Zero-offset Research Articles

Migrating around on hyperbolas and parabolas

The term “migration” in seismic reflection imaging is traditionally used to describe a strategy by which one can transfer primary time‐domain reflections (e.g., as observed in a common‐shot or common‐offset section) into their correct spatial positions on the searched‐for subsurface reflector. It is also employed to describe a strategy to transfer primary reflections into their time‐migrated positions directly in the time domain. More recently, the term migration has also been used to describe the process by which primary reflections, as observed in a common‐offset (CO) section, are transferred to the positions where they would be observed in the zero‐offset (ZO) section. The latter procedure is referred to as migration‐to‐zero offset (MZO). It is the subject to which we want to contribute.

Seismic constant‐velocity remigration

When a seismic common midpoint (CMP) stack or zero‐offset (ZO) section is depth or time migrated with different (constant) migration velocities, different reflector images of the subsurface are obtained. If the migration velocity is changed continuously, the (kinematically) migrated image of a single point on the reflector, constructed for one particular seismic ZO reflection signal, moves along a circle at depth, which we call the Thales circle. It degenerates to a vertical line for a nondipping event. For all other dips, the dislocation as a function of migration velocity depends on the reflector dip. In particular for reflectors with dips larger than 45°, the reflection point moves upward for increasing velocity. The corresponding curves in a Time‐migrated section are parabolas. These formulas will provide the seismic interpreter with a better understanding of where a reflector image might move when the velocity model is changed. Moreover, in that case, the reflector image as a whole behaves to some extent like an ensemble of body waves, which we therefore call remigration image waves. In the same way as physical waves propagate as a function of time, these image waves propagate as a function of migration velocity. Different migrated images can thus be considered as snapshots of image waves at different instants of migration velocity. By some simple plane‐wave considerations, image‐wave equations can be derived that describe the propagation of image waves as a function of the migration velocity. The Thales circles and parabolas then turn out to be the characteristics or ray trajectories for these image‐wave equations.

Migrating around in circles: Part II

This paper proposes (1) simulating a zero‐offset (ZO) section from common‐offset (CO) sections of different offsets and (2) a possibility to find the best velocity to do this simulation. More precisely, the paper discusses the kinematic aspects of a common‐reflection‐point (CRP) stack for an unknown velocity model.

Offset dependent amplitudes: Fault curvature effects and dip move-out corrections

Equations have been presented previously which predict that reflector curvature can affect significantly seismic reflection amplitudes at both zero and nonzero source-receiver offsets. Here the fact that faults are generally concave-upward is used to examine the curvature effect for compaction-driven faults which the sediments have both exponential and logarithmic porosity decreases with increasing depth.

Field measurement of leaf water potential with a temperature‐corrected in situ thermocouple psychrometer

Abstract. The construction and evaluation of a temperature‐corrected in situ thermocouple psychrometer for measurement of leaf water potential (Ψ) is described. The instrument utilized two chromel‐constantan thermocouples which allowed for detection of both the psychrometric zero offset and the temperature differential between the sample and the Peltier measuring junction. The psychrometer was subjected to stable temperature gradients while in contact with reference solutions of sodium chloride, and the effects of thermal gradients were quantified. Regression analysis indicated that temperature differentials were responsible for errors in water potential determinations of approximately –7.73 MPa°C−1. When installed on leaves of field‐grown cotton (Gossypium hirsutum L.), corn (Zea mays L.) and soybean (Glycine max L. Merr) the instrument detected temperature differentials up to 0.1°C (–6.0 μV) which were associated with relatively small shifts in psychrometric zero offsets (–0.05––0.75 μV). Results indicated that substantial errors in apparent Ψ were caused by non‐isothermal conditions between the leaf and the psychrometer measuring junction. The relative magnitude of these errors could be quantified and the corrected results showed good agreement with conventional psychrometric determination of Ψ on excised samples during a diurnal cycle.

