Wave Interferometry Applied to Borehole Radar - UConn School of ...

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 8, AUGUST 2007

Wave Interferometry Applied to Borehole Radar: Virtual Multioffset Reflection Profiling Lanbo Liu and Kuang He

Abstract—Based on wave-interferometry principles, we describe a procedure to synthesize monostatic and multioffset borehole radar reflection data from cross-hole radar tomography data. The procedure is equivalent to placing multiple transmitting sources in the receiving hole and conducting wide-angle reflection and refraction surveys. The procedure is illustrated using transmission and reflection data generated by numerical simulation of electromagnetic waves for a simple fracture model. The numerically simulated reflection data compare favorably with the reflection data synthesized using the wave-interferometry method. An experimental test of this procedure is also applied to a piece of field data from a pair of boreholes in crystalline igneous bedrock. The results demonstrate the potential of practical use of waveinterferometry methods for extracting reflection information from cross-hole radar tomography data. Index Terms—Borehole radar, cross correlation, multioffset reflection, virtual source, wave interferometry.

I. I NTRODUCTION

B

OREHOLE radar methods provide information useful for characterizing the shallow subsurface for hydrologic studies, contamination monitoring, and civil engineering investigations [1]–[6]. Borehole radar surveys are usually carried out in either cross-hole transmission mode, suitable for cross-hole tomography, or single-hole constant-offset (short transmitter–receiver separation) reflection mode. For example, [7] and [8] discussed tomographic imaging of a cylindrical object (e.g., a tunnel) between two boreholes by using crosshole transmission mode. Reference [9] modeled the single-hole constant-offset cylindrical dipole antenna for using reflection mode in oilfield exploration. References [10] and [11] discussed the theory and applications of using single-hole constant-offset reflection mode to detect a planar reflector (e.g., a thin fracture). Single-hole multioffset reflection-mode surveys also involve placement of the transmitter and receiver in the same hole, as done in the single-hole constant-offset reflection mode. Nevertheless, for each source position, the receiver positions are moved over the entire depth of interest, both above and below the transmitter. A multioffset borehole radar reflection survey is analogous to a surface ground penetrating radar wideangle reflection and refraction (WRR) survey [12]. Whereas

Manuscript received September 2, 2006; revised February 9, 2007. L. Liu is with the Department of Civil and Environmental Engineering, University of Connecticut, Storrs, CT 06269 USA (e-mail: [email protected]). K. He is with the Department of Physics, University of Connecticut, Storrs, CT 06269 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2007.900686

it is straightforward to conduct a WRR survey on the Earth’s surface, borehole radar hardware makes it difficult to conduct multioffset reflection-mode surveys in a single hole. Overcoming this difficulty would improve the utility of borehole radar reflection methods for identifying small-scale heterogeneities, including discrete fractures. This paper is motivated by the challenge of imaging hydraulically conductive fractures. Because of the small aperture of most fractures, it is difficult to locate or detect discrete fractures with borehole radar using cross-hole tomography methods. In contrast, fractures can be imaged by borehole radar using reflection methods, which exploit the high reflectivity of fractures (water-saturated fractures have higher relative dielectric permittivity and typically higher electrical conductivity than the rock host) [6]. The radiation pattern of most borehole-radar antennas, however, limits the strongest reflections to a narrow range centered on the plane parallel to the borehole; reflections from fractures perpendicular to the borehole are less likely to be detected. Multioffset reflection surveys, if practical, would improve resolution of fractures with suboptimal orientations. In this paper, we introduce an approach based on wave interferometry that uses cross-hole radar tomography transmission data to synthesize multioffset borehole radar reflection data. Reference [13] demonstrated that the autocorrelation of the seismic P-wave transmission response for layered Earth media yields the superposition of the reflection response and the time reversed transmission response. In recent years, use of wave-interferometry principles based on the correlation of wave responses acquired with a receiver array has increased; the concepts have been applied in physical science and engineering disciplines where they have been variously called “daylight imaging,” “virtual sources,” and “the empirical Green’s functions for arbitrary inhomogeneous media” [14]–[20]. We use the term “wave interferometric virtual source” (WIVS). WIVS has been used for nondestructive testing of construction materials [21], for biomedical imaging [22], and to image structures beneath complex overburden and salt domes [23]. In this paper, we apply the WIVS approach to borehole radar surveys, to synthesize single-hole monostatic and multioffset reflection responses from cross-hole radar tomography transmission data. II. T HEORIES OF W AVE I NTERFEROMETRY Geophysical waves (e.g., radar waves) propagating through the Earth traverse a highly reverberant environment, the result of a medium that contains numerous scatterers. For numerical modeling purposes, the medium can be bordered by 1) a perfectly reflecting medium (the Dirichlet boundary condition)

