Ph. D. dissertation of Jean-Paul Van Gestel; Acknowledgements

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Structure and tectonics of the Puerto Rico-Virgin Islands platform
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Multi-configuration Ground Penetrating Radar data.

Contents Preface Acknowledgements Abstract Chapter 1 Chapter 2 Chapter 3 References Vita pdf-version

Chapter 3: Multi-configuration Ground Penetrating Radar data.

3.1 Introduction

3.1.1 Ground Penetrating Radar

The Ground Penetrating Radar (GPR) is a geophysical instrument, which makes it possible to create an electromagnetic reflection profile, or radargram, of the top-layers of the earth. With this non-destructive method, electromagnetic anomalies can be mapped and shallow geological features can be studied. The main areas of application are geotechnical, archeological, environmental and hydrological investigations. The applications of the instrument can be divided into two categories: 1) the detection and identification of buried objects, 2) local subsurface characterization.

Traditional GPR surveys are conducted using two similar antennas. The first one is dedicated as the transmitting antenna, or transmitter (Davis and Annan, 1989). The transmitter radiates a short pulse of electromagnetic energy into the ground. If this wave encounters an electromagnetic discontinuity it is reflected back to the second antenna, the receiving antenna, or receiver. In traditional recordings the two antennas are held perpendicular to the line of data collection and parallel to each other with a fixed distance between them. The transmitter produces a wave with a polarization parallel to its long axis. In the receiver only the component parallel to the long axis of the receiver is recorded. Other configurations where one or both of the antennas are rotated, are rarely used. In this study it is shown that multiple transmitter-receiver configurations can be used to increase knowledge of buried objects and improve the final image of the target.

The principles of GPR are based on electromagnetic wave theory and can be compared to seismic reflection measurements, which are based on elastic wavefield theory. The traditional method of GPR data collection and processing is similar to single channel seismic reflection methods. One difference between GPR data and seismic reflection data is the higher frequency content of the GPR data, which lies in the range of 100 MHz to 1 GHz, compared to 10-100 Hz for acoustic methods (Davis and Annan, 1989). As a result the penetration depth of the GPR measurements decreases to about 40 meter, and the resolution increases. Another difference is the presence of the conductivity term in electromagnetic wavefield propagation, which makes electromagnetic theory less straightforward. It also causes the wave to attenuate faster, especially in materials where this conductivity is high, like clays and salty water.

The most important difference, which is the main focus of this study, is the difference in source character. The source used in GPR surveys, is an antenna, which is a dipole, while in acoustics the dynamite or airgun source is a monopole. As the electromagnetic source is a dipole, it has directivity, and the recorded wavefield is dependent on the angle of orientation of the transmitter and receiver to the orientation and location of the targets. This characteristic can be used to extract additional information about the target and improve the processing sequence applied to GPR data. This phenomenon has rarely been studied and in this study the value of collecting data in unconventional configurations is shown.

3.1.2 Previous work done in multi-configuration GPR data collection

Some field and theoretical studies were conducted to understand the value of multi-configuration GPR data collection. Differences are observed in the recordings using co-polarized and cross-polarized configurations, and when changing the orientation of the line of data collection to the orientation of the objects. However, these differences are never used to extract information about the targets.

Cook (1970) studied the electrical properties of bituminous coals and concluded that the velocity and the attenuation are dependent on the angle between the orientation of the electromagnetic wave and the orientation of the clay and pyrite veins in the coal. Coon et al. (1981) studied the use of a pulsed radar system in coal seems and concluded that the vertically polarized electric field, i.e. perpendicular to the bedding plane, showed deeper penetration and less attenuation than the horizontally polarized electric field, i.e. parallel to the bedding plane. Tillard (1994) compared reflections of isotropic and anisotropic geological formations (i.e. granite and schists) and concluded the same for the anisotropic schists: the velocity and the attenuation are dependent on the angle between the orientation of the antennas and the orientation of the schistocity. She also showed that radargrams can be improved using non-conventional use of the GPR system. Kovacs and Morey (1978) studied the reflection from radar signal through ice sheets. They concluded that the reflection from the bottom of the ice was strongly related to the orientation of the radar antennas to the ice crystal c-axis, and to the orientation of the subice water movement.

Others studied the reflections of the electromagnetic waves from objects with a distinct polarization. Ruck et al. (1970) give the theoretical expected reflection of various objects, including cylinders. They define the reflection for different sizes and angles of incidence. The amplitude of the reflection is always dependent on the angle of incidence. These reflections are modeled and measured for a GPR setup by Roberts (1994) and Roberts and Daniels (1997).

Sato et al. (1998) and Miwa et al. (1999) also reported a difference in their recordings using the co-polarized and cross-polarized configurations for polarimetric borehole radar. The ratio between these two reflections provides them with a value for the roughness of the surface, which can be related to the number of fractures.

Tsoflias et al. (1999) and Tsoflias (1999) studied the transmission of electromagnetic waves through vertical fractures. Using theory, modeling and field data they show that there is a phase and amplitude difference between the E-polarized wavefield and the H-polarized wavefield transmission. Here E-polarization indicates that the electric wavefield is parallel to the fracture and H-polarization indicates that the electric wavefield is perpendicular to the fracture.

The source character of the GPR antenna is recognized by several authors who took the radiation pattern into account during migration. In general GPR processing sequences, data are migrated using conventional migration methods (e.g. Fisher et al., 1992), but others implemented the directivity of the GPR antenna in their migration algorithms (Campman and Slob, 1999; Moran et al. 1998; Saintenoy and Tarantola, 1998).

3.1.3 Objective of this study

In this study the possibilities of using the different configurations of GPR antennas are investigated. First of all a method similar to Alford rotation (Alford, 1986) and to work done by Ulaby and Elachi (1990) is applied to GPR data. Thereby, GPR data are used that are collected in four configurations, being the configurations obtained by rotating both transmitter and receiver under an angle of 90 degrees. As was pointed out before, GPR recordings are dependent on the orientation of the antennas to the orientation of the subsurface targets. This angle-dependence can be used in combination with Alford rotation to determine the angle of orientation of targets with a distinct directivity (Van Gestel and Stoffa, 1999). In the following section the effects are studied of using multiple antenna configurations, where both antennas are kept parallel to each other, but are rotated simultaneously. Combination of data collected in these configurations using a weighted migration results in a better signal to noise ratio of the final image and a more uniform energy distribution of the radiated wavefield than using a regular migration method (Van Gestel et al., 2000). In the last section, Alford rotation is applied in the migration algorithm to improve the final image and increase the migration speed. Using the radiation pattern in combination with Alford rotation, the location of the origin of the reflected wavefield is found. This information is used to improve the migration scheme (Van Gestel and Stoffa, 2000). In conventional migration schemes, recorded data are migrated to all angles. In this method, as the location of the origin of the reflected wavefield is known, the recorded data are migrated to certain grid points only.

The first part of this chapter consists of theory. The equations for the radiation pattern of the GPR antennas, the reflection from cylinders, transmission of electromagnetic waves through fractures, and Alford rotation are shown. In the following section it is shown that the different methods can be successfully applied to synthetic data. The results are given from field data collected at a GPR testing site in Scheveningen, The Netherlands. The various methods are applied to reflections from a cylinder and good results are obtained with the collected field data. An error analysis is applied to synthetic data to obtain a better understanding of the effects of three different sources of noise. The final part contains a summary and discussion on the practical applications of the various methods.

3.2 Theory

3.2.1 Configuration definitions

An essential start for this study is to give accurate definitions of the different configurations that are used. In GPR surveys, data are collected using two antennas, a transmitter and a receiver. The configuration where the antennas are oriented in the same direction is called co-polarized (Figure 3.1A), and when the antennas have a 90 degree angle to each other the configuration is called cross-polarized (Figure 3.1B). When the antennas are oriented parallel to the line of data collection, the configuration is called parallel (Figure 3.1C), in contrary to the perpendicular configuration, where the antennas are held perpendicular to the line of data collection (Figure 3.1D). In this study, it is assumed that the main axis of the studied objects are perpendicular to the line of data collection, which means that in the perpendicular configuration the antennas are parallel to that axis. When the antennas are pointing with their end points towards each other the configuration is called endfire (Figure 3.1E), and the configuration is called broadfire when the broad sides of the antenna are pointed towards each other (Figure 3.1F).

