Bogdan G. Nita

Modeling bubbles and droplets in magnetic fluids

We develop, test and apply a volume of fluid (VOF) type code for the direct numerical simulation of two-fluid configurations of magnetic fluids with dynamic interfaces. Equilibrium magnetization and linear magnetic material are assumed and uniform imposed magnetic fields are considered, although extensions to nonlinear materials and to fields with spatio-temporal variability are possible. Models are computed for configurations of bubbles of non-magnetic fluid rising in ferrofluid and droplets of ferrofluid falling through non-magnetic fluid. Bubbles and droplets exhibit similar changes of shape in the presence of vertical fields, due to a combination of elongation along the field lines and the fluid dynamics of ordinary rising or falling at small Bond number. Bubbles become more prolate than droplets under the same parameters and are accordingly found to break up more readily than droplets in stronger fields. Indirect effects are observed, such as the change in rise time and the consequent changes in the flow due to increased Reynolds number.

Extension of the non-linear depth imaging capability of the inverse scattering series to multidimensional media: strategies and numerical results

The inverse scattering series (ISS) has proven, and continues to prove, to be a highly effective formalism for the separate and isolated accomplishment of several key tasks of reflection seismic processing and inversion. In particular, Weglein et al. (2000), Shaw et al. (2003), and Shaw (2005) describe the development of an algorithm distilled from the ISS that concerns itself with the location of subsurface reflectors with no prior knowledge, or related intervening estimation, of the medium wavespeed. The specific non-linear data activity that accomplishes this goal has been investigated by Shaw as such for an idealized 1D pre-stack acoustic experiment; we here describe the extension of those ideas to accommodate media with lateral variation. This is a non-trivial step. Nevertheless, beneath the added algebraic complexity, recognizable patterns and mechanisms are visible. Analysis of these terms and patterns suggests that certain portions of the 2D reflector location mechanisms of the ISS are a good starting point for the creation of algorithms for the accurate depth location of reflectors with a moderate level of lateral variability. The partial 2D imaging capability within the ISS is examined in this paper for the special case of a constant density acoustic medium and taking kh=0. We demonstrate numerical implementations of these forms and discuss ongoing work towards capturing further imaging capability residing within the ISS, especially with regards to the accommodation of larger levels of contrast and rapidity of spatial variation in medium properties.

Multi-dimensional seismic imaging using the inverse scattering series

The inverse scattering series (ISS) is a comprehensive multidimensional theory for processing and inverting seismic reflection data, that may be task-separated such that meaningful sub-problems of the seismic inverse problem may be accomplished individually, each without an accurate velocity model. We describe a task-separated subseries of the ISS geared towards accurate location in depth of reflectors, in particular the mechanisms of the series that act in multiple dimensions. We show that some 2D ISS imaging terms have analogs in previously developed 1D ISS imaging theory (e.g., Weglein et al., 2002; Shaw, 2005) and others do not; the former are used to create a 2D depth-only imaging prototype algorithm which is tested on synthetic salt-model data, and the latter are used to discuss ongoing research into reflector location activity within the series that acts only in the case of lateral variation and the presence of, e.g., diffraction energy in the data. Numerical tests are encouraging and show clear added value.

FORWARD SCATTERING SERIES AND SEISMIC EVENTS: FAR FIELD APPROXIMATIONS, CRITICAL AND POSTCRITICAL EVENTS

Inverse scattering series is the only nonlinear, direct inversion method for the multidimensional, acoustic or elastic equation. Recently developed techniques for inverse problems based on the inverse scattering series [Weglein et al., Geophys., 62 (1997), pp. 1975--1989; Top. Rev. Inverse Problems, 19 (2003), pp. R27--R83] were shown to require two mappings, one associating nonperturbative description of seismic events with their forward scattering series description and a second relating the construction of events in the forward to their treatment in the inverse scattering series. This paper extends and further analyzes the first of these two mappings, introduced, for 1D normal incidence, in Matson [J. Seismic Exploration, 5 (1996), pp. 63--78] and later extended to two dimensions in Matson [An Inverse Scattering Series for Attenuating Elastic Multiples from Multicomponent Land and Ocean Bottom Seismic Data, Ph.D. thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver,...

