# Global optimization of data quality checks on 2-D and 3-D networks of GPR cross-well tomographic data for automatic correction of unknown well deviations Page: 3 of 4

avoided to prevent the possibility of inadvertently
removing natural heterogeneity trends. Our model is a
simple straight well with a single deviation angle with the
pivot point located at the surface location of the well.
The model of a network of interconnected tomograms is
most efficiently described in terms of network notation.
Two and three-dimensional networks of GPR tomograms
are described by set of nodes and connections, or edges,
where the nodes represent the boreholes of the GPR TX's
and RX's, and the tomography data represents the
connections between the nodes.
The coordinate of each transmitter and receiver (x',,y',,z'.)
is given by
x =R sin(O ) cos )+x)x
y =R sin(O )cos( )+y
z =Rcos(0O)
where R. is the distance down the borehole of the TX or
RX station, 0. is the angle of deviation of well n, and q0 is
the azimuth of the deviation, and (x,,y,,z,) are the
coordinates of the station when perfectly vertical wells are
assumed. The angle of deviation 0. for each well is
constrained to less than or equal to 100, while the azimuth
is free to rotate though all 3600. In the case of a 2-D
network q, is held constant.
The merit function of the optimization consists of
quantitative checks on the continuity of velocity between
connections of the network, and checks to ensure that there
is minimal correlation between apparent velocity and the
take-off angle. The first quality control is given by,
V.- V
QC,-
nENiEC n
where v.- is the mean apparent velocity of the subset of the
tomography data C. connected to node n, v, is the velocity
of connection i of the subset C., and N is the total number of
nodes. The second quality control is given by,
C
QC2= RU+R d
c=1
where Rc" is the correlation coefficient of connection c for
upgoing take off angles versus apparent velocity and Rcd is
the correlation coefficient of connection c for the down
going rays. The sum of these two measures of data quality
defines the merit function. Together the two measures
provide counter-weight between continuity of velocity
between tomograms and the quality of individual
tomograms. To minimize this merit function, and thus
maximize quality, we utilize a global optimization
algorithm known as particle swarm optimization (PSO).
The PSO algorithm of Kennedy and Eberhart (1995) is a

technique based loosely on the observed behavior of large
swarms. This algorithm optimizes the quality control merit
function by moving "particles", or solutions of the merit
function, around in a search space towards the optimal
solution. The movements of the particles are controlled by
communication between the particles of the best solution of
all of the particle's past positions, and the best current
solution. We utilized the PSO FORTRAN code of Mishra
(2006), which has been extensively tested for finding
global minima of complicated test functions.
Validation on synthetic data for a 2-D network
To test our method we utilized travel-times picked from a
numerical model of electromagnetic propagation through a
geostatistical model of the distribution of dielectric
permittivity and electrical conductivity (Figure 3). The
numerical model is a 2-D TE mode implementation the
ADI-FDTD algorithm of Namiki (1999), and the travel
times were automatically picked with the algorithm of
Crosson and Hesser (1983). The structure and variance of
the geostatistical model is based on high resolution
geophysical logs acquired at our test field site. Source
positions were situated at the location of the open circled
on the edges and the RX locations were situated along the
black lines in figure 3.
0.0 ReIive

MCdel

2.5
-5.0
5.0 C
00

Dielectric

, 25 5.0-
A 7.5-
012.5-
30K
15.0-

I

I

0.0 2.5 5.0
X (meters
Figure 3. The geometry of deviated wells between two vertical
wells.
In this tomography network the two end wells are
unconstrained by neighboring tomography data. Therefore,
the geometry of these end wells was held constant, and
only the central well was allowed to update during the
inversion. The results of our inversion method are
summarized in table 1. To check how robust the technique
is to the presence of other data errors such as static time-
zero shifts, anisotropy, and incorrect estimation of well
separation, additional constant time shifts were added to the
data in subsequent test. Additionally, the possibility of a
change in time-zero between the two tomography data sets

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