Figure 3.1. Simplified representation of the different transmitter-receiver configurations, showing the difference between A. Co-polarized, B. Cross-polarized, C. Parallel, D. Perpendicular, E. Endfire, and F. Broadfire antenna configurations.

The line of data collection is always chosen to be the x-direction (Figure 3.1D). Configurations are sometimes named after the direction of orientation of the transmitter and receiver axes respectively, where the antennas are both placed along the line of data collection. For example the co-polarized, perpendicular, broadfire configuration is also denoted as the yy-configuration, and the co-polarized, parallel, endfire configuration is also denoted as the xx-configuration.

In conventional GPR surveys only the yy-configuration is used (Figure 3.1A). In this study the antennas are rotated, but not moved away from the line of data collection. This means two cross-polarized configurations are used, respectively the yx- (Figure 3.1B) and the xy-configuration, and several co-polarized configurations, ranging from the perpendicular, broadfire, or yy-configuration (Figure 3.1A) to the parallel, endfire, or xx-configuration (Figure 3.1C). As the reflections that are studied, originate from objects that are considered in the far field of the antenna, this last configuration can be replaced by the co-polarized, parallel, broadfire configuration. In theory, the xy-configuration and the yx-configuration should give identical results due to reciprocity (Ward and Hohmann, 1987). However, in practice the reflections are recorded in both configurations and used in the different methods.

3.2.2 Radiation profile

In this study one important aspect of the GPR antenna is used: its directivity. The GPR antenna is a dipole source, which means that the transmitted wavefield has a distinct directivity, which can be used to extract information about the location of the origin of the reflected wavefield. The radiation of a horizontal, electromagnetic dipole along the plane interface of two lossless dielectric, non-magnetic half spaces is described by Engheta et al. (1982). Arcone (1995) shows more numerical and measured results of these radiation patterns. The solutions are formulated in a more convenient spherical coordinate system. The equations for the electric fields in the far zone of the lower half space are:

,(3.1)

for , (3.2)

and

,(3.3)

, for , (3.4)

where r, q, and j are the spherical coordinates, is the critical angle, I_o is the current frequency spectrum of the source, , is the wavenumber in the upper half space, is the velocity of electromagnetic waves in air, is the angular frequency, , , m is the magnetic permeability, and is the electric permittivity. Subscript 0 denotes the properties in air and is the electric permittivity in the lower half space.

Figure 3.2 shows a cross-section of this radiation pattern from a point along the antenna axis and a point perpendicular to the antenna axis. One can observe the strong influence of the critical angle on the radiation pattern. Most of the excited energy is distributed along the plane that is perpendicular to the antenna and crosses the antenna in the middle. Almost no wave is generated in the direction of the endpoints of the antennas. Consequently, the amplitude of the reflected signal is dependent on the angle between the orientation of the antennas and the orientation of the line of data collection, except when the reflection originates from an object, located exactly underneath the two antennas.

Figure 3.2. Theoretical radiation profile of the horizontal electromagnetic dipole on the boundary of two homogeneous interfaces. Contour are posted every 1000. The antenna is located at the origin. A. Cross-section along the x-axis, perpendicular to the antenna. B. Cross-section along the y-axis, parallel to the antenna. For the calculation of this profile the following values are used: frequency is 450 MHz, e_r=1 in the upper half space and e_r==5 in the lower half space, s =0 mS/m, and m_r=1 for both half spaces.

GPR surveys are collected using two antennas, the transmitter and the receiver, and as a result the final recorded data are a combination of the two separate radiation patterns. They are also dependent on the input source function and the contrast function. This equation is given in tensor notation as following:

, (3.5)

where D_m is the scattered data in the scattered domain D^sc, x^rec is the location of the receiver, x^sou is the location of the source, a ^rec is the angle of orientation of the receiver, a ^sou is the angle of orientation of the source, J_p is the input source function, c is the relative permittivity contrast function for isotropic media and G_mr and G_rp are the tensor Green's function given by Equations 3.1 till 3.4 after emission of the source function. Equation 3.5 can be simplified for respectively the xx-, xy-, yx-, and yy-configurations using the following equations:

, (3.6)

, (3.7)

, (3.8)

. (3.9)

Figure 3.3 shows a cross-section of the radiation pattern at a depth of 2.5 meter below two antennas in the yy-configuration. This cross-section has been calculated using Equation 3.9. The antennas have a separation of 30 cm, and are located at the interface of air and a medium with e_r=5, s =0 mS/m, and m_r=1. By rotation of the antennas the focus points of the radiated energy rotate simultaneously and a different part of the subsurface is highlighted by the antenna configuration. Combining different configurations results in better data coverage and improves the final radargram, as the combined coverage of the subsurface is more uniform.

Figure 3.3. Theoretical radiation pattern of two GPR antennas in the yy-configuration on the boundary of two homogeneous halfspaces. Cross-section at a depth of 2.5 m, the antenna separation is 30 cm and their midpoint is located at the origin. The main axis of the antennas is pointed towards the y-axis. For the calculation of this profile the following values are used: frequency is 450 MHz, e_r=1 in the upper half space and e_r=5 in the lower half space, s=0 mS/m, and m_r=1 for both half spaces.

The radiation pattern of the two antennas is even more complicated when the geometry of the subsurface changes. The radiation pattern is also affected by the construction of the shields of the antennas, which are different for every GPR set. Exact knowledge of this radiation pattern is necessary for accurate data processing and interpretation. However, even measurement of the radiation pattern under controlled conditions does not guarantee exact knowledge of this pattern, as the radiation pattern is dependent on the permittivity, permeability and conductivity of the surrounding media and accordingly changes due to changes in terrain. In this study the theoretical radiation pattern of the antennas is used, but one has to keep in mind, that the results can be improved if accurate measurements of the radiation profile are recorded.

3.2.3 Reflections of cylinders

The reflections of various kinds and sizes of cylinders for electromagnetic waves are described by Ruck et al (1970), and Roberts and Daniels (1996). They show that the reflection is dependent on the angle of orientation of the incoming wavefield, which is dependent on the orientation of the cylinder with respect to the orientation of the transmitter and the receiver.

Figure 3.4 shows the relative amplitude values of the reflection from a cylinder as a function of the angle between the orientation of the cylinder and the orientation of the antennas. The amplitude values are given both for the co-polarized and the cross-polarized configuration. Figure 3.4 shows that the amplitude from the reflection of the cylinder is maximum when the antennas are parallel to each other and parallel to the cylinder. For the cross-polarized configuration, the reflection is zero when one of the antennas is perpendicular to the cylinder.

Figure 3.4. Theoretical reflection response of a cylinder versus angle between the orientation of the antennas and the orientation of the cylinder. The co-polarized configuration and cross-polarized configurations are shown with a solid and a dashed line respectively. The amplitude of the reflection is maximum for the yy-configuration, or when the co-polarized configuration is parallel to the cylinder. The reflection is zero for the cross-polarized configuration, when either of the two antennas in perpendicular to the cylinder. For these calculations the perfectly conducting cylinder has a length of 1m, a diameter of 20 cm, and is embedded in sand with e_r=5, s=0 mS/m, and m_r=1.

3.2.4 Transmission through vertical fractures

Tsoflias et al. (1999) and Tsoflias (1999) studied the transmission of electromagnetic waves through vertical fractures. Using theory, modeling and field data they show wavefield polarization dependent phase and amplitude differences introduced by high angle transmission through thin layers, such as vertical fractures. They exploited phase relationships between electromagnetic wavefield polarization and fracture orientation to determine the location and orientation of vertical fractures. These differences occur when the wavefield has a fairly large angle of incidence to the fracture (Figure 3.5B), which is the case for most reflections in conventional GPR surveys (Figure 3.5A).

Figure 3.5. A. Simplified model of GPR data collection across a vertical fracture. Reflections from the transmitter (T) cross the fracture at a high incident angle (q i), before being reflected back to the surface and being recorded in the receiver (R). B. The relative amplitude of the theoretical response of a wavefield transmitted through a vertical fracture for the yy- and the xx-configuration. The response is dependent on the angle of incidence of the wavefield to the fracture. For normal incidence the response is identical, but the difference is considerable for higher angles of incidence. These calculations were made using a background media of e_r=5, s=0.01 mS/m, and m_r=1. The fracture has a width of 1 cm, and is air-filled with an e_r=1, s=0 mS/m, and m_r=1.

