two successive profiles has yielded a theoretical indication of the angle by means of the following
equation:
a s arcsin Rrd ,
1
Ž .
Ž .
Ž where R represents the distance lag after 2D migra-
. tion , in meters, and d the distance between the two
profiles, considering the case of the investigated vertical side. 3D radar processing has already been
Ž studied Grandjean and Gourry, 1996; Grasmueck,
. 1996 , and our equation is merely a simplification in
order to obtain information on the level of inclina- tion of the cracks. As observed in Fig. 1, the shape
of the pillars does not justify the processing of a large number of radar profiles using this hypothesis.
Depending on the shape of the investigated pillars, two or three radar profiles have been developed, at a
spacing of 40 cm. Moreover, a thin carriage, includ- ing a survey wheel, has been built, allowing us to
record accurate scans, at a constant height, from the untreated surface of the pillars. Measurements were
Ž carried out in 1 day by three operators two would
. have sufficed .
2.2. Classical GPR data processing Successive processing steps have been employed
Ž .
with a commercial software WinRad from GSSI in order to localize cracks and damaged zones from the
Ž .
different sides see Fig. 2a . After a vertical high-pass Ž
. filter over 250 MHz on the profiles, the first step
consisted of normalizing the surface in distance by adding an EM velocity. For this, we compared the
thickness of different pillars and the corresponding double travel times. Results from the velocity mea-
surement fluctuated from 11.6 to 11.9 cm ns
y1
; these measurements take into account the possibility
of errors due to the 3D shape of the pillar. We then assumed a constant velocity for each pillar.
Surface normalization enables comparing the per- pendicular, or opposite, radar profiles from the same
pillar section and localizing the cracks detected from the different sides. To accomplish this step, we used
the geometrical data from a surveyor; data which were also necessary for the seismic tomographies.
Afterwards, frequency bandpass filters were ap- plied in order to remove all noise. This step is
Ž focused primarily on the major reflectors see Fig.
. 2b .
The next step involved the use of a time migration to focus the EM energy and establish a relation
between time and distance. A Kirchhoff method was used with a specific hyperbolic width of 2 m, due to
the number of scans per meter. Since the migration attenuates the amplitude of the signals, a constant
Ž .
gain value of 3 was applied on the profiles Fig. 2c . The main limitation of this process concerns the
fact that the migration itself does not take into account the topography, and distort the shape of the
surface. By compensating this distortion with a new surface normalization, we can displace reflectors
slightly from their correct position. This problem is focused mainly in the edges of the pillars, or when
the topography presents an important gradient.
Ž .
Lehmann and Green 2000 have adapted a topo- graphic migration for GPR data based on an algo-
Ž .
rithm proposed by Wiggins 1984 for seismic data collected in mountainous areas, and have shown that
topographic migration should be recommended when surface gradient exceed f 10. For our particular
application, some mere calculations can show that the positioning error remains under 0.5 m, even if
some areas present surface gradient over 10, and which can be considered as an acceptable approxi-
mation.
Finally, we concluded this processing with a Hilbert transform in order to present the reflected
Ž .
energy see Fig. 2d . The result is a map showing dark
plots that
correspond to
fracture zones
Ž .
Grandjean and Gourry, 1996 . All of these steps can be considered as classical
processing in the localization of fractures or dam- aged areas, and they provide the basis for the radar
imaging.
3. Seismic tomographic investigation
3.1. Experimental set-up Even though the geometry of a pillar is essentially
3D, we carried out our measurements in 2D. In the present case, the experimental and processing times
for 3D seismic analysis are indeed prohibitive since many pillars are being studied.
Since the major zone of interest is that around the pillar’s smallest section, it was decided to perform a
horizontal tomography at this level. In most cases, the four sides of the pillar were all accessible, thereby
allowing for good ray coverage. Similarly, we per- formed a vertical tomography with source and re-
ceiver points located on two opposite faces. The objective was twofold: to control the state of the
pillar vertically, and to ascertain whether the hori- zontal tomography plane was located in the area of
the pillar where the velocities were highest. This approach prevented against the misinterpretation of
artifacts that may arise from a 3D velocity distribu- tion where the horizontal tomography plane may be
surrounded by higher horizontal velocity zones. In such a case, the ray paths would not be in the
tomography plane, as presumed in the inversion process, and the calculations performed would be
erroneous due to an incorrect ray geometry assump- tion.