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LIU AND HE: WAVE INTERFEROMETRY APPLIED TO BOREHOLE RADAR

Fig. 1.

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Sketch to illustrate the WIVS problem.

to form a closed medium; 2) a perfectly transparent medium (the absorption boundary condition) to form an open medium; or 3) some combination of 1) and 2) on different sides of the model. We consider a 2-D, open medium case. Extension of the approach described below to three dimensions is straightforward. Consider two receiver points a at xa , and b at xb , and a source point c at xc , and the source is excited with a time function e(t) (Fig. 1). The two receivers and one source are randomly located in this domain with many multiple scatterers. If T (xa , xc , t) is the wave field recorded at xa , and T (xb , xc , t) is the wave field recorded at xb due to the source at xc , respectively, our goal is to recover the response at Location xa caused by a virtual source at Location xb , denoted as R(xa , xb , t), through the responses of T (xa , xc , t) and T (xb , xc , t). This is all that WIVS process concerns about. First, we can express the recorded wavefield at the two locations as T (xa , xc , t) = e(t) ∗ gac (t) T (xb , xc , t) = e(t) ∗ gbc (t)

(1)

where gac (t) and gbc (t) are the impulse responses between points a and c, and points b and c, respectively; and ∗ denotes the process of convolution. Equation (1) implies that the recorded responses at locations xa and xb are the convolutions of the impulse response at the respective point with the source excitation function e(t), i.e., gac (t) and gbc (t) are the Green’s functions for locations xa and xb when it has an excited source at xc in this particular inhomogeneous domain. The cross correlation of the wavefield recorded at a and b is then ∞ T (xa , xc , τ ) · T (xb , xc , t + τ )dτ

R(xa , xb , t) =

S

= TS (xa , t) ∗ TS (xb , −t) = δ(xa − xb )δ(t) − R(xa , xb , −t) − R(xa , xb , t).

= T (xa , xc , t) ∗ T (xb , xc , −t)

(3)

To make the relationship more explicit, it is straightforward to justify that we can drop the first two terms on the right-hand side of (3). The first term with the delta functions is mostly irrelevant to the problem, and the second term is noncausal with t < 0, so that only the third term has physical meanings. Finally, we get R(xa , xb , t) = −TS (xa , t) ∗ TS (xb , −t).

(4)

Equation (4) states that the receiver response at location xa to a virtual source at location xb , and reciprocally, the receiver response at location xb to a virtual source at location xa , is simply the cross correlation of the two received records at xa and xb , from all sources available, true and/or secondary, which contain all the information of the domain, regarding all the inhomogeneities and scatterers. For the case considered here, i.e., to get the reflection response as if the source (virtually, either point a or b) and receiver (point b or a) located in the same borehole, from the transmission response while the source (point c) and receivers (points a and b) reside in separate boreholes, (4) can be understood as more specifically with the letter R standing for reflection, and letter T for transmission as R(xa , xb , t) = −TS (xa , t) ∗ TS (xb , −t) = − T (xa , xs , t) ∗ T (xb , xs , −t).

