The nonuniqueness of the determination of interval velocities from moveout velocities

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1209-1211 ◽  
Author(s):  
Theodor C. Krey
Keyword(s):  
Seismic Profile ◽  
Layered Earth ◽  
Interval Velocity ◽  
Offset Time ◽  
Zero Offset ◽  
Interval Velocities

In earlier papers (Krey, 1976; Hubral and Krey, 1980) I described how to obtain an equation for [Formula: see text], the nth interval velocity in an isovelocity layered earth having interfaces with arbitrary dips and curvatures, provided the velocities [Formula: see text], [Formula: see text], … to [Formula: see text] for the first n − 1 layers and the depths of the first n − 1 interfaces [Formula: see text], K = 1, 2, …, n − 1, are known and have continuous derivatives. Moreover, we assume that the zero‐offset time for the reflection from the base of the nth layer and gradient of the traveltime with respect to the horizontal coordinates are known. Finally, the normal moveout (NMO) velocity [Formula: see text] for the nth interface is observed in one arbitrary azimuth (one only), defined by ϕ, the angle between the x‐axis and the seismic profile.

THE REFRACTION SEISMOGRAPH IN THE ALBERTA FOOTHILLS

Geophysics ◽  
1956 ◽  
Vol 21 (3) ◽  
pp. 828-838 ◽  
Author(s):  
G. J. Blundun
Keyword(s):  
Operating Conditions ◽  
Radio Communication ◽  
Interval Velocity ◽  
Mississippian Limestone ◽  
Interval Velocities ◽  
Definition Of ◽  
Characteristic Interval ◽  
Consequent Improvement ◽  
Continuous Determination

In the Alberta foothills the most valuable use of the refraction seismograph is for the definition of overthrust faulting in the Mississippian limestone which is overlain by a faulted, overthrust, and overturned Cretaceous section. Normally, two refracted arrivals are recorded with characteristic interval velocities of 14,000 ft/sec and 21,000 ft/sec, the former arising from an unknown Cretaceous marker, and the latter from the Mississippian. In contrast to a shot‐range of 65,000 ft required to record the refracted arrival from the Mississippian at a depth of 10,000 ft as the first event, a range of 20,000 ft permits recording it as the later event, with consequent improvement in the quality and reliability of the data, reduces the amount of surveying required together with smaller dynamite charges, and improves radio communication. A geophone spread of 6,300 ft with single geophones at 300 ft intervals recorded on 22 traces is recommended. Both in‐line and broadside refraction with the Mississippian arrival recorded as the later event have been used successfully with certain advantages to each method. The former permits continuous determination of the interval velocity of the refracted events as well as providing two‐way control; the latter is considerably faster, and often faulting may be observed directly on the seismograms without reduction of the data. Specimen seismograms are included to illustrate the two methods. Field operating conditions pertaining to survey tolerances, shot formation, size of dynamite charges, the weathering shot as a polarity check, filtering, geophone frequency, and costs are discussed.

Description of a scanless method of stacking velocity analysis

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1596-1606 ◽  
Author(s):  
Hans J. Tieman
Keyword(s):  
Empirical Studies ◽  
Real Data ◽  
Velocity Variation ◽  
Velocity Analysis ◽  
Velocity Control ◽  
Lateral Velocity ◽  
Interval Velocity ◽  
Offset Time ◽  
Zero Offset ◽  
Traditional Approaches

Stacking velocities can be directly estimated from seismic data without recourse to a multivelocity stack and subsequent search techniques that many current procedures use. This is done as follows: (1) apply NMO to the data (over a window, for a particular common midpoint) using initial estimates for zero offset time and velocity; (2) produce two stacks by summing the data over offset after applying different weighting functions; (3) cross correlate the two stacks; and (4) translate the lag into velocity and time updates. The procedure is iterated until convergence has occurred. Referred to as ARAMVEL (U.S. Patent No. 4,813,027), the method is best implemented as an interactive continuous velocity analysis. Although very simple, both empirical studies and theoretical analysis have shown that it determines velocities more accurately than more traditional approaches based on a scan approach. Convergence is fast, with only one or two iterations usually necessary. The method is robust, as only approximate information is necessary initially. Results with real data show that the method can economically give the detailed velocity control necessary for processing data from areas with considerable lateral velocity variation, as well as provide traveltime information that can be used for sophisticated inversion into interval velocity and depth.

