Adriana Citlali Ramírez

Clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic data

Linearinversionisdefinedasthelinearapproximationofa direct-inverse solution. This definition leads to data requirements and specific direct-inverse algorithms, which differ with all current linear and nonlinear approaches, and is immediately relevant for target identification and inversion in an elastic earth. Common practice typically starts with a directforwardormodelingexpressionandseekstosolveaforward equation in an inverse sense. Attempting to solve a direct forward problem in an inverse sense is not the same as solvinganinverseproblemdirectly.Distinctionsincludedifferences in algorithms, in the need for a priori information, and in data requirements. The simplest and most accessible examples are the direct-inversion tasks, derived from the inverse scattering series ISS, for the removal of free-surface and internal multiples.The ISS multiple-removal algorithms require no subsurface information, and they are independent ofearthmodeltype.Adirectforwardmethodsolvedinaninverse sense, for modeling and subtracting multiples, would require accurate knowledge of every detail of the subsurface the multiple has experienced. In addition, it requires a different modeling and subtraction algorithm for each different earth-model type. The ISS methods for direct removal of multiples are not a forward problem solved in an inverse sense. Similarly, the direct elastic inversion provided by the ISSisnotamodelingformulaforPPdatasolvedinaninverse sense. Direct elastic inversion calls for PP, PS, SS, … data, for direct linear and nonlinear estimates of changes in mechanical properties. In practice, a judicious combination of directandindirectmethodsarecalleduponforeffectivefield dataapplication.

Near offset data extrapolation

Summary We present a method that predicts water bottom primaries at near offsets. The concepts of limited aperture migration and inverse scattering are combined to develop a data driven theory for data reconstruction. Data extrapolation can be particularly challenging in shallow water, where early events of the recorded wavefield are mainly in the postcritical regime. For this kind of data, our results predict water bottom primaries with the correct travel time. When most of the data are precritical, the method accurately predicts primaries and multiples without the need for modeling or inverting for earth properties.

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 elimination: Leading‐order and higher‐order closed forms

Internal multiples are multiply reflected events in the measured wavefield that have experienced all of their downward reflections below the free surface. The order of an internal multiple is defined to be the number of downward reflections it experiences, without reference to the location of the downward reflection. The objective of internal multiple elimination using only recorded data and information about the reference medium is achievable directly through the inverse scattering task specific subseries formalism. The first term in the inverse scattering subseries for first-order internal multiple elimination is an attenuator, which predicts the correct traveltime and an amplitude always less than the true internal multiples’ amplitude. The leading and higher-order terms in the elimination series correct the amplitude predicted by the attenuator moving the algorithm towards an eliminator. Leading-order as an eliminator means it eliminates a class of internal multiples and further attenuates the rest. Adding the leading-order terms in a closed form provides an algorithm that eliminates all internal multiples generated at the shallowest reflector. The generating reflector is the location where the downward reflection of a given firstorder internal multiple took place. The higher-order subseries and its closed form correct the attenuation due to information on the overburden of deeper generating reflectors. A prestack form of the algorithm, which can be extended to a multidimensional form, is given for the leading-order subseries and its closed form.

The Role of the Direct Wave And Green¿s Theorem In Seismic Interferometry And Spurious Multiples.

Techniques to estimate the Green’s function between two measured points using wavefield correlations and/or crosscorrelations are classified as seismic interferometry. In this paper we provide a unifying framework for understanding a broad class of interferometric techniques using Green’s theorem. This framework and foundation allows spurious multiples that occur in certain interferometric approaches to be anticipated and fully explain as a consequence of approximations and compromises made within Green’s theorem. We also develop a set of more effective seismic interferometry methods, where fewer compromises effect in a better result. Standard seismic interferometry is based on far-field and one-way approximations of the Green’s theorem relating two wavefield measurements in the same volume. This method, when applied to surface seismic data, reconstructs the wavefield with a squared source signature and adds spurious multiples, whose amplitudes are comparable to the reconstructed primaries. The artefacts introduced, reduce the method’s value. The spurious multiples come from the approximations made to avoid the need of the wavefield’s normal derivative. We propose and examine various alternative approaches to seismic interferometry which overcome the appearance of spurious multiples and provides an improvement over traditional methods. One method uses Green’s theorem relating a reference Green’s function with the measured wavefield. The data are reconstructed without spurious multiples and with a wavelet due to a single source. A synthesized wavefield ought to have a single factor of the wavelet. Using an analytic reference Green’s function, data can be extrapolated to positions where no receivers or sources were located. In addition, we provide another form of Green’s theorem by imposing a two-surface Dirichlet boundary condition to the reference Green’s function; this method only requires the total wavefield. The normal derivative of the field is not needed.

