Near Surface Geophysics, 2006, 275-283
Imaging subsurface objects by seismic P-wave tomography: numerical and experimental validations Gilles Grandjean* BRGM, BP 6009, 45060 Orléans, France Received January 2005, revision accepted November 2005 ABSTRACT We tackle the problem of characterizing the subsurface, more specifically detecting shallow buried objects, using seismic techniques. This problem is commonly encountered in civil engineering when cavities or pipes have to be identified from the surface in urban areas. Our strategy consists of processing not only first arrivals, but also later ones, and using them both in tomography and migration processes, sequentially. These two steps, which form the basis of seismic imaging, can be carried out separately provided that the incident and diffracted wavefields are separated in the data space. Tomography is implemented here as an iterative technique for reconstructing the background velocity field from the first-arrival traveltimes. The later signals are then migrated by a Kirchhoff method implemented in the space domain. To study the reliability of this methodology, it is first applied to synthetic cases in the acoustic and elastic approximation. Both the background velocity field and the local impedance contrasts are reconstructed as defined in the predicted model. An experimental case, specifically designed for the purpose, is then considered in order to test the algorithms under real conditions. The resulting image coincides well with the predicted model when only P-waves are generated. In the elastic mode, surface waves make P-wave extraction difficult, so that the reconstruction remains incomplete. This is confirmed by the real data example. Finally, we demonstrate the appropriateness of the proposed method under such circumstances, provided that suitable preprocessing of data is carried out, in particular, the removal of surface waves. INTRODUCTION Among the many difficulties encountered in the environmental and civil-engineering fields, characterization of the subsurface from a structural and a mechanical point of view constitutes the major challenge. Seismic sounding is an appropriate geophysical method to tackle this problem because of its ability to produce high-resolution images of the first 100 m of the subsurface. Numerous examples illustrate the diverse applications where seismic methods are effective, whether for detecting cavities (Grandjean et al. 2002), imaging active faults (Pratt et al. 1998) or delineating aquifer structure (Liberty 1998; Grandjean 2006). Such seismic images are generally difficult to obtain, especially when the subsurface is characterized by strong velocity variations with heterogeneities similar to the seismic signal wavelength. Nevertheless, we show that such media can be studied by this means, provided that both the first and the later arrivals are considered. Our work is particularly focused on applications related to civil engineering, such as the detection of shallow buried objects, like cavities or pipes. As Mora (1989) has already described, the complex problem of tomography and migration in subsurface imaging needs to be *
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© 2006 European Association of Geoscientists & Engineers
tackled. The former is dedicated to estimating large wavelength velocity contrasts, here called the background velocity. By extending the normal moveout (NMO) analyses, the tomography concept has the advantage of being applicable to laterally heterogeneous velocity fields. Moreover, seismic antenna geometry can account for complex configurations making the data independently indexed to a source–receiver pair, in contrast to classical seismic profiling where data refer to CDP gathers. Experimental seismology provides good examples of the possibilities, as shown by Nolet (1987). The migration process is often considered for the step following tomography because it relies upon the background velocity field to re-allocate the seismic events to their real positions. Numerous techniques can be used, depending on the complexity of the sounded medium, the kind of data considered, i.e. prestack or post-stack, and the available a priori information on the velocity field. For complex media, the prestack depth migration is generally used for ensuring the focusing of reflectors (Robein 1999). In particular, reverse-time migration has been successfully tested for detecting cavities from radar-wave measurements (Hui Zhou and Sato 2004). New methods have been developed more recently, either for reducing the computation times (Ecoubtet et al. 2002) or improving the inverse problem conditioning. For example, the 275