−∞

(5)

s

= e(t) ∗ gac (t) ∗ e(−t) ∗ gbc (−t) = gac (t) ∗ gbc (−t) ∗ f (t) = gab (t) ∗ f (t)

xb is contained in gab (t), so that is the Green’s function for the response at xa for a source at xb ; and 2) the real response R(xa , xb , t) and the Green’s function gab (t) are proportional to each other and only differed by a factor of f (t). Moreover, it should be noted that R(xa , xb , t) has a length of 2 ∗ N − 1 in time with N the original time length of the records. The above discussion assumes that only one source exists at Location xc . In practical geophysical surveys, we definitely have to use numerous sources at different locations, although with either economical or physiographic limitations. Availability of multiple resources provides better constraints for improving the accuracy of the WIVS extracted Green’s functions. Numerous previous studies (e.g., [3] and [9]) have demonstrated that for an ideal case with many true sources or scatterers/reflectors available in the domain, the following relationship exists: T (xa , xs , t) ∗ T (xb , xs , −t)ds

(2)

where the factor f (t) = e(t) ∗ e(−t) depends only on the excitation function e(t) imposed at the source. Equation (2) indicates that 1) the impulse response between locations xa and

Equation (5) states that the virtual reflection wavefield R(xa , xb , t) between two points a and b in the same borehole can be achieved by the summation of cross correlations of two transmission wavefields T (xa , xs , t) and T (xb , xs , t) with the running index s going through all source points available. Apparently, longer recording time of the records possesses the advantage to allow the waves to pass through more

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Fig. 2. Two-dimensional synthetic DFN model. The size of the entire domain consists of 800 × 1000 grid cells. The white dots represent receiver antenna locations in the two boreholes (A and B). The red square in Borehole A represents a generic location of the transmitter antenna. The oblique conjugate sets of the longer fractures are interconnected and permeable. The horizontal and vertical short fractures are isolated. All fractures are filled with freshwater.

scatterers and bring back more information of the domain to xa and xb Now, let a and b in xa and xb be a pair of running indexes and go through all the desired location pairs, we can generate a new dataset in which there are no physical sources but only the virtual ones. This is fundamentally the way how WIVS works. III. WIVS A PPLIED TO A S YNTHETIC D ATASET We first test the idea of generating WIVS derived multioffset reflection borehole radar profiling with a synthetic dataset. Working on synthetic data possesses the advantage of the availability of both directly modeled and WIVS derived reflection profiles for a comparison to validate the approach. First, we briefly describe the fractured rock model used for this purpose. Then, we describe the procedure to generate the WIVS derived multioffset reflection profiling. A. Synthetic Dataset Generation Fig. 2 is a scheme of a conceptual model representing a cross section between two vertical boreholes (Borehole A on the left and Borehole B on the right with 3-m apart) within a fractured bedrock environment far from land surface. This setting is corresponding to the open medium discussed in the last section on WIVS theory. The discrete fracture network (DFN) model generates two sets of fractures: 1) the hydraulically interconnected fractures (the red thick lines in Fig. 2) between the depth of 1.25 to 3.75 m that allow the injected fluid from one hole to reach the other; 2) the hydraulically isolated fracture, including the smaller horizontal and vertical

Fig. 3. Snapshots of the simulated electric field at 4.17, 12.51, 20.85, and 29.19 ns for a radar source located at 2.5-m depth in Borehole A. All fractures are filled with freshwater. Diffraction and reflections from the major fractures are clearly seen from the snapshots.

fractures and the obligated fractures without connection to the interconnected ones. For the purpose of this paper, all fractures are filled with freshwater. No fluid replacement occurs in the interconnected fractures. Readers are referred to [6] for more detailed information about the model. Fig. 3 shows four temporally sequential snapshots of the radar wavefield generated by the FDTD model for an intuitive visualization of wave-fracture interaction. Reflection, diffraction, and wavefront distortion development can be clearly seen while the radar wave propagates away from the source in Borehole A at the depth of 2.5 m from these snapshots. Fig. 4 shows the modeled time sequences in Borehole A as the directly modeled multioffset reflection model profile [Fig. 4(a)] and the cross-hole transmission mode profile [Fig. 4(b)], for the source location at 2.5-m depth in Borehole A. For the reflection record, the first arrival wave-train is linearly dependent on time and shows they are the direct waves, and later events are the reflections from fractures. For the transmission record, in the proximity of the depth range of the interconnected fracture zone (1.25–3.75 m), the amplitude of the transmitted waves has been diminished due to the fact that a substantial portion of the energy was reflected back to the left of the model. Fig. 5 presents the radar profiles of the multiple-source multiple-receiver single-hole monostatic reflection profile and cross-hole level-run transmission. The monostatic reflection mode profile [Fig. 5(a)] clearly shows the radar signatures of multiple fractures. The most significant energy standouts (large amplitudes) in Fig. 5(a) are the responses to the major