Traveltime inversion using transmitted waves of offset VSP data

Geophysics ◽  
1990 ◽  
Vol 55 (8) ◽  
pp. 1089-1097 ◽  
Author(s):  
Myung W. Lee
Keyword(s):  
Seismic Profile ◽  
Inversion Method ◽  
Earth Model ◽  
Computationally Efficient ◽  
Arrival Times ◽  
Traveltime Inversion ◽  
Interval Velocity ◽  
Vsp Data ◽  
First Arrival ◽  
Interval Velocities

Estimation of layer parameters such as interval velocity, reflector depth, and dip can be formulated as a generalized linear inverse problem using observed arrival times. Based on a 2-D earth model, a computationally efficient and accurate formula is derived for traveltime inversion. This inversion method is applied to offset vertical seismic profile (VSP) data for estimating layer parameters using only transmitted first‐arrival times. As opposed to a layer‐stripping method, this method estimates all layer parameters simultaneously, thus reducing the cumulative error resulting from the errors in the upper layers. This investigation indicates (1) at least two source locations are required to estimate layer parameters properly, and (2) accurate arrival times are essential for computing the dip of a layer reliably. Bulk time shifts, such as static shifts, do not affect the parameter estimation significantly if the amount of shift is not too large. The result of real and modeled VSP data inversions indicates that traveltime inversion using transmitted first‐arrival times from at least two source locations is a viable method for estimating interval velocities, reflector depths, and reflector dips.

Determination of Fresnel zones from traveltime measurements

Geophysics ◽  
1993 ◽  
Vol 58 (5) ◽  
pp. 703-712 ◽  
Author(s):  
Peter Hubral ◽  
Jörg Schleicher ◽  
Martin Tygel ◽  
Ch. Hanitzsch
Keyword(s):  
Fresnel Zone ◽  
Three Dimensional ◽  
Earth Model ◽  
Simple Method ◽  
Interval Velocity ◽  
Stratigraphic Analysis ◽  
Zero Offset ◽  
Fresnel Zones ◽  
Key Parameter

For a horizontally stratified (isotropic) earth, the rms‐velocity of a primary reflection is a key parameter for common‐midpoint (CMP) stacking, interval‐velocity computation (by the Dix formula) and true‐amplitude processing (geometrical‐spreading compensation). As shown here, it is also a very desirable parameter to determine the Fresnel zone on the reflector from which the primary zero‐offset reflection results. Hence, the rms‐velocity can contribute to evaluating the resolution of the primary reflection. The situation that applies to a horizontally stratified earth model can be generalized to three‐dimensional (3-D) layered laterally inhomogeneous media. The theory by which Fresnel zones for zero‐offset primary reflections can then be determined purely from a traveltime analysis—without knowing the overburden above the considered reflector—is presented. The concept of a projected Fresnel zone is introduced and a simple method of its construction for zero‐offset primary reflections is described. The projected Fresnel zone provides the image on the earth’s surface (or on the traveltime surface of primary zero‐offset reflections) of that part of the subsurface reflector (i.e., the actual Fresnel zone) that influences the considered reflection. This image is often required for a seismic stratigraphic analysis. Our main aim is therefore to show the seismic interpreter how easy it is to find the projected Fresnel zone of a zero‐offset reflection using nothing more than a standard 3-D CMP traveltime analysis.