In terms of GPR data collection, this means that the recorded radargram using the yy-configuration are different from the data collected using the xx-configuration. This difference is maximum when the line of data collection is perpendicular to the fracture and is zero when the line of data collection is at a 45 degree angle to the fracture. In this last case the angle between the orientation of the transmitted wavefield and the orientation of the vertical fracture is 45 degrees in both cases.

3.2.5 Alford rotation

Collection of multi-configuration GPR data can be compared to collection of multi-component elastic data. By rotation of the transmitter a situation is obtained similar to the two different shearwave sources and by rotation of the receiver the two different reflected electromagnetic components are recorded, which is similar to recording the two different shearwave components.

Alford rotation (Alford, 1986) is one of the most successful ways to evaluate shearwaves and is used in this study. Alford rotation allows one to construct the response for every angle of orientation, after collection of data in just four different configurations, which are the xx-, xy-, yx-, and yy-configurations. The following equations are used:

, (3.10)

, (3.11)

where q is the angle of rotation, U is the 2x2 matrix with the original recordings in the four different configurations, and V is the 2x2 matrix with the rotated xx-, xy-, yx-, yy-configuration values. These equations can be applied for any possible angle of rotation, thereby calculating the response for all four configurations for every different angle between the orientation of the antennas and the orientation of the line of data collection.

In the elastic case the amplitude and the arrival time of the reflections are dependent on the orientation of the source and receiver to the orientation of the axis of anisotropy of the medium (Alford, 1986). As a result, Alford rotation can be used to find the angle of orientation of the main axis of anisotropy and the average azimuthal anisotropy of the medium (e.g. fractures). It is known that the amplitude of the reflection values for the xy- and yx-configurations is theoretically zero, when the transmitter or receiver is parallel to the main axis of anisotropy of the medium. By plotting the amplitude response versus angle of rotation and observing where the amplitude of the reflection is minimum for the xy- or yx-configuration, the main orientation of the axis of anisotropy of the medium can be found.

This dependence also occurs in electromagnetic media (Keller, 1987) and Alford rotation can be applied to find the main axis of anisotropy for electromagnetic media. However, this study is not focused on anisotropic media, but on reflections from targets with a distinct directivity. For these targets the reflection is also dependent on the orientation of the antennas. It is assumed that the medium in which the wavefield travels is isotropic and as a result the travel path, the velocities and the resulting arrival time are identical for all different configurations. The reflection strength is only dependent on the angle of orientation between the antennas and the main axis of the object. As a result the orientation of this axis can be extracted.

Zebker et al. (1987) applied a technique similar to Alford rotation on electromagnetic recordings. They collected electromagnetic reflection data from airborne radar surveys. In conventional airborne radar methods only one component of the reflected field is transmitted and recorded. They show that collection of four different configurations improves the value of the data, as it allows for easier distinction between different areas on the surface. This method is effective in various situations, for example in differentiation of urban, park and sea areas (Lim et al., 1989), and of different lavaflows (Zebker et al., 1987). Van Zyl et al. (1987) and Schwartz et al. (1988) published more related work to this method, and Ulaby and Elachi (1990) are the editors of a book containing a summary of this work.

There is a difference in the practical application of Alford rotation used in seismic applications and the Alford rotation applied in this study. In seismic applications the angle is determined where the reflections in the cross-polarized configurations are zero. This leads to two different angles, as the reflections in the cross-polarized configurations are not only zero in the case when the transmitter is perpendicular to the orientation of the main axis of the of the origin of the reflection, but also when the transmitter is parallel to this orientation. In the last case the receiver is perpendicular to the orientation of the main axis of the origin of the reflection and the resulting reflection is zero. Therefore, it is better to find the angle where the reflection in the co-polarized configuration is maximum, which results in an explicit angle of orientation.

3.3 Results for the synthetic data

3.3.1 Modeling code

To study the propagation of the electromagnetic wavefield and the reflections from objects, a modified three dimensional, three component finite difference time domain (3DFDTD) modeling code is used, written by Roberts (1994; Roberts and Daniels, 1997). This code makes it possible to study reflections of dipping layers and cylinders, and transmission through vertical fractures. This code was compared to theoretical and measured data and was shown to accurately forward model GPR data (Roberts, 1994; Roberts and Daniels, 1997). A limitation of the code is the fact that it is not possible to model antennas on the boundary of two half spaces. Instead the transmitter is located in the subsurface, which influences the radiation pattern, but not the reflections from the targets. This modeling code is used to show the angle-dependence of the GPR measurements for three different situations: dipping layers, cylinders and vertical fractures. The input source function that was used in the modeling, is given by the following equation:

, (3.12)

where ,, and a is the center frequency of the used wavelet, which is equal to 450 MHz in all the models.

3.3.2 The Alford rotation method

3.3.2.1 Dipping layers

The first method that is studied is the Alford rotation method. It was noted that most of the radiation energy of the antennas was focused into the plane perpendicular to the antennas, crossing in the middle of the two antennas (Figure 3.2). This means that the amplitude of the reflection is maximum when the reflection is coming from a reflection point in this plane. In this case no cross-polarized reflection components are generated. By collecting four antenna configurations and applying Alford rotation the horizontal angle of the origin of any reflection coming from an isotropic object can be determined.

This is clarified with the simplest model, the reflection coming from a dipping plane. The media above and below this plane are isotropic. In Figure 3.6A the relative amplitude of a reflection is shown, coming from a perfectly conducting, dipping surface that has a dip of 20 degrees. The relative amplitude is given for the co-polarized and the cross-polarized configuration. Figure 3.6B shows the difference between the recorded reflection of the yy-configuration, where both antennas are parallel to the strike of the dipping plane, and the recorded reflection of the xx-configuration, where both antennas are perpendicular to the strike of the dipping plane. The amplitude of the reflection from this dipping plane is maximum when the antennas are parallel to the strike of this layer, as in this case the maximum amount of energy hits the surface and is reflected back to the receiver. As a result, Alford rotation can be applied to determine the strike of the dipping plane. Figure 3.7A shows the calculated results of the Alford rotation method for different angles. It is shown that for the synthetic data the strike of the dipping plane can be exactly calculated.

Figure 3.6. A. Relative amplitude for the modeled reflection from a dipping plane versus the angle between the orientation of the antennas and the strike of that plane. The solid line is the co-polarized configuration; the dashed line is the cross-polarized configuration. B. Comparison of the reflection from the dipping plane for the yy-configuration and the xx-configuration. The reflection is coming of the boundary between sand with e_r=5, s=0.01 mS/m, and m_r=1 and a perfectly conducting plane with a dip of 20 degrees.

Figure 3.7. The angles of orientation, calculated using Alford rotation, versus the real angles of orientation for three cases. A. The reflection of a dipping plane. B. The reflection of a cylinder. C. The reflection of a plane, transmitted through a vertical fracture.

3.3.2.2 Cylinders

Figure 3.8A shows the relative amplitude of a reflection coming from a perfectly conducting cylinder with a length of 1 meter, and a diameter of 0.1 meter. The cylinder is embedded in sand with e=5, and s=0.01 mS/m. The relative amplitude is given for the co-polarized and the cross-polarized configuration. The shape of this figure is similar to the theoretically calculated results (Figure 3.4). Figure 3.8B shows the difference between the recorded reflection of the yy-configuration, where both antennas are parallel to the orientation of the cylinder, and the recorded reflection of the xx-configuration, where both antennas are perpendicular to the orientation of the cylinder. Because of the difference in reflection for the two configurations, Alford rotation can be applied to determine the orientation of the cylinder. Figure 3.7B shows the calculated angles of orientation of a cylinder for different angles. For synthetic data, the calculated angle of orientation of the cylinder is exactly the same as the real angle of orientation of the cylinder.

Figure 3.8. A. Relative amplitude for the reflection from a cylinder versus the angle between the orientation of the antennas and the orientation of the cylinder. The solid line is the co-polarized configuration; the dashed line is the cross-polarized configuration. B. Comparison of the reflection from the cylinder for the yy-configuration and the xx-configuration. Cylinder is perfectly conducting, has a length of 1.0 m and a diameter of 0.10 m, and is embedded in sand with e_r=5, s=0.01 mS/m, and m_r=1.