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Fig. 6. Comparison of the FDTD modeled (left column) and WIVS-generated (right column) monostatic reflection profiles for two source positions (upper row: Yt = 1.45 m; lower row: Yt = 1.95 m) in Borehole B. Fig. 4. Synthetic radargrams for (a) single-hole reflection-mode and (b) transmission-mode measurements for the transmitting source located at depth of 2.5 m in Borehole A. The red, vertical broken lines denote the elapse times of the snapshots in Fig. 3. All fractures are filled with freshwater. The arrows indicate the wave trains of the reflected energy arrivals.

Fig. 5. Synthetic radargrams for (a) single-hole monostatic reflection-mode, and (b) cross-hole transmission-mode (also called level-run) measurements. All fractures are saturated with freshwater.

interconnected fractures. In the level-run transmission mode time record [Fig. 5(b)], the travel times of the first arrivals are not a constant, in the depth range of the interconnected, highly permeable fracture zone (1.25–3.75 m), the travel time is delayed, because a relatively large portion of the transmission path is water, the radar velocity in which is much lower than that in the background rocks. B. WIVS Redatuming The word of redatuming comes from seismic migration. It is given the name as if the source and receiver location is shifted

through the process of downward and/or upward continuation. We borrowed the term here to indicate the transform of the source of the transmission survey to a data expression in which it looks like that the source is moved to the receiver hole to form a multioffset reflection profile. Based on the FDN model shown in Section III, we generated the multioffset reflection profiles in the receiving-hole (Borehole B) of the transmission survey, with 45 source locations straddling the entire depth range as shown in Figs. 3 and 4. The WIVS-derived multioffset reflection profiles for two randomly selected source depths of 1.45 and 1.95 m are shown in the right column of Fig. 6. We have also shown the FDTD modeled multioffset reflection profiles for the same source depths (1.45 and 1.95 m) in the left column of Fig. 6 for comparison. For these two source locations, a comparison of the modeled and WIVS-derived multioffset profiles shows that the WIVS derived profiles bear similar features with the directly modeled ones. Only the reflection wave trains, not the direct waves are shown in the WIVS-derived profiles. WIVS redatuming for all the 45 source locations was carried out. Comparison of all the 45 modeled and WIVS-derived multioffset profiles has shown similar features as shown in Fig. 6. We have also compared the zero-offset monostatic reflection profiles obtained by direct modeling and WIVS redatuming, and the results are shown in Fig. 7. Monostatic reflection mode profiles can be obtained in both numerical modeling and field measurements. It is an important means to validate WIVS redatuming process by comparing the monostatic reflection profiles. Mathematically, the WIVS-derived monostatic reflection profiles is the autocorrelation of the transmission profile [13] at the same receiving location, it is a special case for the WIVS-derived multioffset reflection redatuming that involves cross correlation of records at two locations. The left and right panels of Fig. 7 bear numerous similarities in the time domain records, demonstrating that WIVS is a legitimate way to redatum data obtained by the transmission surveys. In addition, the WIVS-derived profile seems to be carrying out

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Fig. 7. Comparison of (a) the FDTD modeled and (b) WIVS-generated monostatic reflection profile in Borehole B.

Fig. 8. Sketch of a bedrock wellfield on University of Connecticut Storrs campus to show the relative position of the two wells (SIMA1 and SIMA2) in which borehole radar data were collected.

some kind of migration process, evidenced by it looks smoother and has fewer hyperbolic diffraction events. Since it takes about 25 ns for the source impulse to arrive at Borehole B [Fig. 5(b)], the total length with formation information is no longer than 30 ns for a total time window of 55 ns. Thus, the WIVS-derived monostatic reflection profile is about 25 ns. This fact reminds us it is better to record data as long as technically possible and economically allowable. It will be advantageous for later data analysis. IV. WIVS A PPLIED TO A F IELD D ATASET

Fig. 9. WIVS-generated multiple offset reflection profile at the depths of (a) 26 and (b) 54 m in the transmission receiver hole SIMA1. The red broken lines denote some reflectors.

