BASIC EQUATIONS FOR SYNTHETIC SEISMOGRAMS USING THE z‐TRANSFORM APPROACH

Geophysics ◽  
1968 ◽  
Vol 33 (3) ◽  
pp. 521-523 ◽  
Author(s):  
Enders A. Robinson
Keyword(s):  
Wave Propagation ◽  
Seismic Wave ◽  
Present Note ◽  
Layered Media ◽  
Seismic Wave Propagation ◽  
Synthetic Seismograms ◽  
The Past ◽  
Basic Equations ◽  
Fundamental Equations

In the past few years several papers have been published on the z‐transform approach to the problem of seismic wave propagation in layered media. Although these papers start with the same fundamental equations, small differences in notation make the study of the synthetic seismograms derived in these papers time consuming. In order to aid the reader, the present note gives the basic equations as well as tables which show the correspondence for the main symbols used in the papers of Goupillaud (1961), Kunetz (1964) Sherwood and Trorey (1965), Treitel and Robinson (1966), Robinson (1967), and Claerbout (1968).

Seismic versus sonic velocities: A vertical seismic profiling study

Geophysics ◽  
1984 ◽  
Vol 49 (8) ◽  
pp. 1153-1168 ◽  
Author(s):  
Robert R. Stewart ◽  
Phil D. Huddleston ◽  
Tze Kong Kan
Keyword(s):  
Wave Propagation ◽  
Layer Thickness ◽  
Seismic Wave ◽  
Short Path ◽  
Velocity Dispersion ◽  
Seismic Wave Propagation ◽  
Synthetic Seismograms ◽  
Seismic Profiling ◽  
Vertical Seismic Profiling ◽  
Sonic Log

Vertical seismic profiling (VSP) techniques provide a method to measure accurately the seismic velocity and lithologic structure near the borehole. The analysis of a VSP survey can also provide insight into seismic‐wave propagation especially when related to sonic measurements. But VSP and sonic log velocities (or traveltimes) are often found to disagree. Recent field evidence of these differences suggests that the VSP traveltimes are delayed with respect to the integrated sonic times, especially in the deep section (>3000 ft), by about 2.0 ms/1000 ft on the average. The VSP has numerous applications in exploration geophysics, such as calibrating the sonic log. It is thus important to understand why the two measurements differ. Differences in the geometries, source frequencies, and instrumental errors of the two surveys are reviewed. More detailed analysis of seismic wave propagation in the VSP shows that short‐path multiples and velocity dispersion can have a significant delaying effect on the seismic traveltimes. One‐dimensional, wide‐band VSP synthetic seismograms are generated in the frequency domain to study these effects. Different parameters (bandwidth, signal‐to‐noise, layer thickness, multiples, attenuation, dispersion) are varied in the synthetic seismograms. A comparative display of synthetic VSP traveltime minus the integrated sonic time is used to view the effects of these parameters on the synthetic traces. Reasonable variation in noise, layer thickness, bandwidth, and picking method have a small effect on traveltimes. Field data from the Anadarko Basin (4 wells) and an East Texas well are examined with the same technique. From the modeling and field examples, it is found that short‐path multiples can cause a seismic pulse delay of up to 2.0 ms/1000 ft with respect to the integrated sonic log in highly cyclically stratified sections. Velocity dispersion associated with attenuation can have a larger effect, causing up to 7.0 ms/1000 ft delay of the VSP traveltimes with respect to the integrated sonic. These wave propagation effects can explain the observed discrepancy between VSP and integrated sonic times in the deep section.

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