Three-dimensional target-oriented pre-stack migration

The main limitations of current 3D migration techniques do not lie in the zero-offset migration algorithm itself but in the preceding stacking process which generates the zero-offset input data. In conventional common-midpoint (CMP) stacking four problems may occur: 1. The members of a CMP gather may have different individual midpoints and, therefore, the concept of 'common midpoint' may not actually exist. 2. In situations with complex geology, reflection times in a CMP gather may not be described by a hyperbolic relationship and so the concept of 'stacking velocity' may not exist. 3. Conflicting structural dips require different stacking velocities. 4. One stacked event may represent information from different reflection points. In the seventies and early eighties refined stacking procedures were developed, based on the dip moveout (DMO) concept (Bolondi et al. 1982; Hale 1984). These DMO-oriented stacking techniques (followed by post-stack migration) can be successfully applied if the aforementioned second problem does not occur. However, in 3D data sets generally all four problems occur and, as a consequence, the quality of post-stack migrated data is often far from optimum. Therefore, particularly for complicated subsurface structures, alternative techniques must be developed. The ideal procedure would be full pre-stack migration by singleshot record inversion (SSRI), followed by zero-offset (ZO) stacking (Berkhout 1984). However, for the 3D case this promising technique is still too laborious. It is obvious that, given the limitations of today's computational power, a more practical approach to 3D prestacking migration is required. In many practical situations seismic interpreters are mainly interested in an accurately positioned and well-resolved image of a prespecified target zone. Hence, much work can be saved by the following two-stage procedure: 1. Apply conventional CMP processing and two-pass migration for an initial 3D evaluation of the total area under investigation. 2. Apply full 3D pre-stack migration to specific areas of interest ('target-oriented' stage). In this paper we propose a full 3D target-oriented prestack migration scheme and discuss various theoretical and practical aspects. Also we demonstrate the validity of the target-oriented approach with the aid of a physical scale model (water tank) data example.

Statistical assessment of some errors in thermocouple hygrometric water potential measurement

A model is presented for the calculation of some errors in the measurement of water potential for individual thermocouple hygrometers used in the dewpoint or psychrometric mode. The model is based on calculation of the relative standard error in measured thermocouple psychrometric water potential as a function of temperature. Sources of error in the psychrometric mode were found in the calibration of the instrument as a function of water potential and temperature and in voltage (due to electronic noise and zero offsets) and temperature measurement in the field. Total error increased as temperature decreased, approaching a value usually determined by the shape of the thermocouple junction, electronic noise (at low voltages < 1 μV) and errors in temperature measurement. At higher temperatures, error was a combination of calibration errors, electronic noise and zero offset voltages. For a measured water potential of — 1250 kPa, field calibration data for a number of leaf psychrometers contained total errors that ranged between 6 (at 0°C) and 2% (at 45°C) for the better psychrometers and between 11 (at 0°C) and 5% (at 45°C) for the worst, assuming zero offsets were 0.5 μV. Zero offsets were < 0.7 μV at all times. The dewpoint errors arose from calibration of the dewpoint hygrometer as a function of water potential, not temperature correcting the calibration slope to other temperatures, setting the dewpoint coefficient incorrectly and errors in voltage and temperature measurement. The total error also increased as temperature decreased, because of the differences in temperature sensitivity between dewpoint and psychrometric calibration constants. Consequently, the major source of error in the dewpoint mode arose from the difficulty in determining the dewpoint coefficient. This error, which is temperature dependent, contains three subcomponent errors; the temperature dependence, random variation associated with determining the temperature dependence and error in setting the correct value of the dewpoint coefficient. Calibration and extrapolation errors were smaller than those of the psychrometric technique. Typically, the error in a dewpoint measurement varied between ∼ 6 and 2% for the best leaf hygrometer and between 10 and 3% for the worst for temperatures beween 0 and 45°C, respectively. At low temperatures, the dewpoint technique often has no advantage over the psychrometric technique, in terms of measurement errors.

The Effect of Zero Offset Changes on Pressure Transducer Performance

The Effect of Zero Offset Changes on Pressure Transducer Performance

Capacitor-fed parallel ‘chopper’ as a phase-sensitive demodulator

The capacitor-fed parallel `chopper? is shown to make a phase-sensitive demodulator with good zero stability, good linearity and a wide dynamic range for carrier frequencies of up to about 100kHz. Design problems are discussed in detail. A balanced version gives very low zero offsets, and allows fast response with relatively simple filters.