Removing free-surface multiples from teleseismic transmission and constructed reflection responses using reciprocity and the inverse scattering series

We develop a new way to remove free-surface multiples from teleseismic P- transmission and constructed reflection responses. We consider two types of teleseismic waves with the presence of the free surface: One is the recorded waves under the real transmission geometry; the other is the constructed waves under a virtual reflection geometry.The theory presented is limited to 1D plane wave acoustic media, but this approximation is reasonable for the teleseismic P-wave problem resulting from the steep emergence angle of the wavefield. Using one-way wavefield reciprocity, we show howtheteleseismicreflectionresponsescanbereconstructed from the teleseismic transmission responses. We use the inverse scattering series to remove free-surface multiples from the original transmission data and from the reconstructed reflection response. We derive an alternative algorithm for reconstructing the reflection response from the transmission data that is obtained by taking the difference between the teleseismic transmission waves before and after free-surface multiple removal. Numerical tests with 1D acoustic layered earthmodelsdemonstratethevalidityofthetheorywedevelop.NoisetestshowsthatthealgorithmcanworkwithS/Nratioaslowas5comparedtoactualdatawithS/Nratiofrom30 to 50. Testing with elastic synthetic data indicates that the acousticalgorithmisstilleffectiveforsmallincidenceangles oftypicalteleseismicwavefields.

Using the inverse scattering series to predict the wavefield at depth and the transmitted wavefield without an assumption about the phase of the measured reflection data or back propagation in the overburden

The starting point for the derivation of a new set of approaches for predicting both the wavefield at depth in an unknown medium and transmission data from measured reflection data is the inverse scattering series. We present a selection of these maps that differ in order (i.e., linear or nonlinear), capability, and data requirements. They have their roots in the consideration of a data format known as the T-matrix and have direct applicability to the data construction techniques motivating this special issue. Of particular note, one of these, a construction of the wavefield at any depth (including the transmitted wavefield), order-by-order in the measured reflected wavefield, has an unusual set of capabilities (e.g., it does not involve an assumption regarding the minimum-phase nature of the data and is accomplished with processing in the simple reference medium only) and requirements (e.g., a suite of frequencies from surface data are required to compute a single frequency of the wavefield at depth when the subsurface is unknown). An alternative reflection-to-transmission data mapping (which does not require a knowledge of the wavelet, and in which the component of the unknown medium that is linear in the reflection data is used as a proxy for the component of the unknown medium that is linear in the transmission data) is also derivable from the inverse scattering series framework.

Inverse Scattering Internal Multiple Attenuation Algorithm: an Analysis of the Pseudo-depth And Time Monotonicity Requirements

Pseudo-depth monotonicity condition is an important assumption of the inverse scattering internal multiple attenuation algorithm. Analysis reveals that this condition is equivalent to a vertical-time monotonicity condition which is different than the total traveltime monotonicity suggested in recent literature/discussions. For certain complex media, the monotonicity condition can be too restrictive and, as a result, some multiples will not be predicted by the algorithm. Those cases have to be analyzed in the forward scattering series to determine how the multiples are modeled and to establish if an analogy between the forward and the inverse process would be useful to expand the algorithm to address these kind of events.

A comprehensive strategy for removing multiples and depth‐imaging primaries without subsurface information: Direct horizontal common image gathers without the velocity or “ironing”

In AVO (Amplitude Variation with Offset) analysis, the amplitudes of reflected waves with different incident angles are studied to deduce lithology information beyond the structure map obtained by seismic imaging algorithms. The quantitative analysis of the amplitude, relies on common-image gathers being flat (or equivalently, at the same depth). But the waves with different incident angles will have different apparent velocities, resulting in different depths for the same image point at different angles, or non-flat common image gathers. In many scenarios, non-flat common-image gather was flattened by trim means at the cost of compromising zero-crossing and polarity-reversal information. This work presents a solution based on the seismic imaging subseries of the inverse scattering series (ISS) that flattens the common image gather without knowing or determining the subsurface velocity, and without any harmful amplitude consequencies.

