Dave Hale

Stable explicit depth extrapolation of seismic wavefields

Stable explicit filters for depth extrapolation of seismic wavefields may be derived through a modification of the conventional Taylor series method. The modified Taylor series method described here yields extrapolators with maximally-flat amplitude spectra in their passband, while ensuring that no spectral components in the wavefield are amplified by any factor greater than one. The price for stability is increased phase error. For low normalized frequencies, implicit extrapolators are more accurate than the 39-coefficient stable explicit extrapolator described here. However, the small spatial sampling intervals required to obtain high phase accuracy in implicit extrapolation imply that this accuracy is rarely achieved in practice. Over the wide range of normalized frequencies likely to be encountered in practice, stable explicit extrapolators outperform implicit ones. The method presented here for deriving stable explicit extrapolators is in no formal sense optimal. It is only guaranteed to yield stable extrapolators. In my limited experience with alternative methods for designing stable extrapolators, the method presented here produced the least phase error while ensuring stability. Nevertheless, a simple method for designing optimal (in some sense) stable explicit extrapolators would be preferred over the method presented here. 7 refs., 15 figs.

3-D depth migration via McClellan transformations

Three‐dimensional seismic wavefields may be extrapolated in depth, one frequency at a time, by two‐dimensional convolution with a circularly symmetric, frequency‐ and velocity‐dependent filter. This depth extrapolation, performed for each frequency independently, lies at the heart of 3-D finite‐difference depth migration. The computational efficiency of 3-D depth migration depends directly on the efficiency of this depth extrapolation. McClellan transformations provide an efficient method for both designing and implementing two‐dimensional digital filters that have a particular form of symmetry, such as the circularly symmetric depth extrapolation filters used in 3-D depth migration. Given the coefficients of one‐dimensional, frequency‐ and velocity‐dependent filters used to accomplish 2-D depth migration, McClellan transformations lead to a simple and efficient algorithm for 3-D depth migration. 3-D depth migration via McClellan transformations is simple because the coefficients of two‐dimensional depth ...

Imaging salt with turning seismic waves

Turning seismic waves, which first travel downward and then upward before (and after) reflection, have been recorded in a 3-D seismic survey conducted over an overhanging salt dome. Careful processing of these turning waves enables the imaging of the underside of the salt dome and of intrusions of salt into vertical faults radiating from the dome.When seismic wave velocity increases with depth, waves that initially travel downward are reflected and may turn so as to travel upward before reflection. A simple geometrical argument suggests that these turning waves are likely to exhibit abnormal moveout in com-mon-midpoint (CMP) gathers, in that reflection time decreases with increasing source-receiver offset. This abnormal moveout and the attenuation of turning waves by most migration methods suggest that conventional seismic processing does not properly image turning waves.The most important step in imaging turning waves, assuming that they have been recorded, is the migration process. Simple and inexpensive modifications to the conventional phase-shift migration method enable turning waves to be imaged for little additional computational cost. The examples provided in this paper suggest that these and other such modifications to conventional processing should be used routinely when imaging salt domes.

A method for estimating apparent displacement vectors from time-lapse seismic images

Reliable estimates of vertical, inline, and crossline components of apparent displacements in time-lapse seismic images are difficult to obtain for two reasons. First, features in 3D seismic images tend to be locally planar, and components of displacement within the planes of such features are poorly resolved. Second, searching directly for peaks in 3D crosscorrelations is less robust, more complicated, and computationally more costly than searching for peaks of 1D crosscorrelations. I estimate all three components of displacement with a process designed to mitigate these two problems. I address the first problem by computing for each image sample a local phase correlation instead of a local crosscorrelation. I address the second problem with a cyclic sequence of searches for peaks of correlations computed for lags constrained to one of the three image axes.