During an initial series of experiments, we deter- mined an optimum spacing for the source and re-
ceiver points such that the information contained on the tomography maps was sufficient to perform the
same diagnostic evaluation as with a larger, Asuper-
Ž abundantB number of rays typically 2000 rays in the
. horizontal tomography . In the horizontal tomogra-
phy, we located nine equidistant sourcerreceiver Ž
. points on each side
see Fig. 3a . Sources and receivers never belong to the same face; hence,
the total number of source ™receiver combinations
was reduced to a maximum of 477. In the vertical tomography, we located 18 equidistant source
points on one side and 18 equidistant receiver points on the opposite side; hence, the total number of
source
™receiver combinations was reduced to a maximum of 324. Afterwards, a surveyor provided
Ž us with all of the NGF French geographic stan-
. dards coordinate points.
A Krenz data-acquisition system of transitory sig- Ž
. nals the TRC 4000 and TRC 4011 model , with
sampling frequencies of up to 1 MHz on 10 channels Ž
. 10 bits , was used to collect and store the seismic
signals on a microcomputer. Since the shortest source–receiver travel times are around 0.1 ms, the
sampling frequency used was 1 MHz, in order to ensure acquiring a sufficient number of points for the
selection of arrival times. The source consisted of a hammer coupled with a
Ž pre-amplified Bruel and Kjaer accelerometer
no. .
4381 , with the trigger being the hammer stroke. The receivers were nine other pre-amplified Bruel and
Kjaer accelerometers. Both the receiver and source signals were recorded on the microcomputer for all
of the possible source
™receiver combinations. The time picking was carried out subsequently in the
laboratory. These arrival times and the coordinates were then fed into the RAI-2D algorithm for inver-
sion.
The tomography algorithm used in this paper, RAI-2D, was developed by the LCPC laboratory
Ž .
Cote et al., 1992 . It has already led to numerous
ˆ
Ž applications in both soil surveying Abraham et al.,
. Ž
1998 and the NDT of structures Cote and Abra-
ˆ
. ham, 1995; Abraham et al., 1996 . RAI-2D has been
inspired by the simultaneous iterative reconstruction Ž
. Ž
. technique SIRT method Gilbert, 1972 . The do-
main of investigation is discretized into a mesh of Ž
points, on which the slowness is defined see Fig. .
3b . One of the key RAI-2D features pertains to its zone of influence which, as opposed to a block-dis-
cretization grid, is used when searching for rays to calculate the slowness at a given grid point. RAI-2D
is also characterized by its use of circular analytical rays. The level of accuracy for civil engineering
purposes of this simple and rapid inversion tech- nique, which has been tested using both synthetic
and field data, is similar to that provided by more standard methods based on complex ray paths.
3.2. Detailed results on Pillar 1 It is recommended to include certain complemen-
tary information with the final velocity map in order to guarantee the quality of the survey and facilitate
its interpretation. First of all, the algorithm’s conver- gence should be tracked from a statistical point of
Ž .
view mean residual, standard deviation . Further- more, the residual statistics of each source and re-
ceiver should be checked so as to eliminate those sources andror receivers displaying out-of-scale sta-
tistical values. Secondly, since both the precision and resolution of the velocity map are linked to the ray
coverage, the plot of the ray should at least be given.
Ž . Ž .
Fig. 3. a Location of the sources and receivers on the pillar. b Discretization grid with the circular zone of influence.
For instance, in zones with few rays, the value of the velocity is less precise than in zones with well-dis-
tributed and large numbers of rays. Fig. 4 shows the horizontal and vertical seismic
tomographic results for Pillar 1. In both cases, the grid size is 0.4 m = 0.4 m, and the results listed are
those obtained after 10 iterations. Both inversions Ž
. did converge see Fig. 4c . The number of rays is
Ž .
maximized 324 in the vertical tomography. In the horizontal tomography, several sources and receivers
were eliminated due to poor statistical values. The out-of-scale values of several source and receiver
statistics can be explained by the heavily damaged surface of the pillar at certain locations. Conse-
quently, the final number of rays is reduced to 350 in the horizontal tomography.
Ž .
The vertical tomography see Fig. 4a shows that the highest velocities are located near the smallest
pillar horizontal section, as would be expected. The information on the top and bottom of the tomogra-
phy plane is less precise than in the middle due to
Ž .
ray bending see Fig. 4b . Indeed, those two areas are crossed by a very small numbers of rays and the
velocity information is here only indicative.
Ž . Ž
. Ž . Ž .
Fig. 4. a Vertical and horizontal seismic tomographies Pillar 1 . b Ray curve density. c Convergence parameters.
The horizontal tomography reveals a large dam- aged zone inside the section extending downwards
Ž .
see Fig. 4a . The rays tend to travel around this damaged area. Apart from a small zone in the upper
right-hand part, the pillar is quite damaged. Its mean Ž
y1
. velocity 3811 m s
is well below the average velocity of mechanically sound pillars at this level
Ž
y1
. around 4500 m s
.
4. Comparison and interpretation