Analytic continuation of perturbative solutions of acoustic 1D wave equation by means of Padé approximants

The forward scattering series is an important and useful tool in constructing perturbative solutions to wave equation and understanding their relationship to their non-perturbative counterparts. When it converges, the series describes the total wavefield everywhere in a given medium as propagations in a reference medium and interactions with point scatterers. The method can be viewed as constructing a mapping between non-perturbative solutions of wave events and their volume point scatterer description. This mapping was shown to be required by the recently developed techniques for inverse problems based on the inverse scattering series with applications to seismic exploration (Weglein, A.B., Gasparotto, F.A., Carvalho, P.M. and Stolt, R.H., 1997, An inverse scattering series method for attenuating multiples in seismic reflection data. Geophysics, 62, 1975--1989, Weglein, A.B., Araujo, F.V., Carvalho, P.M., Stolt, R.H., Matson, K.H., Coates, R., Corrigan, D., Foster, D.J., Shaw, S.A. and Zhang, H., 2003, Inverse scattering series and seismic exploration. Topical Review Inverse Problems, 19, R27--R83). The forward scattering series for a 1D acoustic medium and a normal incidence plane wave was shown in Matson, K.H., 1996, The relationship between scattering theory and the primaries and multiples of reflection seismic data. J. Seis. Expl., 5, 63--78 to converge for a ratio less than between the reference and the actual velocity. Same restricted convergence was obtained in Innanen, K.H., 2003, Methods for the treatment of acoustic and absorbtive/dispersive wavefield measurements, PhD Thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, Canada for a visco-acoustic medium with or without dispersion. In this article, we propose an explanation for this divergence and an extension of the method able to construct the solution of the 1D wave equation for any velocity contrast between the actual and the reference medium for both acoustic and visco-acoustic cases. The method involves the analytic continuation of the forward scattering solution by computing a certain sequence of Padé approximants to the partial sums of the forward scattering series.

Inverse scattering sub-series direct removal of multiples and depth imaging and inversion of primaries without subsurface information: strategy and recent advances

SUMMARY This paper provides: (1) a review of the logic and promise behind the isolated task inverse-scattering sub-series concept for achieving all processing objectives directly in terms of data only, without knowing or determining or estimating the properties that govern wave propagation in the actual earth; (2) the recognition that an effective response to pressing seismic challenges requires understanding that those challenges arise when assumptions, and prerequisites behind current leading edge seismic processing are not satisfied and that those failures can be attributed to: (A) insufficient acquisition, and/or (B) compute power and (C) bumping into algorithmic limitations and assumptions, and (3) the status and plans of the inverse series campaign to address the fundamental algorithmic limitations of processing methods, that are not addressed by more complete acquisition and faster computers. BACKGROUND Scattering theory is a form of perturbation theory. It relates a perturbation in the properties of a medium to the concomitant perturbation in the wave field. L0G0 = d and LG = d represent the equations governing wave propagation in the unperturbed and perturbed media where L0 and G0, and L and G are the unperturbed and perturbed differential operators and Green’s functions, respectively. V = L0 L is the perturbation operator and y = G G0, is the scattered field, and is the difference between the unperturbed and perturbed medium’s Green’s functions. In our seismic application the unperturbed medium is called the reference medium, and will be chosen (in this paper) for the marine case to be water. The perturbed medium in our marine context is the actual earth and the domain of the perturbation, V, the difference between earth properties and water, begins at the water bottom. Scattering equation The relationship between G, G0, and V is given by an operator identity called the scattering equation or the Lippmann-Schwinger (LS) equation (Goldberger and Watson, 2004; Taylor, 1972; Joachain, 1975, e.g.),