We present a new approach to the design of stable and accurate wavefield extrapolation operators needed for explicit depth migration. We split the theoretical operator into two component operators, one a forward operator that controls the phase accuracy and the other an inverse operator, designed as a Wiener filter that stabilizes the first operator. Both component operators are designed to have a specific fixed length and the final operator is formed as the convolution of the components. We utilize this operator design method to build an explicit, wavefield extrapolation method based on the migration of individual source records. Two other features of our method are the use of dual operator tables, with high and low levels of evanescent filtering, and frequency-dependent spatial down sampling. Both of these features improve the accuracy and efficiency of the overall method. Empirical testing shows that our method has a similar performance to the time-migration method called phase shift, meaning it scales as NlogN. We illustrate the method with tests on the Marmousi synthetic dataset. We call our method FOCI which is an acronym for forward operator conjugate inverse.
We present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space-frequency domain. Instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts, one to control phase accuracy and a second to improve stability. The first partial operator is simply a windowed version of the exact operator for a half step. The second partial operator is designed, using the Wiener filter method, as a band-limited, least-squares inverse of the first. The final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second. This resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length. Additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower temporal frequencies. The amount of evanescent filtering is controlled by building two operator tables, one corresponding to significant evanescent filtering and the other to very little evanescent filtering. During the wavefield extrapolation process, most steps are taken with the second table while the first is invoked only for roughly every tenth step. Also, the data are divided into frequency partitions that are optimally resampled in the spatial coordinates to further enhance the performance of the extrapolation operator. Lower frequencies are downsampled to a larger spatial sample size. Testing of the algorithm shows accurate, high-angle impulse responses and run times comparable to the phase shift method of time migration. Images from trial depth migrations of the Marmousi model show very high resolution.
Downward-continuation migration algorithms are powerful tools for imaging complicated subsurface structures. However, they usually assume that extrapolation proceeds from a flat surface, whereas most land surveys are acquired over irregular surfaces. Our method downward continues data directly from topography using a recursive space-frequency explicit wavefield-extrapolation method. The algorithm typically handles strong lateral velocity variations by using the velocity value at each spatial position to build the wavefield extrapolator in which the depth step usually is kept fixed. To accommodate topographic variations, we build space-frequency wavefield extrapolators with laterally variable depth steps (LVDS). At each spatial location, the difference between topography and extrapolation depth is used to determine the depth step. We use the velocity and topographic values at each spatial lateral position to build extrapolators. The LVDS approach does not add more data nor does it require preprocessing prior to extrapolation. We implemented the LVDS method and applied it to a source profile prestack migration technique. We also implemented the previously developed zero-velocity layer approach to use for comparison. For both algorithms, we modeled the acoustic source as an approximate free-space Green’s function, not as a simple extrapolated spatial impulse. Tests on a synthetic data set modeled from rough topography and comparisons with the zero-velocity layer approach confirm the method’s effectiveness in imaging shallow and deep structures beneath rugged topography.
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