Kenneth P. Bube

Tomographic determination of velocity and depth in laterally varying media

Estimation of reflector depth and seismic velocity from seismic reflection data can be formulated as a general inverse problem. The method used to solve this problem is similar to tomographic techniques in medical diagnosis and we refer to it as seismic reflection tomography. Seismic tomography is formulated as an iterative Gauss‐Newton algorithm that produces a velocity‐depth model which minimizes the difference between traveltimes generated by tracing rays through the model and traveltimes measured from the data. The input to the process consists of traveltimes measured from selected events on unstacked seismic data and a first‐guess velocity‐depth model. Usually this first‐guess model has velocities which are laterally constant and is usually based on nearby well information and/or an analysis of the stacked section. The final model generated by the tomographic method yields traveltimes from ray tracing which differ from the measured values in recorded data by approximately 5 ms root‐mean‐square. The i...

Hybrid l 1 /l 2 minimization with applications to tomography

Least squares or l2 solutions of seismic inversion and tomography problems tend to be very sensitive to data points with large errors. The lp minimization for 1 ≤ p < 2 gives more robust solutions, but usually with higher computational cost. Iteratively reweighted least squares (IRLS) gives efficient approximate solutions to these lp problems. We apply IRLS to a hybrid l2/l2 minimization problem that behaves like an l2 fit for small residuals and like an l1 fit for large residuals. The smooth transition from l2 to l1 behavior is controlled by a parameter that we choose using an estimate of the standard deviation of the data error. For linear problems of full rank, the hybrid objective function has a unique minimum, and IRLS can be proven to converge to it. We obtain a robust efficient method. For nonlinear problems, a version of the Gauss‐Newton algorithm can be applied. Synthetic crosswell tomography examples and a field‐data VSP tomography example demonstrate the improvement of the hybrid method over le...

Crosswell traveltime tomography in three dimensions

Conventional crosswell direct-arrival traveltime tomography solves for velocity in a 2-D slice of the subsurface joining two wells. Many 3-D aspects of real crosswell surveys, including well deviations and out-of-well-plane structure, are ignored in 2-D models. We present a 3-D approach to crosswell tomography that is capable of handling severe well deviations and multiple-profile datasets. Three-dimensional pixelized models would be even more seriously underdetermined than the pixelized models that have been used in 2-D tomography. We, therefore, employ a thinly layered, vertically discontinuous 3-D velocity model that greatly reduces the number of model parameters. The layers are separated by 2-D interfaces represented as 2-D Chebyshev polynomials that are determined using a priori structural information and remain fixed in the traveltime inversion. The velocity in each layer is also represented as a 2-D Chebyshev polynomial. Unlike pixelized models that provide limited vertical resolution and may be overparameterized horizontally, this 3-D model provides vertical resolution comparable to the scale of wireline logs, and reduces the degrees of freedom in the horizontal parameterization to the expected in-line and out-of-well-plane horizontal resolution available in crosswell traveltime data. Ray tracing for the nonlinear traveltime inversion is performed in three dimensions. The 3-D tomography problem is regularized using penalty constraints with a continuation strategy that allows us to extrapolate the velocity field to a 3-D region containing the 2-D crosswell profile. Although this velocity field cannot be expected to be accurate throughout the 3-D region, it is at least as accurate as 2-D tomograms near the well plane of each 2-D crosswell profile. Futhermore, multiple-profile crosswell data can be inverted simultaneously to resolve better the 3-D distribution of velocity near the profiles. Our velocity parameterization is quite different from pixelized models, so resolution properties will be different. Using wave-modeled synthetic data, we find that near horizontal raypaths have the largest mismatch between ray-traced traveltimes and traveltimes estimated from the data. In conventional tomography, horizontal raypaths are essential for high vertical resolution. With our model, however, the highest resolution and most accurate inversions are achieved by excluding raypaths that travel nearly parallel to the geologic layering. We perform this exclusion in both a static and model-based manner. We apply our 3-D method to a multiple-profile crosswell survey at the Cymric oil field in California, an area of very steep structural dips and significant well trajectory deviations. Results of this multiple-profile 3-D tomography correlate very well with the independently-processed single profile results, with the advantage of an improved tie at the common well.

Theoretical and numerical issues in the determination of reflector depths in seismic reflection tomography

Seismic reflection tomography obtains an estimate of the subsurface slowness field and the location of strong reflectors by minimizing the difference between measured travel times from seismic reflection events and the corresponding travel times computed from a model of the subsurface. We present some theoretical results for the undiscretized problem regarding the possible ambiguity between slowness and depth. These results indicate that the depths of the reflectors are determined in theory except for edge effects, but a sufficiently large aperture at the reflector is necessary to resolve this ambiguity in practice. The slowness field, however, does have some undetermined features. These results have strong implications for how the tomography problem should be discretized and regularized to compute solutions which are accurate in the features of the model which are well determined from the travel time data. In particular, the slowness model should not be discretized much more coarsely than the reflectors as a way of regularizing the problem because that may force the computed reflector depths to try to match aspects of the travel time data which are caused by features in the slowness field.

