Philip M. Carrion

Dual tomography for imaging complex structures

The geophysical community admits that geotomography is a workable way to obtain accurate estimates of seismic velocity in complex structures. So‐called dual tomography improves the resolution of computed tomograms and thus can be applied to different geophysical and nongeophysical problems. Dual tomography emerges as a generalized approach to linearized constrained inversion. Dual inversion transforms a generalized constrained optimization problem being formulated in the physical space of seismic velocities to a dual unconstrained problem posed in the vector space of Lagrangian multipliers. It is a parametric optimization problem, with unknown solutions that are always perpendicular to the null‐space of a tomographic matrix. Imposed constraints improve the resolution and act as if the angular aperture coverage was extended. Thus dual tomography is able to quash image blurring associated with incomplete angular recording. Dual tomography does not require accurate knowledge of initial model. One starts with...

CROSS-BOREHOLE TOMOGRAPHY IN ANISOTROPIC MEDIA

Anisotropy has significant effect on traveltime cross‐borehole tomography. Even relatively weak anisotropy cannot be ignored if accurate velocity estimates are desired, since isotropic traveltime tomography treats anisotropy as inhomogeneity. Traveltime data in our examples were synthetically generated by a ray‐tracing code for anisotropic media, and the computed quasi‐P‐wave traveltimes were subsequently inverted using the “dual tomography” technique (Carrion, 1991). The results of the tomographic inversion show typical artifacts due to the anisotropy, and that accurate imaging is impossible without taking the anisotropy into account.

Tomographic imaging of opaque and low‐contrast objects in range‐independent waveguides

A feasibility study is presented for acoustic tomographic imaging of arbitrarily shaped objects in range-independent oceanic waveguides. One of the main obstacles in the efficient use of tomographic imaging is that tomographic systems usually allow for a number of different independent solutions which satisfy the same systems of equations. This adverse property of acoustic tomography is especially pronounced in underdetermined systems characterized by null spaces of tomographic matrices which relate measured data to an unknown sound-speed distribution. In fact, acoustic tomography is usually underdetermined and the null space problem becomes critical in many applications. Acoustic tomography in many instances yields blurred images precluding its efficient use for mapping underwater objects in oceanic waveguides. How to improve the resolving power of tomographic imagery in waveguides is shown. A new technique based on the sliding window approach is suggested which uses constrained traveltime inversion. The method will be tested on mapping opaque and low-contrast acoustic objects placed in range-independent waveguides. The method is fast since it does not directly invert tomographic matrices of large rank but instead carries out summation over nonzero elements of tomographic matrices. It can be proved that constrained inversion compensates for limited angular aperture recording and thus can be applied to practical problems with incomplete angular aperture. This work is limited to processing of transmitted signals. A separate study will be devoted to the analysis of reflection data.

A method for computation of velocity profiles by inversion of large-offset records

This paper describes a new method for recovering velocity profiles utilizing both phase and amplitude information including wide‐angle arrivals, post‐ and precritical reflections. This method is based on a double spatial transformation with a minimization procedure. The first transformation is slant stacking of the observed wave field (seismogram). The second is projecting the slant stacked wave field into the domain of horizontal slowness p and depth z. In this domain the inverse problem is reduced to finding the critical path p=V-1(z) where V(z) is the true velocity of the compressional waves. A numerical algorithm based on a minimization technique is used to find the critical path, which is equivalent to the set of turning points of the critically reflected rays. When this path is found, then the following criteria are satisfied: (1) most of the energy is concentrated away from the precritical region; (2) the computed reflection coefficients reach their maximum on this path; and (3) for horizontally st...

