The compressional wave reflection coefficient R(θ) given by the Zoeppritz equations is simplified to the following: [Formula: see text] The first term gives the amplitude at normal incidence (θ = 0), the second term characterizes R(θ) at intermediate angles, and the third term describes the approach to critical angle. The coefficient of the second term is that combination of elastic properties which can be determined by analyzing the offset dependence of event amplitude in conventional multichannel reflection data. If the event amplitude is normalized to its value for normal incidence, then the quantity determined is [Formula: see text] [Formula: see text] specifies the normal, gradual decrease of amplitude with offset; its value is constrained well enough that the main information conveyed is [Formula: see text] where [Formula: see text] is the contrast in Poisson’s ratio at the reflecting interface and [Formula: see text] is the amplitude at normal incidence. This simplified formula for R(θ) accounts for all of the relations between R(θ) and elastic properties first described by Koefoed in 1955.
INTRODUCTIONEstimation of reflector depth and seismic velocity from seismic reflection data can be formulated as a general inverse problem. The method used to solve this problem is similar to tomographic techniques in medical diagnosis and we refer to it as seismic reflection tomography.Seismic tomography is formulated as an iterative Gauss-Newton algorithm that produces a velocitydepth model which minimizes the difference between traveltimes generated by tracing rays through the model and traveltimes measured from the data. The input to the process consists of traveltimes measured from selected events on unstacked seismic data and a first-guess velocity-depth model. Usually this first-guess model has velocities which are laterally constant and is usually based on nearby well information and/or an analysis of the stacked section. The final model generated by the tomographic method yields traveltimes from ray tracing which differ from the measured values in recorded data by approximately 5 ms root-mean-square.The indeterminancy of the inversion and the associated non uniqueness of the output model are both analyzed theoretically and tested numerically. It is found that certain aspects of the velocity field are poorly determined or undetermined.This technique is applied to an example using real data where the presence of permafrost causes a nearsurface lateral change in velocity. The permafrost is successfully imaged in the model output from tomography. In addition, depth estimates at the intersection of two lines differ by a significantly smaller amount than the corresponding estimates derived from conventional processing.Estimation of velocity and depth is often an important step in prospect evaluation in areas where lithology and structure undergo significant lateral change. Depth estimation is usually accomplished by converting zero-offset traveltimes, interpreted from a stacked section, to depth using a velocity field obtained from a normal-movement (NMO) analysis. In areas with complex lateral changes, a depth migration technique may be necessary to obtain the correct depth estimate (Lamer et aI., 1981). Both of these methods require an accurate representation of the root-mean-square (rms) velocity field. However, the stacking velocities used for such analyses can deviate significantly from rms velocities because analysis of stacking velocities assumes that the medium is laterally invariant and that traveltime trajectories for reflection events in CDP gathers are hyperbolic.Media vary laterally due to either reflector dip or curvature, or due to lateral velocity variations, or both. A large portion of the effect of reflector dip or curvature on the stacking velocity can be removed approximately by first migrating commonoffset panels with a first-guess velocity function, and then recalculating the stacking velocity in the migrated common-depthpoint (CDP) gathers (Doherty and Claerbout, 1976). The influence of lateral variations in velocity on the stacking velocity cannot be corrected this way. For lateral vari...
We study aeromagnetic maps from four areas in the Western United States: Utah High Plateaus, Yellowstone National Park, Southern Great Basin, and Uinta Basin. In the first area we try to infer depth-to-bottom of magnetic source by comparison of individual anomalies with theoretical fields of vertical prisms. We conclude that this task is impossible, at least for the few anomalies considered. Assuming a bottomless prism a fit can be obtained which is not significantly worse than the best fit obtainable with any prism model. This is documented by both nonlinear and linear inverse theory.We then turn to a spectral theory like that of Spector & Grant. The advantage of spectral analysis is that the broad, low-amplitude negative flanks of positive anomalies are included even though they may be covered by adjacent positive anomalies. The disadvantage is sensitivity to finite data length: the spectra are interpreted with models derived by Fourier transformation in an infinite domain. Anomalies which are only partly included in the map can degrade the estimated spectrum. We try several techniques to deal with this problem, but none seem universally applicable. These techniques include elimination of part of the map and subtraction of a low-order polynomial. For computation of the spectral estimate our favourite procedure is: (1) average autocovariance over azimuth, (2) extrapolate to large lags as r-5, (3) Hankel transform the radial autocovariance profile to a radial power profile.Our usual model for interpretation of the low-frequency spectra is a truncated right circular cone defined by the four parameters ZT (depth to top), ZB (depth to bottom), RT (top radius) and RB (bottom radius). For the Utah High Plateaus and Uinta Basin we find a distinct preference for sloping sides, i.e. RB > RT. In analysing individual anomalies as well as spectra we find anticorrelation of ZT and ZB. We attribute this to the centroid of the average source body being better determined than its thickness.
We present a new and more detailed derivation of the formula due to O’Doherty and Anstey (1971) for filtering a transmitted wavelet by short‐period multiples. We use a continuous rather than discrete formulation and regard the impedance as a random variable. The mean pressure represents the downgoing wavelet as progressively modified by short‐period multiples, while the deviations from the mean field are essentially the upcoming reflections. Standard procedures and approximations lead to the dispersion relation of the mean pressure field. To describe the stratigraphic filtering, we introduce a dimensionless complex quantity F such that a wavelet which has traveled a time ΔT is modified by the filter [Formula: see text]. From the Kramers‐Kronig relation appropriate for a causal earth, F has real and imaginary parts [Formula: see text] and [Formula: see text] Where Q and δt define the apparent attenuation and time delay, both of which may depend upon frequency, R is the spectrum of reflection coefficients, and [Formula: see text] is the spectrum of the impedance fluctuations. The first equation means that the apparent attenuation depends only on the impedance fluctuations with spatial period half the seismic wavelength; the second means that the stratigraphic filter is minimum‐phase. We also show that changes of impedance on a spatial scale much larger than a seismic wavelength modify the amplitude so as to conserve energy, but they do not filter the waveform.
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