Dan Loewenthal

Reversed time migration in spatial frequency domain

During the past decade, finite-difference methods have become important tools for direct modeling of seismic data as well as for certain interpretation processes. One of the earliest applications of these methods to seismics is the pioneering contribution of Alterman who, in a series of papers (Alterman and Karal, 1968; Alterman and Aboudi, 1968; Alterman and Rotenberg, 1969; Alterman and Loewenthal, 1972) demonstrated the usefulness of such numerical computations for the propagation of seismic waves in elastic media. A clear exposition of these techniques, as well as a comparison of results obtained from them with the corresponding analytical solutions, can be found in Alterman and Karal (1968). This subject was further developed and extended to more complicated models by Boore (1970), Ottaviani (1971), and Kelly et al (1976). Claerbout introduced a somewhat different finite-difference approach (Claerbout, 1970; Claerbout and Johnson, 1971) for modeling the acoustic waves which often dominate the reflection seismogram. In his approach, the original wave equation, which governs the propagation of the acoustic waves, is modified in such a way so as to allow the propagation of either only upcoming or only downgoing waves. By moving the coordinate frame with the downgoing waves, Claerbout showed that one could greatly reduce computation time. Using the same concepts, he showed (Claerbout and Doherty, 1972) how to use a similar scheme for migrating a seismic section by downward continuation of the upcoming waves. This migration method is an interesting extension of the ideas of Hagedoorn (1954) and was found to be extremely useful with real data (Larner and Hatton, 1976; Loewenthal et al, 1976).

Suppressing the unwanted reflections of the full wave equation

In many instances in exploration geophysics we are interested in the so‐called one‐way wave equation. This equation allows the wave fields to propagate in the positive depth direction, but not in the reverse (−Z) direction. Some modeling and migration methods, such as the f-k method (Stolt, 1978) and the phase‐shift method (Gazdag, 1978), produce in a natural way the one‐way wave equation. The main advantage of the one‐way wave equation is that it does not give rise to multiples or interlayer reverberations and enables the observation of primary events only.

Two methods for computing the imaging condition for common-shot prestack migration

This note addresses two methods of computing the imaging condition for prestack migration of common‐shot seismic data; our work is based on the ideas from reverse‐time migration for both poststack (Loewenthal and Mufti, 1983; McMechan, 1983) and prestack data (Chang and McMechan, 1986). In reverse‐time migration of poststack data, the whole stacked section is backward‐extrapolated in time, with half of the medium velocity to time zero. All exploding reflectors are imaged at once at time zero. The time zero is referred to as the imaging condition. In prestack migration, the imaging condition is more involved. Each spatial grid point (treated as a point diffractor) has a different excitation time, which is equal to the one‐way traveltime from the source to that grid point. Each point diffractor is imaged separately at its excitation (the “imaging time”).

On unified dual fields and Einstein deconvolution

In many physical phenomena, the laws governing motion can be looked at as the relationship between unified dual fields which are continuous in time and space. Both fields are activated by a single source. The most notable example of such phenomena is electromagnetism, in which the dual fields are the electric field and the magnetic field. Another example is acoustics, in which the dual fields are the particle‐velocity field and the pressure field. The two fields are activated by the same source and satisfy two first‐order partial differential equations, such as those obtained by Newton’s laws or Maxwell’s equations. These equations are symmetrical in time and space, i.e., they obey the same wave equation, which differs only in the interface condition changing sign. The generalization of the Einstein velocity addition equation to a layered system explains how multiple reflections are generated. This result shows how dual sensors at a receiver point at depth provide the information required for a new deconv...

Matrix polynomial representation of the anisotropic magnetotelluric impedance tensor

Abstract A formalism based upon the equal penetration depth stratification assumption is extended to a layered anisotropic medium, yielding a recursive algorithm for the computation of the magnetotelluric impedance tensor elements. The development of this new procedure requires an appropriate layer-discretization as well as an extension of the reflection coefficient definition for the anisotropic layered model. This method transfers the differential problem into an algebraic one, and is independent of the electric and magnetic field vectors. The technique is a recursive process which gives a rational matrix polynomial representation for the magnetotelluric impedance tensor. The procedure involves the use of second-order matrices, rather than the fourth-order ones normally used for such a case. Using this representation, a separation of the contribution of the model parameters from that of the frequency is achieved. Consequently the elements of the magnetotelluric impedance tensor are computed, for each frequency, using coefficients which are evaluated just once for a given model.

Source wavelet estimation by upward extrapolation

A new deterministic technique for wavelet estimation and deconvolution of seismic traces was recently introduced. This impedance‐type technique was developed for a marine environment where both the source and the receivers are located inside a homogeneous layer of water. In this work, an extension of the theory of source wavelet estimation is proposed. As in previous publications, this method is based on extrapolation of the wave field measured at depth, upward to the free surface. The extrapolation is performed by using the finite‐difference approximation to the full inhomogeneous wave equation. The extrapolation results in a wavelet which generally includes ghosts and can be used for source signature deconvolution and deghosting. The method needs two closely spaced receivers and is applicable for arbitrary locations of the source and the receivers in one‐dimensional multilayered models, provided the source is above the receivers; furthermore, it can be applied to both marine and land data. Application o...

On the automatic interpretation of direct current resistivity soundings

A method for the automatic inversion of resistivity soundings is presented. The procedure consists of two main stages. First, application of linear filters which transforms the apparent resistivity curve into the kernel function, and vice versa. In the second stage the first and second derivatives of the kernel function are calculated and used in a second-order modified Newton-Raphson iterative fitting procedure. The model obtained is optimal in the least squares sense. The method has been tried on some field examples and produced resistivity models which show a good agreement with the geological well logs.

Source signature estimation using fictitious source and reflector

Source wavelet estimation is an important step in processing and interpreting seismic data. In the context of this work, the term (source wavelet) includes the pure source signature (the source response measured in a homogeneous medium) along with certain model‐related events (such as the ghost and interbed reflections). An estimate of the source wavelet can be used to increase the resolution of seismic data by signature deconvolution, deghosting, and dereverberation. However, pure source signature determination is of particular importance as a first step in direct inversion schemes, as demonstrated by Bube and Burridge (1983) and by Foster and Carrion (1985).

On the phase constraint of the magnetotelluric impedance

A simple proof that the phase of the one‐ dimensional magnetotelluric modified impedance lies between −π/4 and π/4 is given. The proof is based upon a rational polynomial representation of the impedance. The formalism from which the rational representation is obtained is outlined. An adjoint model is defined and it is shown to be pertinent for the proof. A numerical example is given.

On dual field measurements using geohydrophones

A new device is suggested to record with a single streamer both pressure and particle velocity dual fields in marine and land environments. Actually, a set of two vertical phones is enough to determine its dual conjugate field. By using finer and finer temporal grids, the vertical offset between the upper and lower phones can be reduced, facilitating construction of physically small dual sensor strings.