On stability of 1D exact inverse methods

It is well known that despite the fact that one-dimensional inverse methods are well developed there are still difficulties when these methods are applied to real seismic data. There are several reasons why these methods usually fail. Real seismic data are corrupted by noise. The data typically do not contain the low and high frequencies. This can dramatically degrade the reconstruction of the acoustic impedance as well as affect the resolution. The stability of one-dimensional exact methods is considered. Despite the fact that all these methods require the source time function to be a Dirac delta-function, the author examines the possibility of applying these methods to an arbitrary source function. It is shown that the results of the inversion procedure are very sensitive to the characteristics of the source wavelet chosen for inversion. In practice (in exploration geophysics, for example) it is almost impossible to measure the source wavelet precisely. Therefore, it appears very important to known in which cases small errors in the source wavelet will not cause large errors in a solution. (In other words, in which cases the problem will be well-posed even though the source time function is not impulsive.) Based on the analysis presented the author draws the conclusion that small uncertainties inherent in the estimate of the source spectrum can lead to severe instabilities in the low-frequency portion of its spectrum whereas small uncertainties in the high-frequency portion of the source spectrum hardly affect the recoverability of the acoustic impedance.

Inversion of seismic data using precritical reflection and refraction data

We present a two‐step inversion procedure to extract subsurface velocity estimates from large‐offset seismic data. The first step is the automatic iterative large‐offset inverse method (AILOIM) which we test in the presence of strong water‐bottom multiples. Since this method is based on finding the locus of critically reflected points, low‐velocity zones cannot be resolved and the reconstruction is only an average estimate. In the presence of low‐velocity zones, the second step of the inversion process is required. This second step is a generalized least‐squares inverse scheme applied to the precritical reflections. We computed the inverse solution using a perturbation technique and determined the reference model from an estimate of the average velocity given by the first step. Two major features of this inversion method are: amplitudes of reflection arrivals are incorporated into analysis, and accurate results can be achieved without human interaction (picking the traveltime curves). Numerical examples u...

Wavefront sets analysis of limited aperture migration sections

Although geophysical literature is quite rich in papers on migration, there are only a few papers that treat LAM (Limited Aperture Migration). Deterministic and stochastic criteria for the reconstruction of reflection interfaces in prestack migration were derived and demonstrated on synthetic and real data examples (a real data set was acquired offshore the Brazilian coast, courtesy of Petrobras). The reconstruction of reflection interfaces in LAM depends on the relative position of “normal bundles,” as defined here, and gradients to the reflection interfaces. The derived stochastic criterion relates the variance of “normal bundles” to the variance of slowness in the medium. These criteria will help the geophysicist to identify directly distorted regions caused by LAM and those regions that are correctly migrated.

Born inversion with a variable background velocity

Direct, or Born, inversion has been presented as an approximate solution of the multidimensional seismic‐acoustic inverse problem (e.g., Cohen and Bleistein, 1979; Phinney and Frazer, 1978; Raz, 1981; Clayton and Stolt, 1981). The most basic assumption of the Born model is that variations, or perturbations, of some reference model remain small. This requirement of smallness is due to the fact that errors are directly related to the value of the perturbations. Both the local and overall values of the perturbation variable contribute to these errors, which are reduced if the known reference model, or background velocity, closely matches the actual medium velocity.

Analysis of wave motion in even‐ and odd‐dimensional acoustic media using Radon transform

In this paper, a solution to the acoustic wave equation using a Radon transform is analyzed. It is shown that the application of the Radon transform leads to the fundamental concept of the Huygens’ rings in multidimensional acoustic spaces. It is also shown that Huygens’ rings occur in odd‐dimensional spaces and determine the localization of acoustic waves in space. It is proved that in even‐dimensional spaces, the concept of Huygens’ rings is no longer valid and thus breaks down the local distribution of acoustic waves in space. These statements are demonstrated by using a numerical example treating an inhomogeneous acoustic medium.

A procedure for inverting seismic data to obtain Q-profiles

An inversion technique which can be used to estimate Q-profiles by inversion of reflection seismic data in one dimension is presented. It is shown that, in general, Q-profiles cannot be uniquely estimated from normal-incidence reflection seismograms if Q is an arbitrary function of frequency. The authors discuss a technique to reconstruct the quality parameter Q as a function of depth under the assumption that Q does not depend on frequency and the velocity profile is known a priori.