# 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

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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 atechnique 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.

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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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Sassen, D. S. & Peterson, J. E. 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, article, March 15, 2010; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1015623/m1/3/?rotate=270: accessed March 26, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.