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.
The ability of reflection seismic data to uniquely determine the subsurface velocity has been uncertain. This paper uses a tomographic approach to study the resolution of typical seismic survey configurations. The analysis is first carried out in the spatial Fourier domain for the case of a single horizontal reflector. It is found that for a ratio of maximum offset to layer depth of one, the lateral resolution is very low for velocity and interface depth variations of wavelengths of approximately two‐and‐a‐half times the layer thickness. The resolution improves with an increase in the ratio of maximum offset to layer depth. The results of the analysis in the Fourier domain are confirmed by results from a least‐squares tomographic algorithm. It is found that regularization of the tomography by adding damping terms suppresses the spurious oscillations resulting from the areas of low resolution at the expense of loss of resolution at the shorter spatial wavelengths. Analysis of the single layer response for 3‐D survey geometry shows that a 3‐D acquisition with multiazimuthal coverage has the potential to significantly improve velocity determination.
The normal equations for the discrete Wiener filter are conventionally solved with Levinson’s algorithm. The resultant solutions are exact except for numerical roundoff. In many instances, approximate rather than exact solutions satisfy seismologists’ requirements. The so‐called “gradient” or “steepest descent” iteration techniques can be used to produce approximate filters at computing speeds significantly higher than those achievable with Levinson’s method. Moreover, gradient schemes are well suited for implementation on a digital computer provided with a floating‐point array processor (i.e., a high‐speed peripheral device designed to carry out a specific set of multiply‐and‐add operations). Levinson’s method (1947) cannot be programmed efficiently for such special‐purpose hardware, and this consideration renders the use of gradient schemes even more attractive. It is, of course, advisable to utilize a gradient algorithm which generally provides rapid convergence to the true solution. The “conjugate‐gradient” method of Hestenes (1956) is one of a family of algorithms having this property. Experimental calculations performed with real seismic data indicate that adequate filter approximations are obtainable at a fraction of the computer cost required for use of Levinson’s algorithm.
Abstract
Seismic and gravity observations were carried out in the active volcanic area of Katmai in the summer of 1965. A determination of hypocenters has been aftempted using S and P arrivals at a station located at Kodiak and two stations located in the Monument. However, in most cases, deviations of travel times from the Jeffreys-Bullen tables were rather large. Therefore hypocenters are not well located. A method based on P- and S-wave arrivals yields a Poisson's ratio of 0.3 for the upper part of the mantle under Katmai. This higher value is probably due to the magma formation. The average depth to the Moho from seismic data in the same area is 38 km and 32 km under Kodiak. Using Woollard's relation between Bouguer anomaly and depth to the Moho, a small mountain root under the volcanoes with a depth of 34 km was found dipping gently up to 31 km on the NW side. The active volcanic cones are located along an uplift block. This block is associated with a 35 mgal Bouguer anomaly. The Bouguer anomaly contour map for the Alaska Peninsula is given and an interpretation attempted.
Assessing the repeatability of reflection seismic data in the presence of complex near-surface conditions CO2CRC Otway Project, Victoria, Australia