Sam T. Kaplan

Derivation of forward and adjoint operators for least-squares shot-profile split-step migration

The forward and adjoint operators for shot-profile least-squares migration are derived. The forward operator is demigration, and the adjoint operator is migration. The demigration operator is derived from the Born approximation. The process begins with a Greens function that allows for a laterally varying migration velocity model using the split-step approximation. Next, the earth is divided into horizontal layers, and within each layer the migration velocity model is made to be constant with respect to depth. For a given layer, (1) the source-side wavefield is propagated down to its top using the background wavefield. This gives a background wavefield incident at the layers upper boundary. (2) The layers contribution to the scattered wavefield is computed using the Born approximation to the scattered wavefield and the background wavefield. (3) Next, its scattered wavefield is propagated back up to the measurement surface using, again, the background wavefield. The measured wavefield is approximated by the sum of scattered wavefields from each layer. In the derivation of the measured wavefield, the shot-profile migration geometry is used. For each shot, the resulting wavefield modeling operator takes the form of a Fredholm integral equation of the first kind, and this is used to write down its adjoint, the shot-profile migration operator. This forward/adjoint pair is used for shot-profile least-squares migration. Shot-profile least-squares migration is illustrated with two synthetic examples. The first uses data collected over a four-layer acoustic model, and the second uses data from the Sigsbee 2a model.

Adaptive separation of free-surface multiples through independent component analysis

We present a three-stage algorithm for adaptive separation of free-surface multiples. The free-surface multiple elimination (FSME) method requires, as deterministic prerequisites, knowledge of the source wavelet and deghosted data. In their absence, FSME provides an estimate of free-surface multiples that must be subtracted adaptively from the data. First we construct several orders from the free-surface multiple prediction formula. Next we use the full recording duration of any given data trace to construct filters that attempt to match the data and the multiple predictions. This kind of filter produces adequate phase results, but the order-by-order nature of the free-surface algorithm brings results that remain insufficient for straightforward subtraction. Then we construct, trace by trace, a mixing model in which the mixtures are the data trace and its orders of multiple predictions. We separate the mixtures through a blind source separation technique, in particular by employing independent component analysis. One of the recovered signals is a data trace without free-surface multiples. This technique sidesteps the subtraction inherent in most adaptive subtraction methods by separating the desired signal from the free-surface multiples. The method was applied to synthetic and field data. We compared the field data to a published method and found comparable results.

Data reconstruction with shot-profile least-squares migration

We introduce shot-profile migration data reconstruction (SPDR). SPDR constructs a least-squares migrated shot gather using shot-profile migration and demigration operators. Both operators are constructed with a constant migration velocity model for efficiency and so that SPDR requires minimal information about the underlying geology. Applying the demigration operator to the least-squares migrated shot gather gives the reconstructed data gather. SPDR can reconstruct a shot gather from observed data that are spatially aliased. Given a constraint on the geological dips in an approximate model of the earth’s reflector, signal and aliased energy that interfere in the common shot data gather are disjoint in the migrated shot gather. In the least-squares migration algorithm, we construct weights to take advantage of this separation, suppressing the aliased energy while retaining the signal, and allowing SPDR to reconstruct a shot gather from aliased data. SPDR is illustrated with synthetic data examples and one ...

Seismic rock physics of steam injection in bituminous oil reservoirs

This case study explores rock physics properties of heavy oil reservoirs subject to the steam-assisted gravity drainage (SAGD) thermal enhanced recovery process (Butler, 1998; Butler and Stephens, 1981). Previous measurements —e.g., Wang et al. (1990) and Eastwood (1993)—of the temperature-dependant properties of heavy oil-saturated sands are extended by fluid-substitution modeling and wireline data to assess the effects of pore fluid composition, and pressure and temperature changes on the seismic velocities of unconsolidated sands. Rock physics modeling is applied to a typical shallow McMurray Formation reservoir (depth of 135–160 m) within the bituminous Athabasca oil-sands deposit in western Canada to construct a rock-physics-based velocity model of the SAGD process. Although the injected steam pressure and temperature control the fluid bulk moduli within the pore space, the effective stress-dependant elastic frame moduli are the most poorly known, yet most important, factors governing the changes of ...

