# Mapping the vertical flank of a salt dome

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 13 485 - 496 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 13.6a

Two media of velocities ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$ are separated by a vertical plane interface. A source is located at ${\displaystyle S}$ on the surface of the high-velocity medium and a geophone at point ${\displaystyle G}$ in a well in the other medium. If ${\displaystyle t_{A}}$ is the traveltime of a wave from ${\displaystyle S}$ to ${\displaystyle G}$ along the path SAG where ${\displaystyle A}$ is on the salt/sediment interface, discuss the locus of points located at the intersection of arcs centered at ${\displaystyle G}$ and ${\displaystyle S}$ having radii ${\displaystyle V_{1}\Delta t_{1}}$ and ${\displaystyle V_{2}\Delta t_{2}}$, respectively, where ${\displaystyle \Delta t_{1}+\Delta t_{2}=t_{A}}$.

### Solution

Any path where the interface is tangent to the locus curve satisfies the observed traveltime. To draw the locus we try a series of points ${\displaystyle {\rm {B}}_{i}}$ such that ${\displaystyle ({\rm {SB}}_{i}/V_{2}+{\rm {GB}}_{i}/V_{1})=t_{\hbox{A}}}$. If a number of loci can be drawn for various geophone (or source) loctions, then their common tangent must locate the interface.

While this problem illustrates the concept, it can also be applied in three dimensions to allow for situations where ${\displaystyle V_{1}\Delta t_{1}}$ and ${\displaystyle V_{2}\Delta t_{2}}$ do not lie in the same plane.

Table 13.6a. Survey to define flank of a salt dome.
${\displaystyle z({\rm {m}})}$ ${\displaystyle t({\rm {s}})}$ ${\displaystyle z({\rm {m}})}$ ${\displaystyle t({\rm {s}})}$ ${\displaystyle z({\rm {m}})}$ ${\displaystyle t({\rm {s}})}$
500 0.44 1250 0.52 2000 0.6
750 0.46 1500 0.56 2250 0.67
1000 0.49 1750 0.60 2500 0.70

## Problem 13.6b

An outcropping salt dome has roughly vertical flanks. A source is located on the salt and a geophone is suspended in a vertical well in the sediments 1600 m from the source point. Determine the outline of the salt dome from the ${\displaystyle t-z}$ data in Table 13.6a. Take the velocities in the salt and adjacent sediments as 5.00 and 3.00 km/s.

### Solution

The usual method of resolving this problem is to prepare a vertical section through the well W and source point S, then draw a series of circles centered at S with radii equal to the distances traveled in salt for convenient time intervals such as 0.2 s, 0.4 s, etc., the circles being labeled with the time value. A second set of concentric circles using the sediment velocity is drawn on a transparency; the center of these circles is then placed over a geophone position, and intersections of the two sets of circles are marked wherever the time values of the two radii add up to give the traveltime to the geophone.

An alternative method is to dispense with the circles and use only the radii. The centers of the circles and the extremities of the radii are marked along the edges of two narrow strips of light cardboard (one for salt, one for sediments), the time values being marked as before (the marks and the centers should be on opposite sides of the two strips to permit accurate determinations of the intersections). Map pins can be used to attach the zero points of the two strips to the vertical section, one at the source point and the other at the geophone location in the well. Intersections are found as before and marked directly on the section. After curves have been plotted for each geophone position, the flank is outlined by the curve through the apices of the curves. The results are shown in Figure 13.6a.

Figure 13.6a.  Construction for mapping the flank of a salt dome. Small circles mark the intersections of arcs with centers ${\displaystyle S}$ and ${\displaystyle G_{\hbox{i}}}$.