3.3.2.3 Fractures

The 3DFDTD code makes it possible to study the transmission of an electromagnetic wavefield through a vertical fracture. The fracture was modeled as a thin layer in a homogeneous background medium. For accurate modeling of a fracture it is not legitimate to simplify an irregular, raw surface fracture as a homogenous plane layer, but it provides useful insight and support to this study.

The results are presented of an air-filled fracture in a nonmagnetic, homogeneous environment with e=5, and s=0.01 mS/m. The fracture was modeled using a fracture aperture of 2 cm. Figure 3.9A shows the relative amplitude for the recorded electric field for an electromagnetic wavefield that was transmitted into the ground, reflected at a horizontal layer at a depth of 85 cm, and recorded on the other side of the fracture. The resulting angle of incidence of the wavefield to the vertical fracture was 80 degrees. The relative amplitude is given for the co-polarized and the cross-polarized configuration. Figure 3.9B shows the difference between the recorded reflection of the yy-configuration, where both antennas are parallel to the orientation of the vertical fracture, and the recorded reflection of the xx-configuration, where both antennas are perpendicular to the orientation of the vertical fracture. One can observe a phase separation between the two recorded wavefields. The wavefield in the xx-configuration has a consistent phase lead of, but attenuates more than the wavefield in the yy-configuration.

Figure 3.9. A. Relative amplitude for the reflection of a plane at a depth of 0.85 m, with a vertical fracture between the transmitter and the receiver, versus the angle between the orientation of the antennas and the orientation of the vertical fracture. The solid line is the co-polarized configuration; the dashed line is the cross-polarized configuration. B. Comparison of the reflection from a plane that was transmitted through a vertical fracture, for the yy-configuration and the xx-configuration. The fracture has a width of 2 cm, is air-filled with e_r=1, s=0 mS/m, and m_r=1. The background media has an e_r=5 and s=0.01 mS/m and m_r=1.

Amplitude and phase separation relationships were consistent in varying geologic settings, varying fracture apertures and fracture fluid content. For larger fracture permittivities, the separation decreased. The phase separation was zero for normal incidence and increased for larger angles of incidence of the wavefield. All these results are consistent with the results that are expected from the theoretical solutions (Tsoflias et al., 1999; Tsoflias, 1999).

Figure 3.7C shows the calculated angles of orientation of a vertical fracture versus the real angle of orientation of the vertical fracture. For synthetic data, the calculated angle of orientation of the vertical fracture is exactly the same as the real angle of orientation of the vertical fracture.

3.3.3 The weighted migration method

For the following two studied methods, synthetic data were generated for the reflection of one scatter point in a homogenous background. Instead of the 3DFDTD code a simple travelpath, reflection amplitude method was used to generate the results. A 3D grid was generated that consists of 51 x 51 shotpoints, with a sample spacing of 10 cm in both directions. A scatter point was located in the middle of the grid at a depth of 2.5 meter. To study the weighted migration method, synthetic data were generated for 12 different configurations for every shotpoint. The angle of orientation between the antennas, with a separation of 30 cm, and the line perpendicular to the line of data collection changed from 0 to 165 degrees, with a 15-degree increase between every configuration. The radiation pattern was generated using the theoretical formulae (Equations 3.1 till 3.4). The radiation patterns of both antennas are combined in the final radargram (Equation 3.5). The same source function is used as for the 3DFDTD modeling, which is given in Equation 3.12. The contrast function is taken to be 1. Figure 3.10 shows this radargram for 12 different configurations at four different locations to the target. The first configuration is equivalent to the yy-configuration, the seventh configuration is equivalent to the xx-configuration and the first configuration is added again at the end for convenience.

Figure 3.10. Synthetic data for a scatterer in a homogeneous background for four different locations and 12 configurations. The configurations start with the yy-configuration and they increase with 15 degrees per configuration. In every position the configuration where both antennas are pointing with their broad side towards the target have the strongest reflection. A. The antennas are located directly above object, which results in no difference in reflection between the different configurations. B. The antennas are moved 2.5 m along the x-axis. C. The antennas are moved 2.5 m along the y-axis. D. The antennas are moved 2.5 m both along the x-axis and the y-axis.

The synthetic data are migrated using both a regular migration technique and the weighted migration method. For the regular migration technique a standard Kirchhoff migration was applied (Yilmaz, 1987). For the weighted migration the strength of the radiation pattern for the specific image grid point was calculated and the recorded signal was multiplied with this factor before migrating the signal to this location. This is written down in the following equation, which is the discretization of Equation 3.6:

, (3.13)

where G₁₁ is the combination of the two Green's functions and the original source input function. Equation 3.13 can be rewritten as an imaging condition to extract the original image from the data as:

, (3.14)

which of course can also applied to all other configurations. The difference between the regular migration and the weighted migration is lain in the Green's function, as the weighted migration includes the radiation pattern in this function.

Figure 3.11 shows the results of both the regular and the weighted migrations. As the original amplitudes are multiplied with a weighted function during the weighted migration, the final amplitude of the signal is scaled and the true amplitudes of the final images can not be compared. Therefore, random noise is added to all generated synthetic data before the migrations were applied and the ratio between the signal and noise in the final images resulting from both methods is compared. These signal to noise ratios are only valuable to make comparisons between the various methods. They are dependent on too many variables to justify qualitative significance.

Figure 3.11. Results of imaging of the 3D synthetic data. A. Using the weighted migration B. Using the regular migration.

Table 3.1 shows the signal to noise ratio for five different migrations and five different number of configurations, respectively 1, 2, 4, 6, and 12. The first migration that is applied, is the weighted migration. The second migration is the regular Kirchhoff migration. The third column is added to compare the migration results to the conventional data collection method. Only the data in the yy-configuration are used to generate the image. A regular Kirchhoff migration is applied for this one configuration, but the same number of measurements were combined as in the first two migration methods. Due to the radiation pattern, the signal to noise ratio of the final image is dependent on its location compared to the line of data collection. As a result, the reflection of a target is stronger when it is located along the axis perpendicular to both antennas than when it is located along the axis parallel to both antennas. The resulting range is given in column four and five, with the highest signal to noise ratio in column four and the lowest in column five.

configurations used (total number of configurations)

weighted migration

regular migration

migration of yy-configuration only

yy-configuration in most preferred direction

yy-configuration in least preferred direction

0 (1)

28.7

22.3

22.3

23.2

21.4

0, 90 (2)

38.8

29.3

29.2

36.0

28.5

0, 45, 90, 135 (4)

54.3

41.2

41.0

43.3

39.2

0, 30, 60�150 (6)

64.8

47.0

46.6

53.4

46.1

0, 15, 30�165 (12)

85.0

62.2

61.3

69.9

53.3

Table 3.1. Signal to noise ratios of the images for different numbers of configurations of the synthetic data.

The weighted migration leads to the best signal to noise ratio, significantly stronger than the signal to noise ratio of the image resulting from the regular migration. Collecting data in all configurations leads to a slight increase in signal to noise ratio, but especially to a much more uniform result, where the signal to noise ratio is more consistent and less dependent on the location of the target.

3.3.4 The Alford migration method

In this section it is shown that Alford rotation can be used to improve migration schemes. Instead of migration of the recorded data to all directions, the migration scheme is limited to a small range of angles, determined using Alford rotation. This results in a faster, cheaper and more accurate migration algorithm. A regular Kirchhoff migration (Yilmaz, 1987) is modified, and is limited to those paths that are within the predicted angle of orientation extracted using Alford rotation. For all migrations in this section the radiation pattern is implemented in the migration as this was shown to have better results in the previous section.

Synthetic data were generated in a similar matter as for the weighted migration method for the yy-, yx-, xy-, and xx-configurations to compare the images resulting from Alford migration and the regular migration. In the Alford migration the location of the target is calculated using Alford rotation for every samplepoint on every trace and the data are migrated in that direction only. The regular migration consists of a conventional Kirchhoff migration of the yy-configuration only. Results of both the Alford and the regular migration are shown in Figure 3.12 for two different noise levels. The Alford migration (Figure 3.12B) results in a more focused image than the regular migration (Figure 3.12A), when no noise is present in the data. The angle of orientation can be exactly determined and the data are only migrated to the correct angle. As a result almost no migration artifacts are present in the final image. However, the result decreases when noise is added to the synthetic data. The regular imaging method is hardly influenced by the noise level of 10% (Figure 3.12C), but the Alford migration method shows considerate distortion (Figure 3.12D).