Generalized form of the Dix equation for the calculation of interval velocities and layer thicknesses

Geophysics ◽  
1989 ◽  
Vol 54 (5) ◽  
pp. 659-661 ◽  
Author(s):  
Ali A. Nowroozi
Keyword(s):  
Root Mean Square ◽  
Seismic Data ◽  
Approximate Equation ◽  
Mean Square ◽  
Interval Velocity ◽  
Reflection Seismic ◽  
Interval Velocities ◽  
Horizontal Layers

Over three decades ago, Dix (1955) derived an approximate equation for the determination of interval velocity from observed reflection seismic data. Assuming a stack of m horizontal layers, with interval velocities [Formula: see text], layer thicknesses [Formula: see text], j = 1, m, and near‐vertical raypaths, Dix (1955) showed that [Formula: see text]where [Formula: see text] and [Formula: see text] are the two‐way vertical times and [Formula: see text] and [Formula: see text] are the root‐mean‐square (rms) velocities to interfaces j + 1 and j, respectively.

A method for direct estimation of interval velocities

Geophysics ◽  
1982 ◽  
Vol 47 (12) ◽  
pp. 1657-1671 ◽  
Author(s):  
Philip S. Schultz
Keyword(s):  
Estimation Procedure ◽  
Estimation Accuracy ◽  
Substantial Contribution ◽  
Interval Velocity ◽  
Direct Estimation ◽  
Enhanced Resolution ◽  
Prior Estimate ◽  
Sensitive Function ◽  
Interval Velocities ◽  
Ray Parameter

The most commonly used method for obtaining interval velocities from seismic data requires a prior estimate of the root‐mean‐square (rms) velocity function. A reduction to interval velocity uses the Dix equation, where the interval velocity in a layer emerges as a sensitive function of the rms velocity picks above and below the layer. Approximations implicit in this method are quite appropriate for deep data, and they do not contribute significantly to errors in the interval velocity estimate. However, when the data are from a shallow depth (vertical two‐way traveltime being less than direct arrival to the farthest geophone), the assumption within the rms approximation that propagation angles are small requires that much of the reflection energy be muted, along with, of course, all the refraction energy. By means of a simple data transformation to the ray parameter domain via the slanted plane‐wave stack, three types of arrivals from any given interface (subcritical and supercritical reflections and critical refractions) become organized into a single elliptical trajectory. Such a trajectory replaces the composite hyperbolic and linear moveouts in the offset domain (for reflections and critical refractions, respectively). In a layered medium, the trajectory of all but the first event becomes distorted from a true ellipse into a pseudo‐ellipse. However, by a computationally simple layer stripping operation involving p‐dependent time shifts, the interval velocity in each layer can be estimated in turn and its distorting effect removed from underlying layers, permitting a direct estimation of interval velocities for all layers. Enhanced resolution and estimation accuracy are achieved because previously neglected wide‐angle arrivals, which do not conform to the rms approximation, make a substantial contribution in the estimation procedure.

SEISMIC VELOCITIES FROM SURFACE MEASUREMENTS

Geophysics ◽  
1955 ◽  
Vol 20 (1) ◽  
pp. 68-86 ◽  
Author(s):  
C. Hewitt Dix
Keyword(s):  
Accurate Determination ◽  
Careful Study ◽  
Seismic Velocities ◽  
Multiple Reflections ◽  
Seismic Reflection Data ◽  
Reflection Data ◽  
Surface Measurements ◽  
Secondary Objective ◽  
Interval Velocities

The purpose of this paper is to discuss field and interpretive techniques which permit, in favorable cases, the quite accurate determination of seismic interval velocities prior to drilling. A simple but accurate formula is developed for the quick calculation of interval velocities from “average velocities” determined by the known [Formula: see text] technique. To secure accuracy a careful study of multiple reflections is necessary and this is discussed. Although the principal objective in determining velocities is to allow an accurate structural interpretation to be made from seismic reflection data, an important secondary objective is to get some lithological information. This is obtained through a correlation of velocities with rock type and depth.

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