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stereotomograhy method (Gosselet et al. 2003) inverts the velocity field and the medium reflectivity at the same time. This technique requires the picking of locally coherent seismic events by determining their arrival time and their slope in common receiver and source gathers. Because this work can be very detailed and demands major computational resources, we opted for a strategy where tomography and migration processes are carried out separately. This two-step approach, taken as the basis of seismic imaging, can be carried out sequentially, provided that the incident and diffracted wavefields are separated in the data space (Dessa et al. 2004). Here, we chose to use a simultaneous iterative reconstruction technique (SIRT) tomography algorithm for reconstructing the background velocity field from the firstarrival traveltimes. The late arrivals are then re-allocated by a Kirchhoff migration technique computed directly in the space domain to image the local velocity anomalies. This imaging technique is tested with both with synthetic cases and with a real example, in order to study its robustness, as well as possible associated difficulties. THEORETICAL APPROACH In the tomography and migration processes, we first need to define and solve the forward problem of wave propagation for computing traveltimes. According to previous work in this field, several alternatives can be considered, for example, ray tracing or the finite-difference time-domain (FDTD) method. Ray tracing (Cˇ everný et al. 1977) is defined within the framework of the asymptotic-ray theory in the high-frequency approximation. Using the Born approximation, ray tracing is a fast and robust way to compute wavepaths, provided that the medium is sufficiently smoothed. This condition rules out media that are too rough, thus making ray tracing unsuitable for our purpose, as we are considering the highly heterogeneous subsurface domain. In our approach, we aim to represent wavepaths by fat rays, i.e. Fresnel volumes, because they depend intrinsically on the wave resolution (Husen and Kissling 2001). Fresnel volumes are computed by solving the eikonal equation, which can be numerically resolved, for example by the well-known FDTD method (Virieux 1986). This scheme uses a numerical integration of the equation to compute the wavefield propagation on a grid that is spatially discretized according to the signal wavelength divided by ten. Each grid node is characterized by the properties of the medium, making it possible to work with highly heterogeneous media. Unfortunately, the FDTD method is time-consuming, so we decided to tackle the forward problem by using another method based on a faster numerical technique. In the following sections, we describe the principles of the proposed imaging technique and establish the specific conditions of our approach: the use of Fresnel wavepaths in the forward step of the inverse problem, a probabilistic reconstruction algorithm for reconstructing the background velocity field, and a Kirchhoff-based migration in the space domain to solve the diffraction imaging problem.
FMM and Fresnel wavepath approach The solution of the eikonal equation on a discrete lattice by a fast tracking scheme was presented by Cao and Greenhalgh (1994). More particularly, the fast marching method (FMM), proposed by Sethian and Popovici (1999), solves this equation by an upwind scheme for reducing the computing times. Implemented numerically to the second order (Grandjean and Sage 2004), the algorithm consists of successively searching for the lattice gridpoint to which the wavefront has to be propagated. This procedure, which mimics a monotonously expanding wavefront, is repeated until the lattice has been completely scanned. In fact, this procedure can be considered as another expression of the Huygens’ principle. Cˇeverný and Soares (1992) used a ‘paraxial ray theory’ to define Fresnel volumes. In a medium between source S and receiver R, the Fresnel volume is defined by the set of points M, where the waves are delayed after the shortest traveltime tSR by less than half a period. These waves can thus be added constructively to form the first arrival of the wave. This approach considers the centre frequency of the wave in the analysis, thus enabling the evaluation of the tomography resolution and reducing the sparseness of the ray distribution. Frequency band-limited waves propagating in the ground are thus affected not only by 1 Compute the traveltimes from S to R through each gridpoint M of the grid: tSMR = tSM structures along the raypath, as assumed by the ray theory, but also + tRM.by structures located in the vicinity of the raypath. Using this approach, Watanabe from et al.S(1999) proposed algorithm 1 Compute the traveltimes to R through eachan gridpoint M offor the grid: tSMR = tSM calculating a Fresnel volume: 2 At each gridpoint M, estimate the difference 't (Harlan 1990) between tSMR and tSR; + tRM . 