Figure 3.12. Results of the imaging of the synthetic data for the regular migration method and the Alford migration method. Results are shown for two different levels of noise. A. Regular migration method without noise. B. Alford migration method without noise. C. Regular migration method with 10 % noise. D. Alford migration method with 10 % noise.

To improve the results of Alford migration the data are migrated to a range of angles around the calculated angle instead of migration to the calculated angle only. When noise is added to the synthetic data, the calculation of the angle is incorrect. By sweeping through a range of angles the exact angle of orientation of the target is covered, but the method is still faster and more accurate method than the regular migration to all angles. Figure 3.13 shows the images of the Alford migration using different ranges of angles. It can be observed that when more angles are included in the migration, more migration artifacts are generated in the final image and the final image becomes more similar to the regular migration (Figure 3.12A).

Figure 3.13. Results of the imaging of the synthetic data for the Alford migration method for different ranges of angles. A. Range of angles is zero degrees. B. Range of angles is 10 degrees. C. Range of angles is 20 degrees. D. Range of angles is 40 degrees.

Figure 3.14 and Table 3.2 show the signal to noise ratios for seven different migrations and four different noise levels i.e. 0 %, 0.1 %, 1% and 10 %. The first six migration methods that are applied, are Alford migrations with different kind of ranges of angles. The last method is the regular migration for the yy-configuration only. For low noise levels the Alford migration results in a better signal to noise ratio than the regular migration. However, when more than about 1 % of noise is added to the synthetic data, the regular migration has a higher signal to noise ratio. Further, for low noise levels in the synthetic data the Alford migration to a small range of angles is better than Alford migration to a wider range of angles. However, when the noise level in the synthetic data increases the results get closer and finally the Alford migration with wider ranges has better results than the Alford migration with smaller ranges.

Figure 3.14. Plot of the various signal to noise ratios of the images resulting from the different migration methods for different levels of noise. Dashed line is the result of the regular migration method. Six solid lines are the results of the Alford migration method for six different ranges.

noise level = 0 %

noise level = 0.1 %

noise level = 1 %

noise level = 10 %

time (min.)

range is 0 degrees

2147.4

610.3

99.3

25.8

2

range is 10 degrees

622.4

421.0

98.0

31.6

38

range is 20 degrees

349.3

267.3

93.8

32.6

71

range is 40 degrees

191.2

156.3

86.2

33.9

141

range is 90 degrees

91.8

84.3

64.5

35.7

330

range is 180 degrees

41.1

40.8

41.4

36.6

663

regular migration

80.1

80.0

79.9

75.1

58

Table 3.2. Signal to noise ratios of the images resulting from the different migration methods for different levels of noise. Also the time to run the migration algorithm is given.

In Table 3.2 the necessary time to run the migration can be found. These numbers are only valuable to make comparisons between the various methods. The Alford migration is 29 times faster than the regular migration. However, by applying the method where the data are migrated to a sweep of angles instead of one angle, the method slows down significantly, with the Alford migration over a range of 20 degrees already being slower than the regular migration.

3.4 GPR survey in Scheveningen, The Netherlands

3.4.1 Description of the field data

3.4.1.1 GPR test site

In the spring of 1998 several lines of GPR data were collected in Scheveningen, The Netherlands (Figure 3.15). At this location a GPR testing site was built by the Section of Applied Geophysics and Petrophysics of the Delft, University of Technology and TNO-FEL, which makes it possible to collect GPR data under perfectly idealized conditions. The GPR testing site is 10 meter by 10 meter and filled with dry sand, with known permittivity, permeability, and conductivity values. Objects can be buried in the sand to a depth of 3 meter. The relative permittivity of the sand was measured and is e_r=4.59. This leads to an electromagnetic wavefield velocity of 0.14 m/ns and a wavelength of 31 cm for the 450 MHz antennas.

Figure 3.15. Picture of the GPR testing site. Note the absence of any metallic objects in the area. Picture is shot from the east.

For the measurements the pulse EKKO system 1000 was used, which has shielded GPR antennas. All measurements were done using the 450 MHz antennas, and an antenna separation of 30 cm.

In the GPR testing site several objects were buried at a depth of 1 meter (Figure 3.16). The measurements are focused on the westmost cylinder that is made of iron, has a length of 1.012 meter, and a diameter of 22.3 cm. The cylinder has a distinct polarization, a length that is larger and a diameter that is slightly smaller than the used wavelength.

Figure 3.16. Map of the GPR testing site area, showing the lines of data that were collected and the location of the buried objects. The 2D line runs east west and covers all three cylinders. The lines of the 3D grid completely cover the first cylinder. Dashed box shows outline of figures 3.22 and 3.29.

3.4.1.2 Setup

Seven GPR 2D lines were collected in a straight line over the middle of the cylinder with a trace separation of 10 cm (Figures 3.15 and 3.16). For the seven lines there is a different angle between the orientation of the line of data collection and the orientation of the transmitter. This angle was respectively 0, 15, 30, 45, 60, 75 and 90 degrees. For every trace on every line seven different configurations were recorded, where every configuration has a different angle between the orientation of the line of data collection and the receiver. This angle was again respectively 0, 15, 30, 45, 60, 75 and 90 degrees. All in all, data were collected in a total of 49 configurations for every trace.

From these 49 collected lines of data the configurations of four lines can be extracted, with respective angles of 0, 15, 30 and 45 degrees between the orientation of the line of data collection and the transmitter with four configurations: the yy-, yx-, xy-, and the xx-configurations. These configurations are used to show the results of the Alford rotation method, where an attempt is made to find the orientation of the cylinder. For the weighted migration method, seven configurations are extracted, where the antennas are held parallel to each other, but have a different angle of orientation to the line of data collection.

Also a 3D grid was collected that covered the whole cylinder oriented at a 30 degree angle to the cylinder (Figure 3.16). The 3D grid consisted of 13 lines with a 25 cm line spacing. Trace spacing was 10 cm and data were collected using four different configurations: the yy-, yx-, xy-, and the xx-configurations. The data collected in this grid were used to show the results of the Alford rotation method, the weighted migration method and the Alford migration method.

A simple, standard data processing sequence was applied to the data, which consists of trace-editing, filtering, spherical gain correction, and resampling. As the cylinders are buried in a homogeneous background a constant velocity wave equation migration can be applied (Gazdag, 1978). Figure 3.17 shows the data before and after migration. This migration method is used for the Alford rotation methods. Further in this study, the data are also migrated using a Kirchhoff migration method (Yilmaz, 1987) and compared to several newly proposed migration methods based on this Kirchhoff migration method.

Figure 3.17. Example of the data that were collected for the first 2D line with the antennas under a 90-degree angle to the line of data collection. A. Data results before migration. B. Data results after migration.

3.4.1.3 Data quality

Figure 3.18 shows the recordings of the yy-, yx-, xy-, and the xx-configurations of line 6 of the 3D grid. The measurements are excellent for the various methods that are studied, as the cylinder in the homogeneous background results in a clear single reflection, which can be easily recognized, and there is no interference from other reflections. At first, the data recorded in the xy- and the yx-configuration appear to be buried in the background noise, which has an average amplitude level of 1.1 % of the maximum reflection in the yy-configuration of the cylinder. However, the results show that the reflection is still strong enough to be observed.

Figure 3.18. Example of the data that were collected for line six of the 3D grid. A. Data collected using the yy-configuration. B. Data collected using the yx-configuration. C. Data collected using the xy-configuration. D. Data collected using the xx-configuration.

A difference can be noted between the xy-configuration and the yx-configuration recordings, although theoretically they should be identical (Ward and Hohmann, 1987). The difference is caused by inaccuracy of the position and angle estimates, which result from the manual positioning of the antennas. The recorded data would be improved if the data were collected using a frame, which keeps the distance and angle between the two antennas constant. Such a frame is currently only available for the broadfire, co-polarized configuration.

3.4.1.4 Radiation pattern and reflection from cylinders

Figure 3.19 shows a comparison between the recorded radiation pattern and the theoretical radiation pattern. In this figure the reflection from the cylinder is shown with both antennas parallel (Figure 3.19A) or perpendicular (Figure 3.19B) to the cylinder. In these two cases the reflection of the cylinder is angle-independent and only depends on the radiation pattern of the antennas. Figure 3.19A and Figure 3.19B show the relative amplitude of the reflection for the yy-configuration and the xx-configuration respectively. There is a good match between theory and recorded data for the xx-configuration. For the reflection for the yy-configuration the theoretical radiation pattern appears wider than in the recorded data. But they are similar in the fact that the maximum reflection is not directly underneath the two antennas, but at the position where the excited wavefield is reflected at the critical angle.