fx Gabor Seismic Data Reconstruction

We introduce an fx Gabor reconstruction algorithm that can regularize seismic data in the presence of strong variations of dip. The available data in the fx domain are modeled via a Gabor discrete expansion. The coefficients of the Gabor expansion are

Perturbation methods for two special cases of the time‐lapse seismic inverse problem

ABSTRACT Scattering theory, a form of perturbation theory, is a framework from within which time‐lapse seismic reflection methods can be derived and understood. It leads to expressions relating baseline and monitoring data and Earth properties, focusing on differences between these quantities as it does so. The baseline medium is, in the language of scattering theory, the reference medium and the monitoring medium is the perturbed medium. The general scattering relationship between monitoring data, baseline data, and time‐lapse Earth property changes is likely too complex to be tractable. However, there are special cases that can be analysed for physical insight. Two of these cases coincide with recognizable areas of applied reflection seismology: amplitude versus offset modelling/inversion, and imaging. The main result of this paper is a demonstration that time‐lapse difference amplitude versus offset modelling, and time‐lapse difference data imaging, emerge from a single theoretical framework. The time‐lapse amplitude versus offset case is considered first. We constrain the general time‐lapse scattering problem to correspond with a single immobile interface that separates a static overburden from a target medium whose properties undergo time‐lapse changes. The scattering solutions contain difference‐amplitude versus offset expressions that (although presently acoustic) resemble the expressions of Landro (2001). In addition, however, they contain non‐linear corrective terms whose importance becomes significant as the contrasts across the interface grow. The difference‐amplitude versus offset case is exemplified with two parameter acoustic (bulk modulus and density) and anacoustic (P‐wave velocity and quality factor Q) examples. The time‐lapse difference data imaging case is considered next. Instead of constraining the structure of the Earth volume as in the amplitude versus offset case, we instead make a small‐contrast assumption, namely that the time‐lapse variations are small enough that we may disregard contributions from beyond first order. An initial analysis, in which the case of a single mobile boundary is examined in 1D, justifies the use of a particular imaging algorithm applied directly to difference data shot records. This algorithm, a least‐squares, shot‐profile imaging method, is additionally capable of supporting a range of regularization techniques. Synthetic examples verify the applicability of linearized imaging methods of the difference image formation under ideal conditions.

Two Dimensional Shot-profile Migration Data Reconstruction

We introduce two dimensional shot-profile migration data reconstruction (SPDR2). SPDR2 reconstructs data using leastsquares shot-profile migration with a constant migration velocity model. The velocity model is chosen both for efficiency and so that minimal assumptions are made about earth structure. At the core of least-squares migration are forward (de-migration) and adjoint (migration) operators, the former mapping from model space to data space, and the latter mapping from data space to model space. SPDR2 uses least-squares migration to find an optimal model which, in turn, is mapped to data space using de-migration, providing a reconstructed shot gather. We apply SPDR2 to real data from the Gulf of Mexico. In particular, we use SPDR2 to extrapolate near offset geophones.

Direct Inversion of Differenced Seismic Reflection Data For Time-lapse Structural Changes

Scattering theory is a framework from within which a range of inversion methods suited to the time-lapse/4D seismic reflection problem may be derived. A consistently posed linear inverse scattering treatment of time-lapse difference data, for instance, provides a framework both for deriving algorithms for imaging of regions of 4D structural change, and for explaining their behavior. Amongst the many implementations of linearized inverse scattering that could in principle be applied to difference data, a recent least-squares, shot-profile methodology provides the additional wherewithal to regularize the imaging problem, which may be a useful property when coping with the various repeatability issues particular to 4D. Synthetic testing demonstrates the use of difference data, under a small 4D change/small contrast assumption, to determine regions of time-lapse variation.

The essence of phase in seismic data processing and inversion

The quest of inversion is to delineate the object, the “what”, of interest. Before doing so, one must first answer the question of “where?” (e.g., migration/inversion (Weglein and Stolt, 1991)). The information concerning the “where?” lies in the phase. This abstract deals with this issue. We begin by applying POCS (projection onto convex sets) to only-phase reconstruction. We then describe a novel approach to the high resolution processing of thin layer reflection data using the cepstral approach together with a new technique of phase unwrapping. Finally, we discuss the importance of noise in the design of synthetic data sets used in the testing of algorithms and briefly remind the reader of the importance of phase in the separation of signals.