1 Compute the traveltimes from S to R through each gridpoint where tSR the is the shortest traveltime S to R; M of grid: tSMR = t + tRMfrom . SM 22 AtAt each gridpoint M, M, estimate the the difference 't (Harlan 1990)1990) between tSMR and tSR; each gridpoint estimate difference ∆t (Harlan 3 Atbetween each gridtSMR point whether is less than half a period:from andM,tSRtest ; where tSR 't is the shortest traveltime where tSR is the shortest traveltime from S to R; S to R; 1 M, test whether ∆t is less than half a 3 At each grid point t each tSM grid t RM point tSR M, test , whether 't is less than half a period: ( 3'At 2f period: 1 't tSMf is the t RM centre tSR frequency , where of the propagating wave. 2f
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et al.frequency (1999) alsoofproposed a numerical definition of Fresnel volume where Watanabe f is the centre the propagating wave. where f is the centre frequency of the propagating wave. Watanabe et al. (1999) also proposed a numerical definition characterized by a weighting functionbyZathat dependsfunction linearly on of Fresnel volumes, characterized weighting ω the thatdelay expressed in Watanabe et al. (1999) also proposed a numerical definition of Fresnel volume depends linearly on the delay expressed in (1): (1): characterized by a weighting function Z that depends linearly on the delay expressed in
§ 1 · (1): °1 2 f 't , ¨ 0 't ¸, 2f ¹ ° © Z ® (2) ( § 1 · °0, §¨ · 1 't . °°1 2 f 't , ¨©02 f 't ¸¹ ¸ , ¯° 2f ¹ © ( Z ® § 1 · °0, ' . t formulation will usedininthe thetomographic tomographic reconstruction ¨ ¸bebeused This formulation will reconstruc- to compute the ° This ©2f ¹ ¯ tion to compute the seismic path from a source to a receiver at seismic path from a source to a receiver at each iteration. One important point in the each iteration. One important point in the inverse problem of This formulation will be used in the tomographic reconstruction to compute the seismicproblem imaging is related to underconditioning. Because theBecause the seism inverse of seismic imaging is related to underconditioning. seismicpath wave interacts subsurface objects onlyOne from the point in the seismic from a sourcewith to a receiver at each iteration. important wave interacts with subsurface objects only from the surface, their shadow side is inverse problem of seismic imaging is related to underconditioning. Because the seism poorly investigated, creating divergence in the solution space. This effect can be interacts & with subsurface only from the 2006, surface, their shadow side is © 2006 European Association ofwave Geoscientists Engineers, Nearobjects Surface Geophysics, 4, 275-283 damped by using regularization techniques during the reconstruction (Gosselet 2004). poorly investigated, creating divergence in the solution space. This effect can be With our approach, regularization is straightforward since the spatial influence of the damped by using regularization techniques during the reconstruction (Gosselet 2004). local velocity variations on Fresnel wavepaths depends on the centre frequency of the
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Imaging subsurface object with seismic 277can be written as k 1 whereatN iteration represents the total is number ofgiven Fresnelby wavepaths crossing The updated cellwhere slowness k+1, then stomography s k 'cell s j ,j. which j N represents the total number of Fresnel wavepaths crossing cell jj. The updated cell slowness at iteration k+1, is then given by s kj 1 s kj 's j , which can be written as cell slowness at iteration k+1, is then given by s kj 1 s kj 's j , which can be written as
§ · surface, their shadow side is poorly investigated, creating diver§N · ¨§ ·Zij ¸ ¸ ¨ ' t 1 k 1 gence in thewave: solution space.theThis effect can damped by using (7) (7) ¨1s s ¨'1t 1Zi ' ¸ t Z ¸ .¸ . s kj ¨more ( 1 ¦ j ¦ propagating the lower frequency, the be fatter the Fresnel wavepath sand the M C ¸t . s s ¨¨1 N¦i¨ 1 tN (7) ¸¸ ¨ N ¨t i ¸ Z¦ Z ¸ regularization techniques during the reconstruction (Gosselet ¨¨ © ¦ Z¦ ¸ ij ¸ ¹ j ¹1 © ¹ the velocity far from ray are taken into account in since the reconstruction. 