Figure 3.19. Comparison of the measured versus the theoretical radiation pattern. As the antennas and the cylinder stay in the same orientation to one another, the relative amplitude of the reflection is only dependent on the radiation pattern. A. Radiation pattern for the yy-configuration. B. Radiation pattern for the xx-configuration.

Figure 3.20A shows the maximum amplitude of the reflections for the different angles between the orientation of the antennas and the orientation of the cylinder, both for the co-polarized and the cross-polarized configurations. The amplitude of the reflection of the cylinder is maximum when both antennas are held parallel to the cylinder, and the amplitude of the reflection is minimum when the orientation of one of the two antennas is perpendicular to the orientation of the cylinder. This is similar to the expected theoretical and the modeling results (Figures 3.4 and 3.8) and essential for successful application of the Alford rotation method. Figure 3.20B not only shows an amplitude difference between the recordings for the yy- and the xx-configuration, but also a phase difference.

Figure 3.20. Relative amplitude of the measured reflection from a cylinder versus the angle between the orientation of the antennas and the orientation of the cylinder. The solid line is the co-polarized configuration; the dashed line is the cross-polarized configuration. B. Comparison of the reflection from the cylinder for the yy-configuration and the xx-configuration.

3.4.2 Results of the Alford rotation method

The Alford rotation method is applied both to the 2D line recordings and the recordings of the lines of the 3D grid. The results of the calculated angles of orientation for the cylinder are shown in Figures 3.21 and 3.22 and in Table 3.3. Figure 3.21 shows the results for the first 25 traces of the four 2D lines, where the cylinder is located at trace number 12. For every trace on every line the reflection of the cylinder is picked and the angle of orientation of the cylinder is calculated for a window of about the length of a wavelength around this reflection. The mean of the 12 traces around the middle of the cylinder is given in Figure 3.21 and Table 3.3.

Figure 3.21. Calculated results of the angle of orientation of the cylinder for the traces of the four different 2D lines. Dashed line shows the correct angle of orientation of the cylinder. Mean value of the calculated values of the 12 traces closest to the cylinder is also given. A. 2D line where the angle between the orientation of the cylinder and the orientation of the line of data collection is zero degrees. B. 2D line where the angle between the orientation of the cylinder and the orientation of the line of data collection is 15 degrees. C. 2D line where the angle between the orientation of the cylinder and the orientation of the line of data collection is 30 degrees. D. 2D line where the angle between the orientation of the cylinder and the orientation of the line of data collection is 45 degrees.

angle of orientation of the cylinder

unmigrated

migrated

2D line: 0 degrees

-0.8

-2.0

2D line: 15 degrees

11.1

12.0

2D line: 30 degrees

29.3

31.0

2D line: 45 degrees

43.0

44.0

3D line 5: 30 degrees

28.6

28.0

3D line 6: 30 degrees

20.7

26.6

3D line 7: 30 degrees

26.4

27.0

Table 3.3. Mean of the results of the calculated angles of the 12 traces around the cylinder for the different lines.

Figure 3.22 shows the results of the data collected in the 3D grid. The figure shows the topview of the cylinder, which is shown in light gray in between line number 4 and line number 7. For every sample on every trace of every line in this 3D grid the angle of orientation of the reflection is calculated. Again a window of about the length of a wavelength around that sample is used. Then a line is plotted in the direction of this calculated angle of orientation of the cylinder. The length of this line is as long as the amplitude of the reflection for the yy-configuration at that sample of that trace on that line. For most samples the amplitude of the reflection is small and the calculated angle of orientation of the cylinder is random. However, where the amplitudes are higher, and the reflection of the cylinder is studied, the calculated angle of orientation of the cylinder is similar to the actual angle of orientation of the cylinder.

Figure 3.22. Calculated results of the lines of the 3D grid. The figure shows a topview of the cylinder, which is shown in light gray in between line number 4 and line number 7. For every sample on every trace on every line in this 3D grid the angle of orientation of the cylinder is calculated. Then a line is plotted in the direction of this calculated angle of orientation of the cylinder. The length of this line is as long as the amplitude of the reflection at that sample of that trace. The lines for the positive and negative angles are red and blue respectively.

Table 3.3 summarizes the results for the four 2D lines, and for three of the lines of the 3D grid. The results are shown for two methods. In the first method the data are not migrated, but the angle of orientation of the cylinder is calculated for every single trace. After calculation of the angle of orientation of the cylinder the mean is calculated for the 12 traces closest to the cylinder. In the second method the angle of orientation of the cylinder is calculated using the migrated data. The difference between the two methods is that in the first method the angle of orientation is calculated first for every trace and afterwards the average is taken over all the traces. In the second method, as a result of the migration, the average reflection of the four different configurations is taken first, before the angle of orientation is calculated.

The results appear to be in close agreement with the actual angle of orientation of the cylinder, both for the 2D lines as well as for the 3D lines. However, all the calculated angles of orientation of the cylinder seem to underestimate the actual angles of orientation of the cylinder. This can be especially well observed in the lines of the 3D grid (Figure 3.22). Further in this study, the different sources of error are studied to obtain a better understanding of the origin of this underestimate.

The Alford rotation method where the angles are determined before migration is preferred as supposed to the method where the angles are determined after migration, even though the results in the first case are slightly worse than in the second case. The results of the case without migration are several estimates of the angle of orientation and it is easier to identify and eliminate the erroneous values. For the migrated data this is not possible, as there is only one data point available. However, this might be different in other situations, where several values of the same object can be obtained after migration, like e.g. dipping planes. Another method, which was not applied here, but is expected to give good results in other, more noisy situations, would be to undermigrate the data. In that case several estimates for the angle of the orientation are obtained. Consequently, erroneous values can be eliminated, but the signal to noise ratio and the accuracy of the resulting angle are increased by the partial migration.

3.4.3 Results of the weighted migration method

The weighted migration method was applied to both the 2D and the 3D field data. The data of the different configurations are migrated and combined into a new image. For the 2D field data seven different configurations were combined, where the antennas have an angle of 0, 15, 30 � 90 degrees to the line of data collection. For the 3D field data all four collected configurations are combined into a final weighted image. The following four migrations are applied for comparison: 1) the weighted migration method, 2) a regular Kirchhoff migration, 3) Kirchhoff migration of just the yy-configuration and 4) Kirchhoff migration of just the xx-configuration. Figure 3.23 shows the images of the first two migrations for the 2D field data. Figure 3.24 shows results of the images of the migration of the 3D data. A topview of the cylinder is shown for four different migration methods as the mean amplitude for a window around the depth of the cylinder is displayed.

Figure 3.23. Results of imaging of the 2D field data. A. Using the weighted migration B. Using the regular migration.

Figure 3.24. Topview of the migrated image for different migration methods of the field data. Dashed line shows exact outline of the cylinder. The mean amplitude of the image at the depth of the cylinder is shown. A. Weighted migration method, using all four configurations. B. Regular migration method, using all four configurations. C. Regular migration method, using yy-configuration only. D. Alford migration method, using a range of zero degrees.

Table 3.4 shows that just as for the synthetic data, the weighted migration method leads to the best signal to noise ratio. In the field data, migration of just the yy-configuration results in a better signal to noise ratio than the regular migration of all configurations, although there are three configurations less to construct the image. This is probably due to location inaccuracies due to manual positioning of the antennas. Further, a larger difference is noted in between the migration of the data in the least and in the most preferred direction than was present in the synthetic data. This is due to the difference in reflection amplitude of the elongate cylinder, which has a higher reflection amplitude in the configuration when both antennas are parallel to the main axis of the cylinder.

configurations used (total number of configurations)

weighted migration

regular migration

migration of yy-configuration only

migration of xx-configuration only

2D field data 0, 15, 30�90 (7)

23.3

17.6

19.3

12.6

3D field data yy-, yx-, xy-, xx- (4)

37.1

24.7

26.6

11.2

Table 3.4. Signal to noise ratios of the images of the field data.