Image‐guided blended neighbor interpolation of scattered data

Uniformly sampled images are often used to interpolate other data acquired more sparsely with an entirely different mode of measurement. For example, downhole tools enable geophysical properties to be measured with high precision near boreholes that are scattered spatially, and less precise seismic images acquired at the earth’s surface are used to interpolate those properties at locations far away from the boreholes. Image-guided interpolation is designed specifically to enhance this process. Most existing methods for interpolation require distances from points where data will be interpolated to nearby points where data are known. Image-guided interpolation requires non-Euclidean distances in metric tensor fields that represent the coherence, orientations and shapes of features in images. This requirement leads to a new method for interpolating scattered data that I call blended neighbor interpolation. For simple Euclidean distances, blended neighbor interpolation resembles the classic natural neighbor interpolation.

A nonaliased integral method for dip moveout

Integral (Kirchhoff-style) methods for dip moveout (DMO), while possessing several advantages over Fourier transform methods, are prone to problems of spatial aliasing. DMO methods with spatially aliased operators yield significant processing errors, even when applied to data that are not spatially aliased. In particular, such DMO methods may alter the amplitude and phase of horizontal reflections, to which DMO should do nothing.A simple test can be used to detect spatial aliasing in a DMO implementation. First, compute the responses of DMO to an impulse for several source-receiver offsets and recording times. Second, sum the traces in each of these impulse responses to obtain the corresponding horizontal-reflection responses. These horizontal-reflection responses should be identical to the original impulse for all offsets and times. Integral DMO methods with spatial aliasing problems fail this test.Integral DMO methods typically apply a sequence of time compressions to an input seismogram so that each input sample forms an elliptical impulse response. These methods typically yield horizontal-reflection responses with significant errors in amplitude and phase that vary with source-receiver offset and recording time. Perhaps surprisingly, the worst errors may occur for small offsets and late times, for which DMO action should be inconsequential.The integral DMO method proposed in this paper applies a sequence of time shifts to each input seismogram. These shifted input traces are mapped to output traces along curved trajectories that enable each sample to form the appropriate elliptical impulse response. The proposed method, by design, passes the test described above. Even the most approximate, highly efficient implementation of this method will improve the imaging of dipping reflections without altering horizontal reflections.

Full Waveform Inversion With Image-guided Gradient

Full waveform inversion (FWI) with an image-guided gradient improves convergence by reducing the number of parameters in the subsurface model. We represent the subsurface model with a sparse set of values, and from these values, we use image-guided interpolation (IGI) to compute finelyand uniformly-sampled gradients of the data misfit function in FWI. IGI makes models more blocky than finely-sampled models, and this blockiness from the model space mitigates the absence of low frequencies in recorded data. A smaller number of parameters to invert also reduces the number of iterations required to converge to a solution model. Tests with a synthetic model and data demonstrate these improvements.

Fast Local Cross-correlations of Images

Consider two multi-dimensional digital signals, each with Ns samples. For some number of lags N l Ns, the cost of computing a single cross-correlation function of these two signals is proportional to Ns × N l. By exploiting several properties of Gaussian windows, we can compute Ns local cross-correlation functions, again with computational cost proportional to Ns × N l. Here, local means the cross-correlation of signals after applying a Gaussian window centered on a single sample. Computational cost is independent of the size of the window.

Structure‐oriented bilateral filtering of seismic images

Bilateral filtering is widely used to enhance photographic images, but in most implementations is poorly suited to seismic images. A bilateral filter consists of two (domain and range) filter kernels. By replacing the domain kernel with a smoothing filter that conforms to image structures, we obtain a bilateral filter suitable for seismic image processing. Examples and comparison with conventional edge-preserving smoothing illustrate advantages of structure-oriented bilateral filtering. The only significant disadvantage is a relatively high (roughly 10 to 40 times higher) computational cost.

Seismic interpretation using global image segmentation

Summary A rst step in seismic interpretation is seismic image segmentation. For most seismic images, with incompletely or poorly imaged faults and horizons, global methods for segmentation are more robust than local event tracking or region growing methods commonly used today. The disadvantage of global image segmentation methods has been their relatively high computational cost. We reduce this cost by applying these methods to a space-lling mesh aligned automatically with features in seismic images. The mesh makes global segmentation of 3-D seismic images feasible.