging.

Exploiting Surface Consistency For Ground-roll Characterization And Mitigation

A tomography-like-method is used to invert surfaceseismic data to estimate variable surface-wave properties. Surface consistency is exploited to decompose the data in the frequency domain into frequency-dependent propagation (i.e., velocity and attenuation) effects and variable sourceand receiver-coupling effects. The inversion can be performed simultaneously for single modes (linear optimization) or for multiple modes (nonlinear optimization). Including sourceand receivercoupling variations improves the ability to estimate velocity and attenuation from multi-source multi-receiver data. Further improvements in the estimation are made by constraining the parameters to be a smooth function of frequency. The estimated model parameters can be used to predict the multi-mode ground roll and subtract it from the data with little damage to reflections.

Surface Consistent Surface-Wave Inversion

A tomographic method (SWIPER) is used to invert surface-seismic data to estimate variable surfacewave properties. Surface consistency of multi-source, multi-receiver data is exploited to decompose the data in the frequency domain into frequency-dependent propagation (i.e., velocity and attenuation) effects and variable source- and receiver-coupling effects. The inversion can be performed for single modes (linear optimization) or simultaneously for multiple modes (nonlinear optimization). Including source- and receiver-coupling variations improves the ability to estimate velocity and attenuation. Further improvements in the estimation are made by constraining the parameters to be a smooth function of frequency. The estimated model parameters, such as velocity dispersion relations, can be used to predict the multi-mode ground roll and subtract it from the data with little damage to reflections or to invert for a near-surface depth model. The properties of the near-surface vary rapidly in both vertical and horizontal directions. Consequently, the behavior of ground roll also varies both with frequency and with position along the surface. Traditional methods of surface-wave analysis, such as the MASW, have limited resolution because they effectively average properties over the maximum source-toreceiver distance in the gather. For example, because slant stack sums over the traces in the gather, variability within the gather is averaged. Additional limitations arise from the difficulty in picking the maximum amplitudes in the transform domain because of limited velocity resolution, noise, and interference of multiple modes. Fully exploiting the multi-shot and multi-receiver aspect of 3-D surface seismic data allows the determination of variable surface-wave parameters within a fine grid of surface cells using direct-ray tomography. At the higher frequencies, however, multi-mode behavior and their interference must be included. Amplitude and phase effects are coupled and must be determined by a nonlinear optimization method. We show the ability to determine for a receiver-interval sized grid multi-mode velocity and attenuation parameters, which after horizontal smoothing match those determined with a beam-forming method.