3.4.4 Results of the Alford migration method

The Alford migration method was applied to the 3D data. Figure 3.24 shows a topview of the resulting image of the cylinder for four different migration methods, including the Alford migration (Figure 3.24D). The resulting image of the regular migration method is in better agreement with the known location and shape of the cylinder, which is shown by a dashed line. The Alford migration results in a better signal to noise ratio at certain points of the cylinder, but the image is very inconsistent and at other points the signal of the cylinder is weaker.

Table 3.5 shows the signal to noise ratio for six different migrations. The first four migrations that are applied, are Alford migrations with different kind of ranges. The fifth migration is the regular migration. The last migration is a conventional Kirchhoff migration using all four configurations. It appears that the Alford migration has a better signal to noise ratio than the regular migration, but the image resulting from the regular migration is better (Figure 3.24). Table 3.5 and Figure 3.25 show that using a range of angles instead of one angle, leads to a better image and a higher signal to noise ratio. Table 3.5 also shows the time that was required to run the migration. These numbers are only valuable to make comparisons between the various methods. The Alford migration is again about 30 times faster than the regular migration.

signal to noise level

time (sec.)

range is 0 degrees

19.78

35

range is 10 degrees

22.36

140

range is 20 degrees

25.4

240

range is 40 degrees

30.84

550

regular migration of four configurations

11.27

1115

regular migration of yy-configuration only

8.64

295

Table 3.5. Signal to noise level of the images resulting from the different migration methods and the time that was needed to run the migration algorithms.

Figure 3.25. Results of the imaging of the field data for the Alford migration method for different ranges of angles. A. Range of angles is zero degrees. B. Range of angles is 10 degrees. C. Range of angles is 20 degrees. D. Range of angles is 40 degrees.

3.5 Error analysis for the Alford rotation method

After showing that the Alford rotation method can be applied to field data, a better understanding of the sensitivity of this method is desirable. Three different effects are studied: the noise level, a time shift, and the radiation pattern of the two antennas.

3.5.1 Noise level

The first source of error that was studied is the noise level. In the recordings it appears that there is a considerable amount of background noise in the data. The recordings for the cross-polarized configurations are weak and seem to be buried in the background noise. Therefore different amounts of noise were added to the theoretical data and then an attempt is made to calculate the direction of orientation of the cylinder.

Figure 3.26 shows the calculated angles of orientation of the cylinder for 10 different angles between the orientation of the cylinder and the orientation of the line of data collection. These measurements are repeated for ten different cases, in which a different amount of random noise is added for every case. The mean angle of orientation of the cylinder was also calculated for the ten different measurements. The calculations become less accurate, when more noise is added. However, for the measured noise level in the data, which is 1.1 %, the results are still good. More important, the average error of the calculated values is still close to the real values, as the error averages out by observing enough measurements. It is concluded that the noise level is not a concern for this method, as long as several recordings of the same object are collected, as was the case in the field measurements.

Figure 3.26. The effect of adding noise to the data on the accuracy of determining the angle of orientation of a cylinder. These figures show the calculated angle of orientation of a cylinder for ten different angles for ten different cases. For every case different random noise is added to the synthetic data. Stars are plotted at the mean value of the ten different measurements. A. The added noise is zero for all ten cases, as a result the ten cases are identical, and the calculated angle of orientation of the cylinder is exact. B. The added noise level is 1 %. C. The added noise level is 2 %. D. The added noise level of 5 %.

3.5.2 Time shift

The second source of error that was studied is a time shift. A problem with recording data using multiple configurations is the fact that the different measurements need to be accurate to combine them into a new radargram. In reality there is often a small error in location and it is hard to record with a correct angle of orientation, which is so important in these measurements. In the field measurements, another problem was the fact that the GPR testing site, where all the data were recorded, has an uneven top surface, which led to small differences in elevation. All these effects result in data where the recorded reflection from a cylinder has a different arrival time for the four different configurations. This effect can be observed in the data.

Figure 3.27 shows the effect of these time shifts. The figure shows the calculated angles of orientation of the cylinder versus the real angles of orientation of the cylinder for ten different angles. Again this measurement is repeated for ten different random time shifts applied to the data. For all these figures a noise level of 1 % was applied, comparable to the field data. The time shift leads to fairly big errors. Also, the figure shows that as a result of this time shift the method tends to underestimate the real angles of orientation when the real angle of orientation is smaller than 45 degrees, and to overestimate the real angles of orientation when the real angle of orientation is bigger than 45 degrees. This can be a cause for the underestimate of the angle in the field data, which all show an underestimate of the real angle of orientation, which is less than 45 degrees. To prevent the errors caused by the time shift, more accurate data collection is essential. Preferable is a frame in which the two antennas can be held, in contrast to the manual positioning of the antennas as was done in the field measurements. In that case the time shift can be eliminated and should not cause any problems.

Figure 3.27. The effect of inaccurate data collection on the accuracy of determining the angle of orientation of a cylinder. These figures show the calculation of the angle of orientation for ten cases, where the synthetic data are shifted up or down with a random time shift. The noise level is still 1 %. Stars are plotted at the mean value of the ten different measurements. A. No time shift applied to the data. B. Maximal random time shift is 1/8 times the wavelength. C. Maximal random time shift is 1/4 times the wavelength. D. Maximal random time shift is 1/2 times the wavelength.

3.5.3 Radiation pattern versus reflection of cylinder

The last source of error that was studied is the effect of the radiation profile. The radiation pattern is angle-dependent and the strength of a reflection is different dependent on the location of the origin of the reflection to the transmitter and receiver. There is a problem to distinguish between the angle-dependence of the reflection of the object itself and the angle-dependence due to the radiation pattern of the antennas. To obtain the correct angle of the orientation of the targets, the effect of the radiation pattern needs to be eliminated. For example Figure 3.28 shows the calculated angle of orientation of a cylinder versus the real angle of orientation of a cylinder located 1 meter along the line of data collection and 1 meter off that line. As a result of the radiation pattern, the calculated angle of orientation of the cylinder is different from the real angle of orientation of the cylinder. An underestimate is observed for angles of orientation smaller than 45 degrees and an overestimate for angles of orientation bigger than 45 degrees.

Figure 3.28. Effect of the radiation profile on the accuracy of determining the angle of orientation of a cylinder. The calculated angle of orientation of the cylinder is plotted versus the real angle of orientation of the cylinder when the cylinder is located 1 meter along the line of data collection and 1 meter off the line of data collection.

An attempt was made to eliminate the effect of the radiation pattern in the field data. Figure 3.29 shows the results for the calculated angles for the 3D grid after correction for the radiation pattern, in a similar representation as Figure 3.22. Before applying Alford rotation the measured traces are corrected for the theoretical expected radiation pattern. The correction resolved the underestimate of the angle of orientation of the cylinder. The calculated values seem in better agreement with the actual angle of orientation of the cylinder than the calculated values before the correction. However, there are more calculated values that are incorrect or seem to be at a 90 degree angle to the real angle of orientation of the cylinder.

Figure 3.29. Calculated results of the lines of the 3D grid, corrected for the radiation pattern. The figure shows a topview of the cylinder, which is shown in light gray in between line number 4 and line number 7. For every sample on every trace on every line in this 3D grid calculated the angle of orientation of the cylinder was calculated. Then a line is plotted in the direction of this calculated angle of orientation of the cylinder. The length of this line is as long as the amplitude of the reflection at that sample of that trace. The lines for the positive and negative angles are red and blue respectively.

This source of error is the hardest to remove in reality. In most cases one does not know the exact origin of the reflection, or the exact radiation pattern of the antennas at that specific site. Both have to be estimated, which can be hard. Specific cases have to be studied to decide which is the best way the handle these problems. The field data show that it is possible to extract the orientation of the cylinder, even without applying a radiation pattern correction, as the effect from the reflection of the cylinder seems bigger than the effect of the radiation pattern. In other situations all the available information has to be combined. For example, the theoretical radiation pattern is known for the four different configurations in the case when the object is located directly beneath the line of data collection. In the case when of vertical fractures identification and orientation estimation, the reflection in most cases originates from a point directly underneath the antennas. In the case when the object itself has an angle-independent reflection, like dipping planes, the angle-dependence of the reflection is completely dependent on the radiation pattern and again this method can be applied.