2004). Withvariations our approach, regularization is straightforward propagating wave: the lower thethe frequency, the fatter the Fresnel wavepath and the ©more This SIRT reconstruction algorithm waswas initially implemented in JaTS (Java the spatial influence of the local velocity variations on Fresnel ThisThis SIRT reconstruction algorithm initially implementSIRT reconstruction algorithm was initially implemented in JaTS (Java the velocity variations far the fromcentre the rayfrequency are taken into account in the reconstruction. SIRT reconstruction algorithm was Grandjean initially in JaTS (Java Tomography Software; Grandjean and Sage 2004). It computesimplemented the average slowness wavepaths depends on of the propagating ed Tomography inThis JaTS (Java Tomography Software; and Sage Software; Grandjean and Sage 2004). It computes the average slowness Tomography of background velocities wave: the lower the frequency, the fatter the Fresnel wavepath 2004). It update computes update value onthecell value onthe cell j,average and assumesslowness that every Fresnel wavepath having samej, total Tomography Software; Grandjean and Sage 2004). It computes update value on cell j, and assumes that every Fresnel wavepath having the same totalthe average slowness and the more the velocity variations far from the ray are taken and assumes that every Fresnel wavepath having the same total Tomography of background weight ¦ Z has an equal probability of explaining the traveltime difference 't In theaccount following, the indices i, jvelocities and k are related to the source–receiver number, the cell into in the reconstruction. weight hasancell an probability the traveltime update on j,equal and assumes that of every Fresnel wavepath having the same total equal probability of explaining theexplaining traveltime difference 't weightvalue ¦ Z has difference ∆t between the observed and the calculated travelnumber and the iteration number, respectively. fundamentals of thenumber, proposedthe SIRT between the observed and the calculated traveltimes. Rather than using an arithmetic M cell In the following, the indices i, j and k are relatedThe to the source–receiver between the observed and the calculated traveltimes. Rather than using an arithmetic Tomography of background velocities times. Rather than usingprobability an arithmetic mean, Watanabe et al. Zij has an equal of explaining the traveltime difference 't weight ¦ mean, Watanabe et al. (1999) averaged the slowness update over the sum of all the javeraged 1 mean, Watanabe et al. the (1999)slowness averaged the slowness update overthe the sum of all theall the state that a slowness perturbation 's in cell j generates a traveltime perturbation 't, In the following, the indices i, j and k are related to the source– (1999) update over sum of number and the iteration number, respectively. The fundamentals of the proposed SIRT receiver number, the cell number and the iteration number, Fresnel weights in cell j,j,i.e. Z .. Consequently, Consequently, low-weight low-weight cells are updated in the Fresnel weights in cell i.e. ¦ cells are updatedthan in the using an arithmetic Fresnel the weights in cell j, i.e. ¦ Zthe. Consequently, between observed and calculatedlow-weight traveltimes. Rather according state that atoslowness perturbation 's j generates a traveltime respectively. The fundamentals of in thecell proposed SIRT state thatperturbation a cells 't, are updated in the same manner as high-weight cells. same manner as high-weight cells. Because low-weight cells are defined as low slowness perturbation ∆s in cell j generates a traveltime perturBecause low-weight cells are defined low probability zones sameWatanabe manner as high-weight cells. Because low-weight are defined as low over mean, et al. (1999) averaged theascells slowness update the sum of all the O toC according ' s ' ' t t t t j bation ∆t, according to that can explain the differences between observed and calculated probability (3) zones that can explain the differences between observed and calculated 's j s j C , C C N 9 sj t t t traveltimes, logically relateZtothe to the low-weight slowness pertur9weight Consequently, cells are updated in the Fresnel weights in we cell j,we i.e. traveltimes, logically relate the weight slowness perturbation applied to each ¦ ij .the 's j t O t C 't 't ithe 1 , (3) ' s s bation applied to each cell: lower the probability, the weaker (3) j j C C C cell: the lower the probability, the weaker the slowness perturbations. With JaTS-SIRT, sj t t t where the superscripts O and C refer respectively to observed and calculated thefirst slowness perturbations. With JaTS-SIRT, the cell weight ωij same manner as high-weight cells. Because low-weight cells are defined as low is normalized the all sum the of allweights the weights ¦ FresnelFresnel the cell weight Zijby where the superscripts O and C refer respectively to observed is normalized the sumbyof Z ininthatthat arrivals. We can now state thatC the Fresnel