The underlying unity of distinct processing algorithms for: (1) the removal of free surface and internal multiples, (2) Q compensation (without Q), (3) depth imaging, and (4) nonlinear AVO, that derive from the inverse scattering series

Pressing seismic exploration challenges are frequently caused by the inability to satisfy the requirements for sub-surface information required by many seismic processing methods. Since the inverse scattering series promises to achieve all processing objectives without the need for subsurface information, it is a natural path to explore for addressing that type of challenge. This paper reports further understanding, perspective and progress towards achieving our processing goals by developing and mining that promise. The items reported here include: (1) new ways to understand both the broad perspective and promise of the inverse scattering series (ISS), as well as the similarities and differences between the distinct methods that derive from the inverse scattering series, and that address, for example, the removal of free surface multiples, the removal of internal multiples, Q compensation (without Q) and depth imaging and AVO, (2) how the directness of the methods derived from the ISS is often under-appreciated, since the directness not only provides the tremendous benefit of direct methods for achieving a specific processing goal, but it also provides a unique and clear and unavoidable message and framework for defining, for the very first time, the precise data required to achieve processing objectives, e.g., for AVO analysis or on-shore or ocean bottom data processing; it is entirely direct, and unique, both in terms of the solutions it provides as well as how it explicitly communicates the types of data needed to achieve its goals, and (3) that for the modelling and generation of synthetic data, e.g., modelling primaries and multiples, that scattering theory actually agrees with everyone else in the area of seismic modelling, that is, the need in modelling for precise subsurface information, and (4) we exemplify recent progress with highlights from the projects within the M-OSRP research program, and discuss open issues and plans.

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.),

Quasi‐Monte Carlo integration for the inverse scattering internal multiple attenuation algorithm

Removal of multiples from seismic reflection data is an important preprocessing step before conventional seismic imaging and inversion in most onshore and offshore environments. While many methods have been developed and successfully used to remove free-surface multiples, internal multiple attenuation remains a challenge when working in land and complex marine environments. The inverse scattering series (ISS) internal multiple algorithm is a data-driven tool to predict all orders of internal multiples for all horizons at once, without requiring subsurface information. However, use of the multidimensional version of this algorithm has been limited due to high computational cost, which increases with the maximum output frequency in the prediction. Even with the recent advances in computer hardware, the cost of the multidimensional algorithm remains expensive. To overcome this problem, we use the quasi-Monte Carlo integration technique that can significantly improve the computational efficiency of the multidimensional ISS internal multiple algorithm. The efficiency is improved by reducing the number of samples being evaluated and combining multiple integrals into a single summation.

Deriving, explicating, and extending interferometric methods using Green's theorem

Interferometry has its foundation in Green’s theorem. Dual measurements (pressure data and its normal derivative) are required to satisfy this theorem. Interferometry makes one approximation for each normal derivative in the theorem to avoid needing dual measurements, compromising the theory and giving rise to artifacts or spurious multiples. In this paper, an analytic example is provided to explicitly show the creation of spurious multiples. An analogous analytic example using Green’s theorem will demonstrate its inner workings and the fact that the information in the normal derivative is necessary to avoid creating artifacts. A different form of Green’s theorem, where the normal derivatives are not required, will also be provided.

Green's theorem as the foundation of interferometry

A prerequisite for applying full wavefield theory to marine exploration is the completeness and proper sampling of recorded data, which can be satisfied with data extrapolation/interpolation techniques. Greens theorem can be applied to acoustic measurements of the Earths subsurface to obtain exact equations that incorporate boundary conditions for the retrieval of the Earths Greens function in positions where it was not measured. Recently, a number of papers on seismic interferometry have shown methods to reconstruct the Greens function between a pair of receivers by using data cross correlations. Current interferometry methods require dual measurements (pressure field and its normal derivative) which are not always available. The lack of dual measurements has encouraged the arrival of algorithms using high frequency and one‐way wave approximations to the normal field derivative. The approximations are compromises to the exact theory and, hence, produce artifacts. We present a unifying framework for ...