Migration of Ground Penetrating Radar data in heterogeneous and dispersive media

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1 New Strategies for European Remote Sensing, Oui (ed.) 25 Mipress, Rotterdam, ISBN X Migration of Ground Penetrating Radar data in heterogeneous and dispersive media Armando R. Sena, Pau L. Stoffa & Mrina K. Sen University of Texas, Austin, TX, USA Keywords: Ground Penetrating Radar, migration, dispersion, heterogeneous media ABSTRACT: Migration enabes us to correcty interpret the GPR data from structuray compex areas and determine the geometry and correct ocation of refectors (or scatterers) in the subsurface. The effects produced by heterogeneities, dispersion and attenuation are very important; they may cause mis-positioning of refectors, degradation of resoution and reduction of the effective depth of observation. To address these issues, we present a modification of the Spit Step Fourier migration technique that takes into account, in an efficient and natura way, the dispersion and attenuation effects in the media. A homogeneous wave approximation gives greater numerica stabiity to the spit step operator, aowing the migration of the data through thicknesses equivaent to two or three times the equivaent skin depths of the media (computed at the dominant frequency of the radar signa). The resuts obtained from rea and synthetic data demonstrate that the deveoped migration technique improves the resoution of the fina image and recovers the correct ampitude of the refections. 1 INTRODUCTION Ground Penetrating Radar is a geophysica technique based on the propagation and refection of eectromagnetic waves in the subsurface. The technique has gained broad appication in areas such as archeoogy, civi engineering, gacioogy, environmenta sciences and many others. Davis and Annan (1989), present a genera review of this technique. GPR data are generay coected as common offset gathers. Standard processing of radar data does not incude migration. However, in structuray compex areas it becomes necessary to migrate the data in order to obtain a reiabe image and corresponding interpretation. The theory and techniques of migration methods have been deveoped for seismic appications over severa decades (e.g., Yimaz, 21). Migration invoves focusing of the recorded scattered wave fied to the subsurface ocations where the refected energy originated. The ampitude, form and phase of the migrated image are intrinsicay reated to the refection coefficients of the surfaces or interfaces where the refected waves were generated. Therefore, migration not ony aows us to obtain information about the geometry of the refectors, but aso gives us quantitative information on the refectivity coefficients at those surfaces or interfaces, which are reated to the physica properties of the rocks on both sides of such interfaces. However, to obtain the correct refectivity vaues the migration process must take into account a the effects produced by the media during the propagation of the eectromagnetic waves through them. 711