weightstodefined in (2) represent the where the superscripts O and refer respectively observed and calculated first and calculated first arrivals. We can now state that the Fresnel wavepath. This increases the influence of small-scale Fresnel wavepath. This increases the influence of small-scale Fresnel wavepaths compared to weights defined in (2) represent the probability of a velocity wavepaths makes intuitive probability a velocity perturbation delaying the wave propagating from source arrivals. Weofcan now state that the Fresnel weights defined in (2) represent the to compared to larger-scale ones, which 9 larger-scale ones, which makes intuitive sense. perturbation delaying the wave propagating from source to sense. receiver 't∆t. .aThus, thethe probability that a traveltime difference 't isfrom generated probability velocity perturbation delaying wave propagating sourcebytothe receiverby byof Thus, probability that a the traveltime difference Diffraction imaging ∆t is generated by the cell j crossed by a Fresnel wavepath i Diffraction imaging cell j crossed a Fresnel wavepath ithat becomes receiver by 'tby . Thus, the probability a traveltime difference 't is generated by the becomes After the JaTS-SIRT algorithm has been used to reconstruct the After the JaTS-SIRT algorithm has been used to reconstruct the background velocity, background velocity, we can re-allocate late arrivals by using a cell j crossed by a Fresnel wavepath i becomes we can re-allocate late arrivals by using a migration process. We consider a point (x,z) Z migration (4) process. We consider a point (x,z) located in an elastic Pij't M ij , (4) located in an elastic homogeneous medium bounded by a surface 6. The homogeneous medium bounded by a surface Σ. The backpropa¦ ZZij 't j 1 ij Pij , (4) pressure backpropagation of the pressure fieldPP from from 6 to is given by the gation of the field Σthe topoint the(x,z) point (x,z) is given by M where¦MZijrepresents the number of grid cells. To reduce the theKirchhoff Kirchhoff (Berkhout 1987): integralҏintegral (Berkhout 1987): j 1 M represents the∆t number of grid To reduce thecalculated traveltime difference 't where traveltime difference between the cells. observed and the § r· r · § (8)(8) P v traveltimes,observed the quantity the sj has to be modified. Thethe updated slow modified. ³ D P ¨W ¸ E v n ¨©W V ¸¹ d6 , between calculated traveltimes, quantity sj hasdifference to be where M the represents theand number of grid cells. To reduce the traveltime 't © V ¹ ness s+∆s to be applied to cell j crossed by a Fresnel wavepath i wherei τ Wcan isis the the propagating time, r is thetime, wavepath, the velocity of the medium, The slowness s+'s to product be applied j crossedperturbation by a Fresnel wavepath can updated now be expressed as the of to thecell slowness where propagating r VisP is the wavepath, VP is the between 't Zij the observed and the calculated traveltimes, the quantity sj has to be modified. 'sij s defined (5) in. (3) and the probability defined in (4): velocity of the medium, P and v ⋅ n are respectively the pressure P(5) and v n are respectively the pressure field and the normal component of the j C M 6 Σ Σ t updated now be Z expressed as the product the slowness (3)field and and the The slowness s+'s to beofapplied to cell jperturbation crossed by adefined Fresnelinwavepath i canthe normal component of the velocity vector recorded ¦ ij velocity vector recorded at the surface, and D and E are weighting parameters Zij j 1 't at the surface, 'sij s j C defined (5) . in (4): (5) and α and β are weighting parameters introduced M probability now be expressed as the product of the slowness perturbation defined in (3) and the in the Green’s function. This expression can be simplified by considering the t Zij in introduced the Green’s function. This expression can be simplified by ¦ 'sij as a discrete random variable We now consider j 1variable nsider 'sij as a discrete random having as many values as having as many values as acoustic case and neglecting amplitude effects in the propagation. The reflectivity R at considering the acoustic case and neglecting amplitude effects in probability defined in (4): point (x,z) is then deduced a double summation, related(x,z) to the is then consider ∆s as a the discrete variable having as thethepropagation. The by reflectivity R atrespectively the point ij areWe Fresnel i crossing cellslowness j. In random order to determine the slowness avepaths ithere crossing cellnow j. Inwavepaths order to determine We nowasconsider 'sFresnel randomi crossing variable having values asby a double summation, respectively related to the ij as a discrete many values there are wavepaths cell j. as In