A. Borehole Radar Field Survey WIVS redatuming is also applied to a real world crosshole transmission dataset obtained from a bedrock well field consisting of two wells (SIMA1 and SIMA2, see Fig. 8) penetrating 90 m into crystalline igneous bedrock (mostly gneiss and schist) on the University of Connecticut, Storrs, campus. The distance between the two holes is 25 m. Constant-offset reflections surveys were conducted in both SIMA1 and SIMA2; and the cross-hole transmission survey was conducted with the transmitting antenna in SIMA2, and the receiving antenna in SIMA1. B. WIVS Redatuming The WIVS-redatumed multioffset reflection profiles in SIMA1 at the source depths of 26 and 54 m are shown in Fig. 9 as the examples of WIVS redatuming for the real-world data. The profile at the source depth of 29 m shows clearly a strong reflector in a later time, and the calculation indicates it should be located out of the space domain between SIMA1 and SIMA2. Fig. 10 compares the measured and WIVS-derived monostatic reflection profile. There are numerous vertical events in the WIVS-derived profile, indicating the site at the depth of radar survey (the depths of 10–90 m) may be dominated by vertical and horizontal fractures and cracks. The measured monostatic

Fig. 10. Comparison of the (a) measured and (a) WIVS-derived monostatic reflection profiles in the receiving-hole (SIMA1) based on cross-hole transmission survey.

profile [Fig. 10(a)], is contaminated by the scattering hyperbolas originated from the tips of openings and fractures. V. D ISCUSSION From the examples of applying WIVS to the synthetic and measured borehole radar data, we can see that WIVS redatuming may provide us some advantages. 1) Compensate the technical difficulties to run single-hole multioffset surveys that are useful for detecting fractures


oriented not in favor of zero-offset monostatic reflection surveys. 2) All reflectors identified from WIVS redatuming should have a normal in the plane formed by the two wells for running the cross-hole transmission measurements. This is advantageous to determine the azimuthal orientation of reflectors obtained by single-hole monostatic reflection survey using omnidirectional transmitting antenna. VI. C ONCLUSION WIVS provides a new angle to visualize the data. It is complementary to raw cross-hole transmission data and shows formation features in a more convenient way. From the examples presented above, we can conclude: 1) WIVS is a data-based redatuming or transformation process. It is advantageous for characterizing certain kinds of subsurface features such as fractures; 2) longer recording time of field data will be helpful, since longer record contains more multiple reflections and diffractions from more formation features, thus contains richer formation information. In the near future, we plan to carry out migration for both profiles and compare them in the space domain other than the time domain. This comparison may provide more insights for improving the WIVS redatuming process. ACKNOWLEDGMENT The authors would like to thank P. Joesten and L. Zhu, through the Branch of Geophysics, U.S. Geological Survey (USGS), who acquired the borehole radar dataset used in this paper. The authors would also like to thank the careful review made by A. Waxman, F. Day-Lewis, and J. Lane (all with USGS).

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Lanbo Liu received the B.S. and M.S. degrees in geophysics from Peking University, Beijing, China, in 1978 and 1981, respectively, and the M.S. degree in civil and environmental engineering and the Ph.D. degree in geophysics from Stanford University, Stanford, CA, in 1992 and 1993, respectively. He was the Carnegie Fellow of the Carnegie Institution of Washington from 1993 to 1995, and was with the faculty of the University of Connecticut, Storrs, in 1995. He is also a Consulting Expert in geophysics for the Cold Regions Research and Engineering Laboratory, U.S. Army Engineering Research and Development Center. He has more than 80 publications in peer-refereed journals, conference proceedings, and technical reports. He served as an Associate Editor for Geophysics during 2003–2005, and a Guest Editor for the special issue on urban geophysics in Journal of Geophysics and Engineering in 2006–2007. Dr. Liu is a member of American Geophysical Union, Acoustic Society of America, Society of Exploration Geophysicists, Seismological Society of America, and Society of Environmental and Engineering Geophysics, and an invited speaker at numerous professional conferences and various educational, industrial, and governmental institutions.

Kuang He, photograph and biography not available at the time of publication.