2 Among the techniques commony used in seismic migration, the Spit step Fourier technique (Stoffa et. a, 199) is based on wave fied extrapoation in the frequency-wave number domain and takes into account the effects of weak atera heterogeneities. Even though it is vaid for weak heterogeneities, the agorithm has been found to work we in cases of fairy strong ateray heterogeneous media. Since the wave fied extrapoation is carried out entirey in the frequency domain, it is rather straightforward to extend the agorithm to take into account dispersion and attenuation effects of EM waves when a proper formuation of the spit-step is deveoped for the EM wave equation. The conductive and dispersive characteristics of sois and rocks have been extensivey studied in the iterature (e.g., Hoekstra and Deaney, 1974; Scott et. a., 1963; and Sherman, 1988). Most sois and rocks are conductive, and therefore, attenuation of the GPR signa is amost ubiquitous of this technique. The magnitude of the conductivity can vary widey and depend on diverse factors such as water and cay content, porosity, concentration of eectroytes, temperature, etc. Most of these factors may aso induce dispersion of the eectric permittivity of the soi and rocks, making the phase veocity dependent on frequency and causing dieectric absorption. The dispersion of the eectromagnetic waves is important because it produces a broadening of the radar puse that degrades the resoution of the resuting GPR image. Carcione (1996) and Xu and McMechan (1997) presented numerica simuation of GPR data in ossy and dispersive media, confirming the attenuation and corresponding reduction of the resoution of the GPR data. Here we deveop the Spit step Fourier technique for migration of GPR data. The technique takes into account the heterogeneities, conductivity and dispersion effects of the media. We introduce an approximation of homogeneous waves that is coherent with the main approximation of the spit step Fourier technique and gives greater stabiity to the migration agorithm. This aows us to migrate the GPR data through thicknesses greater than two or three times the equivaent skin depth of the media computed at the dominant frequency of the radar signa. Migration of synthetic and rea data shows that the deveoped migration agorithm recovers the correct ampitudes and increases the resoution of the resuting image, thus improving the interpretation of the GPR data. 2 DISPERSION AND ATTENUATION OF THE GPR SIGNAL Attenuation of the GPR signa can be caused by conductivity osses and dieectric or magnetic absorption. Conductivity osses are the most common in geoogica materias and the effects of the variation of the conductivity with frequency are, in genera, very sma at the frequency range of the GPR technique (except when the conductivity is very high so that σ ω, which is the imit between the so caed diffusion and propagation regimes). On the other hand, dieectric dispersion (and its reated absorption) has been observed at different range of frequencies in sois and rocks (e.g., Sherman, 1988). In particuar, Ohoeft and Capron (1994) and Hoekstra and Deaney (1974) have reported on dieectric dispersion at the frequency range of the GPR technique (MHz to GHz). The dispersion mechanisms appear to be reated to interfacia (doube ayer) and moecuar reaxations. Carcione (1996) and Xu and McMechan (1997) modeed the propagation of eectromagnetic waves in the frequency band of the GPR technique indicating that dieectric dispersion in the subsurface can cause an important attenuation and reduction of the resoution of the GPR data. Ohoeft and Capron (1994) anayzed different soi mixtures of cay and siica with different percentages of water content. The Coe-Coe parameters obtained for one of the sampes are shown in tabe 1. Note that when the water content increases the resistivity decreases abrupty making the conductivity osses to rapidy overcome the dieectric absorption. 712 A.R. Sena, P.L. Stoffa & M.K. Sen

3 Tabe 1. Coe-Coe parameters for an engineering size-fraction cay soi (Ohoeft and Capron, 1994). Water content (weight %) τ (µsec) α σ -1 (ohm-m) > * x1 3 * Interpoated data. We interpoated this data to estimate the Coe-Coe parameters at different water contents. A the parameters where interpoated using quadratic functions, except the resistivity, which was interpoated using an exponentia function. In tabe 1, the ine with interpoated set of parameters is indicated with an asterisk. Ohoeft and Capron (1994) have attributed the dieectric dispersion, observed in these sampes, to the so-caed doube ayer interfacia poarization mechanism. Figure 1. a) Rea and imaginary parts of the product of the anguar frequency by the sowness as a function of frequency for a medium with dieectric dispersion and a medium with non dieectric dispersion. b) Phase veocity as a function of frequency for the same media shown in (a). Figure 1(a) shows the rea and (minus) the imaginary parts of the product of anguar frequency and sowness, computed with the Coe-Coe equation and corresponding to a soi with 2.5% water content (see tabe 1). We aso show the curves obtained with three Debye s mechanisms (e.g., Xu and McMechan, 1997) and the curves corresponding to a conductive media with not dieectric dispersion, reative eectrica permittivity equa to 5 and resistivity of 213 ohm-m. The imaginary part of the product corresponds to the so-caed attenuation factor α (the inverse of α is the socaed skin depth δ ). The rea part of the product is reated to the propagation phase and for nondispersive media its sope shoud be equa to one. Notice how the superposition of three Debye mechanisms approximates the Coe-Coe curve in this frequency band, and aso how for frequencies above 3 MHz, the attenuation factor of the media without dieectric dispersion practicay do not change with frequency and the sope of the rea part of the product approaches asymptoticay to one. On the other hand, for the dieectric dispersive medium the attenuation factor changes in an order of magnitude between 2 MHz and 3 MHz, and the sope of the rea part of the product does not approach asymptoticay to one. Figure 1(b) shows the corresponding phase veocities computed for the same cases presented in figure 1(a). Notice how the phase veocity changes in the entire frequency band for the medium with dieectric dispersion. On the other hand, for the medium Migration of Ground Penetrating Radar data in heterogeneous and dispersive media 713