Because this radiation pattern is so important for exact data processing and handling, accurate recording of this pattern is recommended by measuring the reflection of a small, buried, metal sphere that guarantees an angle-independent reflection. This should improve the understanding of the radiation pattern, and the application of Alford rotation. However, it should be noted that this radiation pattern changes as subsurface parameters change.

3.6 Discussion and conclusions

In this study the value of multi-configuration data is evaluated. Several methods are applied to improve the final image of the data and to increase the available information on the target. The most important conclusion of this work is the importance of taking the radiation pattern into account when processing GPR data. In general, GPR data are processed using methods copied from seismic reflection data processing. However, it is shown that the radiation pattern of the GPR antenna strongly influences the recorded data and has to be acknowledged in the processing sequence. In this section further results and practical applications of the studied methods are discussed.

3.6.1 Alford rotation method

First the Alford rotation method is applied to multi-configuration GPR data to extract the angle of orientation of different subsurface targets with a distinct orientation. Theoretically it is possible to extract the exact orientation of various objects with a distinct orientation, like dipping planes, cylinders and vertical fractures.

Using 3DFDTD modeling the angle-dependence of the different reflections is demonstrated, and this angle-dependence is used to extract the angle of orientation of the targets. Examples are shown of dipping planes, cylinders, and vertical fractures. In all situations the angle of orientation of the different objects can be exactly predicted. Field data were collected to show that the method works for field data situations. The data show good results for determining the orientation of a cylinder.

The Alford rotation method can be applied in various other situations where the target is an object with a distinct orientation. For example the angle of orientation of targets like pipelines, cables, rebar constructions, and tunnels can be determined. This would add useful information about targets that are hard to image using conventional data collection. The possibility to determine the orientation of vertical fractures would be useful in shallow hydrological and geological situations. Multi-component borehole radar could prove to be useful in reservoir fracture and permeability orientation detection, which can be used for horizontal well design. GPR methods may have greater potential as they have better resolution than acoustic methods and deeper penetration than the regular electric logs. Fracture detection and orientation determination are also important for failure control of concrete structures like tunnels, bridges and buildings. Finally, this method can be applied to determine the angle of orientation of the main axis of anisotropy in media like coal seems (Cook, 1970; Coon et al., 1981), ice sheets (Kovacs and Morey, 1978), and schists (Tillard, 1994).

Error analysis shows that noise is not a major problem in this kind of data handling. Time shifts due to inaccurate data measurements could be a problem, but this problem can be prevented using better, more accurate data collection. The field data in this study were collected by manually placing the antennas and can be significantly improved by using a frame for the two antennas in all configurations. Even better results would be obtained using a radar system that allows one to collect all four configurations at one time without having to move the antennas.

Defining the difference between the angle-dependence of the radiation profile of the antennas and the angle-dependence of the actual reflection coming from the target might cause the biggest problem for this method. A correction has to be applied for the radiation pattern of the antennas to separate the true angle-dependent reflection from the target to determine its orientation. Consequently, accurate knowledge of the radiation pattern is a requirement. This can be hard as the exact radiation pattern of the antennas is only known in theory and varies with different subsurface parameters. Accurate measurement of this radiation pattern is recommended to improve the method. However, it is encouraging that even without the correction good results are obtained in the field data.

3.6.2 Weighted migration method

The weighted migration method was successful as well. Implementation of the radiation pattern into a weighted migration significantly increases the signal to noise ratio of the final image. This is shown both in a synthetic data and in the field data. As the source for the GPR data is highly directional and therefore significantly different from acoustic data it is necessary to change the migration algorithms used for GPR data to include the radiation pattern. This weighted migration method should also be included in processing of shearwave seismic data, which also has a highly directional radiation pattern. A similar signal to noise ratio improvement can be achieved as was found in the results of the GPR data.

Creating a GPR antenna set with a more focused radiation pattern would strengthen this effect. The more focused the radiated energy, the better one knows where the signal originates from and the better this knowledge can be used to improve the signal to noise ratio of the final image.

Just as in the Alford rotation method, accurate knowledge of the radiation pattern is a requisite for the weighted migration method. Again, it is encouraging that even with the theoretical radiation pattern good results are obtained in the field data.

Conversely, this method can be applied to find the radiation pattern. Therefore measurements are needed of several configurations over a localized reflection point, which forms a distinct hyperbola. A focusing analysis can be implemented in which the images of various migrations are compared using different radiation patterns. Thereby, the radiation pattern can be found that results in the best image. Combining different configurations makes the final image more uniform and less dependent on the location of the target. Signal to noise ratio can be improved when the target is located in the plane parallel to the main axis of the two antennas. It should be noted that in the case when GPR is used for object localization and data are collected using one configuration only, it is better to collect data using the xx-configuration than the commonly used yy-configuration. Using the xx-configuration sends most of the radiated signal into the plane perpendicular to the line of data collection instead of into the plane of the line of data collection, which is already covered by other shotpoints. As a result a larger part of the subsurface is covered by using the xx-configuration.

3.6.3 Alford migration method

It is shown that in theory the Alford rotation makes it possible to improve the migration algorithms. By collecting data in four different configurations and applying Alford rotation, the location of the origin of reflection can be found. This can be used to improve migration scheme. However this method is very vulnerable to noise present in the data. Presence of noise results in a wrong estimate of angle of and consequently in an incorrect migration. As a result this method is only effective for noise levels below 1 %. The Alford migration has been applied to field data. Although the data were collected under controlled conditions the amount of noise in the data is too high to make this method effective. Another problem with the field data is the fact that the data were collected using two antennas, which are positioned by hand, and the data were collected in four steps. Collection of multi-configuration data in one step using four antennas and a frame to hold the antennas can significantly improve the method.

In a situation where the horizontal and vertical dipoles have the same radiation pattern, this method can be expanded. By collecting GPR measurements in nine configurations: the yy-, yx-, yz-, xx-, xy-, xz-, zx-, zy-, and zz-configurations, not only the horizontal angle of the origin of the target, but also the vertical angle can be determined. In theory the location of the origin of the reflection can be exactly determined. This can be implemented in a migration scheme, which again leads to a faster and more accurate migration. However, this is not possible for current GPR surface measurements, as the radiation patterns of the dipoles in the horizontal and vertical directions are different (Enghata et al., 1982) and can not be combined in the Alford rotation. However, in air or other homogenous environment, where the radiation pattern is always the same, in theory it is possible to determine the exact location of the target. In that case the method can be applied to situations, where GPR is used for target identification and localization.

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configurations used (total number of configurations)	weighted migration	regular migration	migration of yy-configuration only	yy-configuration in most preferred direction	yy-configuration in least preferred direction
0 (1)	28.7	22.3	22.3	23.2	21.4
0, 90 (2)	38.8	29.3	29.2	36.0	28.5
0, 45, 90, 135 (4)	54.3	41.2	41.0	43.3	39.2
0, 30, 60�150 (6)	64.8	47.0	46.6	53.4	46.1
0, 15, 30�165 (12)	85.0	62.2	61.3	69.9	53.3

	noise level = 0 %	noise level = 0.1 %	noise level = 1 %	noise level = 10 %	time (min.)
range is 0 degrees	2147.4	610.3	99.3	25.8	2
range is 10 degrees	622.4	421.0	98.0	31.6	38
range is 20 degrees	349.3	267.3	93.8	32.6	71
range is 40 degrees	191.2	156.3	86.2	33.9	141
range is 90 degrees	91.8	84.3	64.5	35.7	330
range is 180 degrees	41.1	40.8	41.4	36.6	663
regular migration	80.1	80.0	79.9	75.1	58

angle of orientation of the cylinder	unmigrated	migrated
2D line: 0 degrees	-0.8	-2.0
2D line: 15 degrees	11.1	12.0
2D line: 30 degrees	29.3	31.0
2D line: 45 degrees	43.0	44.0
3D line 5: 30 degrees	28.6	28.0
3D line 6: 30 degrees	20.7	26.6
3D line 7: 30 degrees	26.4	27.0

	signal to noise level	time (sec.)
range is 0 degrees	19.78	35
range is 10 degrees	22.36	140
range is 20 degrees	25.4	240
range is 40 degrees	30.84	550
regular migration of four configurations	11.27	1115
regular migration of yy-configuration only	8.64	295

Structure and tectonics of the Puerto Rico-Virgin Islands platformandMulti-configuration Ground Penetrating Radar data.