Fast line searches for the robust solution of linear systems in the hybrid ℓ1 ∕ ℓ2 and Huber norms

Linear systems of equations arise in traveltime tomography, deconvolution, and many other geophysical applications. Nonlinear problems are often solved by successive linearization, leading to a sequence of linear systems. Overdetermined linear systems are solved by minimizing some measure of the size of the misfit or residual. The most commonly used measure is the l2 norm (squared), leading to least squares problems. The advantage of least squares problems for linear systems is that they can be solved by methods (for example, QR factorization) that retain the linear behavior of the problem. The disadvantage of least squares solutions is that the solution is sensitive to outliers. More robust norms, approximating the l1 norm, can be used to reduce the sensitivity to outliers. Unfortunately, these more robustnorms lead to nonlinear minimization problems, even for linear systems, and many efficient algorithms for nonlinear minimiza-tion problems require line searches. One iterative method for solving linear ...

A continuation approach to regularization of ill-posed problems with application to crosswell-traveltime tomography

In most geometries in which seismic-traveltime tomography is applied (e.g., crosswell, surface-reflection, and VSP), determination of the slowness field using only traveltimes is not a well-conditioned problem. Nonuniqueness is common. Even when the slowness field is uniquely determined, small changes in measured traveltimes can cause large errors in the computed slowness field. A priori information often is available — well logs, initial rough estimates of slowness from structural geology, etc. — and can be incorporated into a traveltime-inversion algorithm by using penalty terms. To further regularize the problem, smoothing constraints also can be incorporated using penalty terms by penalizing derivatives of the slowness field. What weights to use on the penalty terms is a major decision, particularly the smoothing-penalty weights. We use a continuation approach in selecting the smoothing-penalty weights. Instead of using fixed smoothing-penalty weights, we decrease them step by step, using the slowness...

A continuation approach to regularization for traveltime tomography

In most geometries in which seismic traveltime tomography is applied (e.g., surface reflection, cross-well, or VSP), the slowness field is not well-determined from traveltimes alone. Nonuniqueness is common. Even when the slowness field is uniquely determined, small changes in the measured traveltimes can lead to large errors in the computed slowness field. A priori information is often availablewell-logs, initial rough estimates of the slowness from structural geology, etc. This a priori information can be incorporated into a traveltime inversion algorithm using penalty terms. To further regularize the problem, smoothing constraints can also be incorporated using penalty terms by penalizing derivatives of the slowness field. A major decision to be made is the of the weights on the penalty terms, particularly the smoothing penalty weights. We use a continuation approach for selecting the smoothing penalty weights. Instead of fixing the smoothing penalty weights; we decrease the smoothing penalty weights in a step-by-step fashion, using the slowness model computed using the previous (larger) weights as the initial slowness model for the next step using the new (smaller) weights. A surprising outcome in synthetic problems is that the model error continues to decrease as we continue to decrease the smoothing penalty weights even after the data error has leveled off at the noise level. This continuation approach can solve synthetic problems more accurately than with fixed smoothing penalty weights, and appears to yield more features of interest in real-data applications of traveltime tomography.

Values of capture cross sections of metastable defects in hydrogenated amorphous silicon

Measurements of defect density, photoconductivity, and dark conductivity are used to obtain information about the values of the electron capture cross sections of charged and neutral metastable dangling‐bond defects in high‐quality, undoped, hydrogenated amorphous silicon at room temperature. Sixty measurements, obtained in the process of optical degradation experiments as a function of time at four different temperatures, have been analyzed using photoconductivity models corresponding to either one or two types of discrete‐level, multivalent defects. A model with two types of defects is able to accurately describe both dark conductivity and photoconductivity results, and gives the following average values: an electron capture cross section of about 1×10−16 cm2 for neutral centers of both higher‐lying (density not increased by light) and lower‐lying (density increased by light) defects, of about 2×10−16 cm2 for positively charged higher‐lying defects, and of about 20×10−16 cm2 for positively charged lower...

Wave tracing: Ray tracing for the propagation of band-limited signals: Part 2 — Applications

Modeling seismic propagation is critically important to our work; unfortunately, we often must trade simulation accuracy for reduced computational expense. We present a new seismic-modeling method that is as simple and computationally efficient as Snell’s law ray tracing but provides propagation paths and arrival times more consistent with finite-bandwidth data. We refer to this modeling method as wave tracing and apply it to nonlinear traveltime tomography and depth imaging. By replacing Snell’s law ray tracing with wave tracing, we get better ray coverage, more robust and faster ray bending (fewer iterations), and a much more robust and faster algorithm for nonlinear tomography (fewer iterations, too). A very significant benefit is increased stability and robustness of tomographic inversion with respect to small changes in model parameterization and regularization. A related benefit is the increased stability of depth images with respect to small changes in velocity, which can increase confidence in int...

Wave tracing: Ray tracing for the propagation of band-limited signals: Part 1 — Theory

Many seismic imaging techniques require computing traveltimes and travel paths. Methods to compute raypaths are usually based on high-frequency approximations. In situations such as head waves, these raypaths minimize traveltime but are not paths along which most of the energy travels. We have developed a new approach to computing raypaths, using a modification of ray bending that we call wave tracing; it computes raypaths and traveltimes that are more consistent with the paths and times for the band-limited signals in real data than the paths and times obtained using high-frequency approximations. Wave tracing shortens the raypath while keeping the raypath within the Fresnel zone for a characteristic frequency of the signal.