4 without dieectric dispersion, the phase veocity reduces appreciaby for frequencies beow 6 MHz, approximatey. This expicit dispersion due to the conductivity of the media is important when the dominant frequency of the radar signa approaches to the imit between the diffusion and propagation regimes. If we use the Coe-Coe equation to express the dieectric dispersion, the compex eectric permittivity can be written as, ( ω) = + ( ) σ i, α 1+ ( iω / ω ) ω where and are the reative eectric permittivities for ow and very high frequencies, respectivey; is the eectric permittivity of the vacuum, σ is the eectric conductivity of the medium, α is the reaxation exponent, and ω τ is the reaxation frequency, which is reated to the characteristic reaxation time τ by ω = 1/ τ. The Coe-Coe equation can be approximated by a discrete superposition of Debye s mechanisms. The advantage of this representation is that it eads to time domain agorithms to mode wave propagation in dispersive media that are simpe and efficient, invoving the use of the so caed memory or hidden variabes, circumventing the costy convoution that is reated to the dispersion phenomenon (e.g., Carcione 1996). The representation of the eectrica permittivity as a superposition of discrete Debye s mechanisms is given by (Xu and McMechan, 1997), L a σ ( ω) = + i, 1+ = 1 ( iω / ω ) ω where ω are the reaxation frequencies of the Debye s mechanisms reated to the characteristic reaxation time τ by ω = 1/ τ, and a are the weighting factors of each mechanism. In practice, most of the energy of the radar signa is contained within a determined frequency band and three Debye s mechanisms constitute a very good approximation of the dispersion curve in such a frequency band (see figure 1). Adding more Debye s mechanisms wi improve the representation of the dispersion curve but has the drawback of increasing the number of memory variabes (and reated parameters) in the modeing agorithm. Tabe 2 shows the Debye s reaxation frequencies and weighting factors corresponding to the fitting of the Coe-Coe curve shown in Figure 1. Tabe 2. Debye s parameters corresponding to the fitting of the dispersion curve shown in Figure 1. a1 a2 a3 ω 1 (Hz) ω 2 (Hz) ω 3 (Hz) x x x SPLIT STEP FOURIER MIGRATION OF GPR DATA In most appication of the GPR technique, the survey ine is aigned aong a direction approximatey perpendicuar to the strike of the geoogic structure in the subsurface (or the principa orientation of the scatterer targets), and therefore, a 2D approximation is generay vaid. Under this approximation and for isotropic and smooth ateray varying media, the propagation of the eectromagnetic waves can be reduced to the foowing scaar wave equation (Sena et. a, 23), 714 A.R. Sena, P.L. Stoffa & M.K. Sen

5 2 2 2 E + ω u E =, where the compex sowness u is given by u = µγ, γ = iσ ω is the compex permittivity and ω is the anguar frequency. Dividing the medium into horizonta ayers of sma thickness dz and considering ony sma atera variation of the eectrica and magnetic properties inside each ayer, the eectric and magnetic properties of the media can be expressed as, µ ( z, x) = µ + µ ( x) ; ( z, x) = + ( x) and σ ( z, x) = σ + σ ( x), where µ, and σ are the mean vaues of the properties in each ayer, and µ ( x), ( x) and σ ( x) are their atera variations. Substituting these expressions in the wave equation and operating we obtain, E + ω u E = ω 2u ue, where 2u u µ ( iσ ω) + µ ( i σ ω) u u, 2 and u = µ ( iσ ω) = µ γ is the mean sowness. With these definitions of u and u the appication of the Spit step Fourier technique (Stoffa, et. a., 199) is straightforward. The Spit step Fourier technique is based on the successive appication of a phase shift in the frequency- wave number domain to extrapoate the wave fied through the ayer, and a phase correction in the frequency-space domain to takes into account the atera variation of the sowness. At each depth interva, the imaging condition is appied to obtain the migrated wave fied at that specific depth. A scheme of the migration technique is shown in figure 2. Figure 2. Scheme of the Spit step Fourier technique. 3.1 Homogeneous wave approximation When the sowness of a homogeneous ayer is compex, an impinging homogeneous pane wave wi originate a transmitted heterogeneous pane wave (e.g., Chen, 1983). To compute the compex vertica wave number k z in the Spit step Fourier technique we suppose that the ayer is homo- Migration of Ground Penetrating Radar data in heterogeneous and dispersive media 715

