the characteristic ‘‘hyperbola’’ scattering shape caused by the metal pipe. If the problem were
utility location, then the problem is solved at this point by noting the presence of a metallic
reflector at the horizontal position of the top of the hyperbola. By rotating the antenna electric
field orientation while centered above the pipe location indicated by the hyperbola, the az-
imuthal strike orientation of the pipe may also be quickly determined from the change in polar-
ization response. By measuring the change in polarization with different antenna geometries
Žoften called HH, VV and HV polarizations;
. van Zyl and Ulaby, 1990 and solving for the
Stokes–Mueller polarization matrices, both the strike and dip of the pipe may be determined.
For the remainder of this example, polarization will be ignored as the data acquisition was setup
to be maximally coupled between the antenna electric field and the long axis of the pipe.
3. Estimating the pipe depth and size
However, if more information is required, the first processing step will be to remove the arti-
facts in the data. In Fig. 2, the time zero has been set at the first energy of arrival. Time zero
is a function of the system timing, cable lengths, and antenna positioning. The horizontal black
line is positioned at time zero and also shows the position of the line plotted across the top of
the image, which now shows the uneven mark locations. The average of all the scans has been
Ž accumulated and removed
background re- .
moval to eliminate the antenna ringing and
horizontal banding across the image. The back- ground removal also removes other horizontal
features such as flat lying geology and the surface of the Earth, so time zero had to be
located first.
Ž In Fig. 3, a median gradient filter Paeth,
. 1990; Pratt, 1991; Weeks et al., 1993 has been
applied to remove the radio frequency interfer- ence from nearby wireless phone and portable
Ž .
radio transmissions see Appendix A . In Fig. 4,
Fig. 2. The data of Fig. 1 with time zero determined and Ž
. the average scan removed background removal .
an image processing contrast enhancement Ž
. stretch Pratt, 1991; Zuiderveld, 1994 has been
applied to bring out details in the image. This last step loses all the absolute amplitude infor-
Ž .
mation that will be recovered later and en- Ž
hances not only geological details note the
appearance of several small hyperbolas caused .
Ž by rocks but also noise the radio frequency
interference as diagonals across the bottom of the image from the computer inside the SIR-
. 10A q control box . These steps are done to
improve the ability to clearly see the tails of the pipe hyperbola, to be used in determining veloc-
ity and size of the pipe.
In Fig. 5, the image has changed size and shape slightly as a spline rubber sheeting pro-
Ž .
cess Bochicchio, 1988 is used with the loca- tions of the marks to correct the horizontal
geometry of the image. This allows the scans numbered across the bottom of the image to
Fig. 3. The data of Fig. 2 with a median gradient filter applied to remove radio noise.
also be labelled as horizontal traverse distance in meters. This may also be done in 3D to
correct for topographic relief, but requires the vertical time scale first be converted to a depth
scale. The survey in this case is across flat ground. In the process of doing the rubber
sheeting, it was found that some scans were missing in making a uniformly spaced image, so
an interpolation was used to fill in, with the locations of the filling indicated across the top
Ž of the plot as small spikes two on the left half
. and seven on the right half of the image . This
interpolative filling is just to make a better looking image.
In Fig. 6, a mathematical function has been fitted to the hyperbola shape in the data. The
function is a slice from a conic section that is often called ‘‘hyperbola’’. In the earlier figures,
note the hyperbola contains considerable varia- tion in amplitude along its locus as well as some
deviation from the ideal hyperbola shape. These are caused by geological heterogeneity and may
be used to describe that heterogeneity. These are also causes of problems in attempting to
refocus the hyperbola by synthetic aperture, phase unwrapping, or migration processing
ŽYilmaz, 1987; Figs. 43 and 44 of Powers,
. 1995; Ghiglia and Pritt, 1998 . The slopes of the
asymptotes on either side are controlled by the velocity of propagation and thus calibrate the
dielectric permittivity between the antenna and the pipe, and give a calibration and conversion
of the two-way travel time into depth. The radius of curvature at the peak of the hyperbola
Ž and the lengths of the asymptotes by taking
. into account the antenna pattern give the size
of the object causing it. The ellipse drawn within the hyperbola indicates the size of the object,
assuming the object is a circular cylinder with axis perpendicular to the plane of the data im-
Fig. 4. The data of Fig. 3 with an image processing histogram contrast stretch applied.
Fig. 5. The data of Fig. 5 after spline rubber sheeting to the marks along horizontal traverse and in fill interpola-
tion.
age. The circle is distorted into an ellipse by vertical exaggeration, as no correction has been
performed to make the vertical and horizontal axes the same scale dimension. Also indicated
in Fig. 6 is the position of the near-field of the antenna, as the hyperbola shape is fit with a
far-field ray tracing model assumption. At this point, the data processing and hyperbola fitting
indicate a pipe centered 2.78 m from the begin- ning of the traverse, at a depth of 0.34 m in a
soil with relative dielectric permittivity of 4.0, and with a diameter of 0.41 m. These numbers
can be further refined.
In Fig. 7, an image processing hyperbola mask has been applied to the data to collapse or
Ž focus the hyperbola using a process similar to
. migration; Yilmaz, 1987 . The image now shows
only the scattering cross-section of the visible radius of curvature of the pipe. By looking at
Ž other hyperbolas in the image such as those
. from the rocks , their over or under migration
Ž focusing residual hyperbolic shapes pointed up-
. wards or downwards indicates the variability of
the velocity and hence dielectric permittivity throughout the section.
4. Refining the pipe depth