manydeduced Kirchhoff integral and to the data redundancy due to the multifold acquisition (Robein correction to be to cell we compute the mathematical expectation the Kirchhoff integral and to the data redundancy due to the multiplied to cell j, we compute the mathematical expectation of the order to applied determine thej,slowness correction be applied to cellthe j,ofslowness 1999). For Nj sources and Ni receivers, we have there are Fresnel wavepaths i crossing cell j. Into8order to determine we compute the mathematical expectation of the discrete random fold acquisition (Robein 1999). For Nj sources and Ni receivers, random variable and correction: obtain the average slowness correction: iable and discrete obtain the average slowness 8 mathematical correction to be applied cell j, we computecorrection: the expectationwe of have the variable and obtain thetoaverage slowness 10
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where (6) X and Z are the coordinates of the receiver i at the surface, , where Xand and tZ are the receiver i at the surface, and tSj and tij are the where N represents the total number of Fresnel wavepaths crossand t are the thecoordinates traveltimesoffrom the source j to the point Z ¦ where Nofrepresents the total number ofijcell Fresnel ij the total number Fresnel wavepaths crossing j. Thewavepaths updated crossing cell j. The updated Sj j 1 ing cell j. The updated cell slowness at iteration k+1, is then (x,z) and from point to i, respectively. traveltimes fromthis the source j tothe thereceiver point (x,z) and from this For pointconto the receiver i, k 1 k k 1 k slowness at iteration k +1, is then given by s s ' s , which can be written as ation k+1,cell is then given by s s ' s , which can be written as given by which can be written as venience, (9) can be rearranged as j j j j j the total j where N represents number of Fresnel wavepaths crossing cell j. The updated 's j
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respectively. For convenience, (9) can be rearranged as
k 1 cell s kj 's j , which can be written as § slowness at iteration · k+1, is then given by s j · 1 t ! t 2T ¨ ¸ N 'ti ZAssociation 'ti Zij ¸¸ k 1 Rx , z Pij275-283 tij x, z 0 , (1 ij of Geoscientists & Engineers, Near Surface Geophysics, 4, ¦¦ ¨ 20061 European ¸. s.j s k© (7) Nj 2006, (7) ¦ j 1 C M j i C M ¨ N i§ 1 ti ¸ ¸ ti · Zij ¸ ¦ Zij ¸ ¦ ¨ ¨ ¸ j 1 1 j N1 'ti¹ Zij ¸ ©k 1 ¹ k ¨ sj s j 1 ¦ C M . (7) t !t 2T where tij x, z 0 is the traveltime map computed from the source j to points (x,z) of ¨ N i 1 ti ¸ Zij ¸ ¦ ¨ This SIRT 1 j algorithm construction algorithm was©reconstruction initially implemented JaTSinitially (Java implemented in JaTS (Java ¹ in was
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FIGURE 3 (a) Background velocity field inverted from the first-arrival traveltimes:
FIGURE 1 (a) Synthetic velocity model. (b) Fresnel wavepath from the source (circles) #1 to the receiver (dots) #15, showing two possible paths for the wave to reach the receiver at the same time. (c) Synthetic seismogram computed in the acoustic approximation at source #1 and recorded at 25 receivers; the diffraction patterns are indicated by arrows.
Rx , z
1 ¦¦ Pij X i , Z Nj j i
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these were computed from the model in Fig. 1(a) in the acoustic approximation; the two layers and their respective velocities are clearly visible. (b) Anomaly reconstructed from the late arrivals using the velocity field shown in (a); despite some ‘smiles’, the boundaries are well reconstructed.
In order to discard the direct wave, the summation only considers amplitudes later than first arrivals, i.e. t0 plus two periods T. This operation is equivalent to performing an automatic muting above the 2T value. By definition, the summation of the late arrivals as proposed here requires that only diffractions or reflections of P-waves are present. This important point affects the applicability of this technique and is discussed in the following sections. For filtering the surface waves, known to be the strongest signals in surface seismics, we can use either frequency filtering or the Karhunen–Loève technique, the latter being well-suited to separating dispersive waves from others (Bitri and Grandjean 2004). In the following sections, we consider three cases of increasing complexity. Two synthetic cases, realized in the acoustic and elastic approximations, are used to investigate the efficiency of the algorithms when either guided waves or wave conversions are not generated but are observed; a discussion presents some (9) filtering non-acoustic wave components. The last solutions for case presents results obtained for real data.