6 geneous. Sena et. a. (23) have shown that computing and appying the vertica wave number under this condition aows a stabe migration of the data ony through thicknesses smaer or comparabes to one skin depth. Since the ayers are, in genera, heterogeneous a more pausibe approximation can be proposed to compute the vertica wave number. We suppose that the pane waves are homogeneous, and therefore the changes in phase and ampitude take pace aong the same direction of propagation. With this approximation, the vertica wave number is given by, k = ωu 1 k / a, where z 2 2 x a= ω Im( u) 2Re( u)( 1 + (Im( u) / Re( u)) 1) 1/2 = Re(u ). 4 RESULTS AND DISCUSSIONS 4.1 Synthetic data The synthetic data have been generated with a Finite Difference Time Domain agoritm in which we have incuded the dispersion effects by using memory variabes associated to the Debye s dispersion mechanisms (Carcione, 1996; and Xu and McMechan, 1997). We aso used the expoding refector modeing technique (e.g., Yimaz, 21). Figure 3(a) shows the mode used to generate the synthetic data. The mode is composed of two different homogeneous media with an interface that is not horizonta. The medium I corresponds to the 2.5% water content soi discussed above whose Debye s mechanism parameters are shown in tabe 2. On the other hand, the medium II is non-dispersive and non-conductive, and has a reative eectric permittivity equa to 4. The ines shown on figure 3(a) are the refector surfaces introduced in the mode. One of the refector surfaces coincides with the rea interface between the two media. The waveet is the derivative of a hanning-squared function with a duration of 1 nanoseconds (and so, a dominant frequency of 1 MHz). Figure 3(b) shows the zero offset section obtained with the modeing agorithm. Notice how the ampitude, trave time and wave form of the received signa is affected by the dispersion mechanisms in the medium I. Figure 4(a) shows the migrated sections obtained when the conductivity and dispersion of the media are not taken into account, whereas Figure 4(b) shows the resuting migration when the conductivity and dispersion of the media are taken into account. In figures 4(a) and 4(b), the atera variations of the mode have been taken into account. Notice how the resoution, ampitude and position of the refector are appreciaby improved by introducing the attenuation and dispersion effects of the media. Figure 3. a) Subsurface mode used to generate the synthetic data. b) Zero offset section generated with the mode presented in (a). Note that in figure (a) the vertica and horizonta scaes are different. 716 A.R. Sena, P.L. Stoffa & M.K. Sen

7 Figure 4. a) Migration without taking into account the dispersion and conductivity effects in the medium I (see figure 3(a)). b) Migration taking into accounts those effects; note the overa improvement of the image and the correct ampitude and ocation of the refectors. 4.2 Rea data Figure 5(a) shows a common offset section (raw data) corresponding to a GPR ine acquired on a paeo-karst area near to Georgetown, Texas. The broadside-perpendicuar mode and a 5 MHz radar system where used for this survey. Figure 5(b) shows the eectric permittivity mode obtained through the veocity anaysis of four CMP gathers acquired at some ocations on the ine (indicated by arrows in figure 5(a)). Figure 5. a) Common offset 2D ine GPR data acquired on a paeo-karst area Georgetwon, Texas. The arrows indicate the ocation where common midpoint (CMP) gathers where aso recorded. b) Eectric permittivity mode obtained from the veocity anaysis of the CMP gathers. The mode was created by interpoation of the permittivity data obtained at the CMP ocations. A conductivity mode was aso estimated from these data using Archie s aw and a genera and empirica reation between water content and eectric permittivity (Topp et. a., 198). The estimated conductivity mode is strongy correated with the eectric permittivity mode and has a mean vaue of 3.5 ms/m. Unfortunatey, data on the dieectric dispersion in these rocks were not avaiabe, and so, we migrated the data supposing that no dieectric dispersion is present in the media. Figure 6(a) shows the migrated section without taking into account the conductivity of the media, whereas Figure 6(b) shows the migrated section obtained by taking into account the conductivity of the media. In both the cases, the atera variations of the media are taken into account. Notice the genera improvement of the resuting image when the conductivity of the media is incuded in the Migration of Ground Penetrating Radar data in heterogeneous and dispersive media 717