where X and Z are the coordinates of the receiver i at the surface, and tSj and tij are the FIGURE 2 Synthetic cases traveltimes fromchart the source j to the the point (x,z) and from to the receiver Wei,first consider a synthetic case in the acoustic approximation. Processing flow summarizing different steps ofthis the point proposed seismic imaging method. In this model, a high velocity (and density) object is embedded in respectively. For convenience, (9) can be rearranged as
a two-layer medium, 5 m long and 1 m deep. This case represents a scaled-down model of a pipe located in the subsurface. The low1 t ! t 2T velocity upper Rx , z (10) , P t x z , (10) layer, velocity 200 m/s (density 1800 Kg/m3) and ¦¦ ij ij Nj j i thickness 0.1 m, represents the weathered zone. The high-velocity lower layer, i.e. the host material, has a velocity of 250 m/s (denwhere tij(x,z)t>t0+2T is the traveltime map computed from the t ! t 2T x, zpoints is the of traveltime map computed sourcetoj to points (x,z) of where sourcetijj to (x,z) the medium and fromfrom thesethepoints sity 1900 Kg/m3). A circular anomaly of velocity 700 m/s (density the receiver i. If the background velocity field can be differenti2200 Kg/m3), 0.4 m deep with a radius of 0.2 m, is introduced into the medium and from these points to the receiver i. If the background velocity field can the host material (Fig. 1a). A 600 Hz Ricker wavelet is used as ated from the velocity tomography, the traveltime map can be easily computed by FMM as described in the previous section. source signal in order to maintain the ratio of approximately unity
0
0
be differentiated from the velocity tomography, the traveltime map can be easily
computed by FMM as described in the previous section. In order to discard the direct © 2006 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2006, 4, 275-283 wave, the summation only considers amplitudes later than first arrivals, i.e. t0 plus two periods T. This operation is equivalent to performing an automatic muting above the 2T
velocity 700 m/s (density 2200 Kg/m ), 0.4 m deep with a radius of 0.2 m, is introduced into the host material (Fig. 1a). A 600 Hz Ricker wavelet is used as source signal in order to maintain the ratio of approximately unity between wavelength and the anomaly
Imaging depth, as is commonly observed in reality (Grandjean and Leparoux 2004). Figure 1(b) subsurface object with seismic tomography
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shows the Fresnel wavepath related to source #1 and receiver #15. Note that two distinct
between wavelength and the anomaly depth, as is commonly For each simulation, vertical displacements were recorded in observed in reality (Grandjean and Leparoux 2004). Figure 1(b) time at receiver points spaced every 0.15 m. The generic processing flow applied to the data is summashows the Fresnel wavepath related to source #1 and receiver #15. anomaly. rized in Fig. 2, except that no surface-wave suppression was Note that two distinct waves arrive at the same time: one is refracted in the secondrecords layer, were the other from by thethe anomaly. because the synthetic data were computed in the acousSeismic simulated FDTD code implemented inapplied, JaTS in the Seismic records were simulated by the FDTD code impletic approximation. Processing was thus reduced to trace ampliacoustic described by Keiswetter et al. (1996). The mented approximation in JaTS in theand acoustic approximation and described by modelling tude normalization and manual first-break picking. Using the Keiswetter et al. (1996). The modelling scheme is based on the source and receiver positions, picks were used to reconstruct the scheme is based on the scalar wave equation in two dimensions: background velocity field (Fig. 3a). After muting the first arrivscalar wave equation in two dimensions: als, the late arrivals were migrated, assuming the background ª w § 1 wp · w § 1 wp · º w2 p , (11) previously reconstructed by tomography. This step x , z f ( t ) G velocity field 2 UVP2 « ¨ (11) S S ¸ ¨ ¸» wt ¬ wx © U wx ¹ wz © U wz ¹ ¼ allows the reconstruction of local impedance contrasts related to the local anomaly, as shown in Fig. 3(b). This resulting image where p is the acoustic pressure, ρ = ρ(x,z) is the density, where p is(x,z) the acoustic pressure, U = U(x,z) is,the density, VP = Vdelta P-wave P(x,z) is the VP = V is the P-wave velocity, δ (x z ) is the Dirac can be compared with the predicted model of Fig. 1(a). We note P s s function, which takes the value of unity at the source position that the upper velocity layer has been correctly imaged, even if velocity, G(xs, zs) is the Dirac delta function, which takes the value of unity at the source and zero elsewhere, and f(t) is a time-dependant source signal the transition to the deeper one is not as sharp. This lack of resodefined and for azero given numberand of f(t) iterations. lution from the limited frequency bandwidth of the seisposition elsewhere, is a time-dependant source signal defined forcomes a The finite-difference solution of (11) uses a second-order difmic signal, as is commonly observed in subsurface seismics. Its given number of iterations. ference scheme. Here, the second derivative with respect to time origin lies in the band-limited source frequency content. The is approximated using a standard central differencing equation. local velocity anomaly is correctly imaged, as we can identify its To reduce numerical dispersion, the sampling interval is reduced shape and depth, but the lateral contours remain blurred. This to ten gridpoints per wavelength. For numerical solutions to be artefact, also commonly observed in surface seismics, is the 12 stable, stability criteria are imposed according to Kelly et al. result of the distortion induced by the source and geophones (1982). The boundary conditions are those described by Reynolds geometry, which is classically spread on the surface side only, (1978). making diffractions from the lateral sides undetectable. This The seismic shot corresponding to source #1 is shown in synthetic case demonstrates the stability of the algorithms preFig. 1(c). The direct and refracted waves clearly appear as first sented and their ability to image subsurface velocity structures arrivals, and the diffracted signal related to the presence of the when only P-waves are generated. Nevertheless, we note certain P-wave velocity anomaly is indicated by red arrows. A total of limitations, essentially due to wave resolution and source/geonine sources, spaced every 0.5 m, were shot using this model. phone geometry. These limitations are well-known in seismic waves arrive at the same time: one is refracted in the second layer, the other from the
FIGURE 4 (a) KL-filtering of the diffracted signals from a synthetic seismic shot computed in the elastic approximation. This filtering separates the surface-wave signals (b) and the volume-wave signals (c) from the whole seismic waves. The filtering uses the dispersion image (d) to characterize the ground-roll propagation. The arrows numbered 1, 2 and 3 refer to direct, surface and diffracted waves, respectively.