8 migration process. The correct ampitude of the refectors is recovered and the resoution of the image has been improved. The geoogica interpretation of this data is enhanced by the correct recovering of the refection ampitudes, improving the geoogica correation of the ayers and structura interpretation of the fauting and fractures present in the system. Notice that without migration, the detaied interpretation of these data woud not be possibe. Finay, notice that some diffraction signas, coming from scatterer objects on the surface, are present at the bottom of the section. The ampification of these signas and noise present in the data reduce the quaity and resoution of the image at the bottom of the section. Further improvement of the instrumentation combined with the new deveoped migration agorithm wi increase the resoution and effective depth of observation of this technique. Figure 6. Migration of the data shown in figure 5(a), a) without taking into account the conductivity of the media, and b) taking into account the conductivity. 5 CONCLUSIONS We have deveoped the Spit step Fourier technique for migration of GPR data taking into account, in a natura and efficient manner, the effects of heterogeneities, dispersion and attenuation in the media. The homogeneous waves approximation, used to compute the vertica wave number, makes the migration agorithm numericay more stabe, aowing the extrapoation of the wave fied through thicknesses up to two or three times the characteristic skin depth of the media (computed at the dominant frequency of the radar signa). Exampes of rea and synthetic data show that the new migration agorithm effectivey recovers the correct refection ampitude, improves the ocation, shape and continuity of the refectors and increases the resoution of the fina image. ACKNOWLEDGEMENTS We thank Dr. Roustam Seifouaev and Dr. Jay Banner for their suggestions and technica support in panning and acquiring the GPR data at the paeo-karst site. Armando Sena thanks CDCH-UCV and UTIG for the financia support received during his PhD studies at the University of Texas at Austin. 718 A.R. Sena, P.L. Stoffa & M.K. Sen

9 REFERENCES Carcione, J.M., Ground-penetrating radar: Wave theory and numerica simuation in ossy anisotropic media: Geophysics, 61, No. 6, Chen, H.C., Theory of eectromagnetic waves: a coordinate-free approach. McGraw-Hi, New York. Davis, J.L. and Annan, A.P., Ground-penetrating radar for high-resoution mapping of soi and rock stratigraphy: Geophysica Prospecting, 37, Hoekstra, P. and Deaney, A., Dieectric properties of sois at UHF and microwave frequencies: J. Geoph. Res., 79, No. 11, Ohoeft, R.G. and Capron, E. D., Petrophysica causes of eectromagnetic dispersion: Proc. of the 5th Internt. Conf. on ground-penetrating radar, Scott, J.H., Carro, R.D. and Cunningham D.R., Dieectric constant and eectrica measurements od moist rocks: A new aboratory methods: J. Geoph. Res., 72, No. 2, Sena A.R., Stoffa P.L. and Sen M.K., 23. Spit Step Fourier Migration of Ground Penetrating Radar Data: 73th Annua Internat. Mtg., Soc. Exp. Geophys., Expanded Abstracts, Sherman, M.M., A mode for the frequency dependence of dieectric permittivity of reservoir rocks: The Log Anayst, 29, No. 5, Stoffa, P.L., Fokkema, J.T., Freire, R.M. and Kessinger, W.P., 199. Spit-step Fourier migration: Geophysics, 55, No. 4, Topp, G.C., Davis, J.L. and Annan, A.P., 198. Eectromagnetic determination of soi water content; measurement in coaxia transmission ines: Water Resources Research, 16, Xu, T. and McMechan G.A., GPR attenuation and its numerica simuation in 2.5 dimensions: Geophysics, 62, No. 1, Yimaz, Ö., 21. Seismic data anaysis. Vo. 1. Ed: Doherty, S. M., Soc. Exp. Geophys., Tusa, USA. Migration of Ground Penetrating Radar data in heterogeneous and dispersive media 719

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