© 2006 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2006, 4, 275-283
we note certain limitations, essentially due to wave resolution and source/geophone res when only P-waves are generated. Nevertheless, we note certain limitations, essentially due to wave resolution and source/geophone geometry. These limitations are well-known in seismic engineering and can be tially due to wave resolution and source/geophone geometry. These limitations are well-known in seismic engineering and can be attenuated under two conditions: working with a signal bandwidth as wide as possible in well-known in seismic engineering and can be attenuated two conditions: working with a signal bandwidth as wide as possible in 280 G.under Grandjean high frequencies, and using additional seismic sources and receivers gathered around working with a signal bandwidth as wide as possible in high frequencies, and using additional seismic sources and receivers gathered around the study zone, for example in boreholes. ional seismic sources and receivers gathered around the study zone,and for can example in boreholes. engineering be attenuated under two conditions: workIn aorder to present a more now consider reholes. ing with signal bandwidth as realistic wide as scenario, possible we in high frequen-an elastic In order present a more realistic scenario, we now gathered consider an elastic cies, and usingtoadditional seismic sources and receivers propagation in the same velocity model. This model had shear-wave velocities that were realistic scenario,around we nowthe consider an elastic study zone, for example in boreholes. propagation in the same velocity model. This model had shear-wave velocities that were 0.7 times In order to present a more realistic scenario, wecoverage, now consider the P-wave velocities. To maximize the ray we doubled the number model. This model had shear-wave velocities that were an times elasticthepropagation in the To same velocity This model 0.7 P-wave velocities. maximize themodel. ray coverage, we doubled the number of geophones. Thevelocities FDTD scheme used 0.7 to compute seismograms was adapted from had shear-wave that were times the P-wave velocTo maximize the ray coverage, we doubled the number of geophones. The FDTD scheme used to compute seismograms was ities. To maximize the ray coverage, we doubled the number ofadapted from Virieux (1986), Levander (1988) and Juhlin (1995), and is based on the stress–velocity e used to computegeophones. seismogramsThe was FDTD adaptedscheme from used to compute seismograms Virieux (1986), Levander (1988) and Juhlin (1995), and is based on the stress–velocity wave equation from in twoVirieux dimensions. In the 2D Cartesian system, P-SV equations of was adapted (1986), Levander (1988) and the Juhlin and Juhlin (1995), and is based on the stress–velocity wave equation two dimensions. In the 2D Cartesian system, in thetwo P-SV equations of (1995), and isinbased on the stress–velocity wave equation motion are . In the 2D Cartesian system, the equations of system, the P-SV equations of dimensions. In P-SV the 2D Cartesian motion are motion are wu wW xx wW xz ww wW zx wW zz and U t , (12a) U t wwutt wwWxxx wwWzxz wwwt wwWxzx wwWzzz and U t , (12a) U (12a) wt wx wz wt wx wz wW zx wW zz , (12a) with the constitutive equations for an isotropic medium: wx wz with the constitutive equations for an isotropic medium: FIGURE 5 with the constitutive equations for an isotropic medium: (a) Background velocity field inverted from the first-arrival traveltimes: r an isotropic medium: wu ww ww wu § wu ww · computed from the model in Fig. 1(a) in the elastic approximaW xx O 2P O , W zx P ¨ ¸ and W zz O 2P (12b) O they , were (12b) wux www www wux z z x· § wuz ww W xx O 2P O , W zx P ¨© ¸¹ and W zz O 2P O tion; , (12b) the two layers and their respective velocities are clearly visible. (b) x wz wx ww wuz © wz wx ¹ § wu ww · The anomaly reconstructed from the late arrivals using the velocity field O , (12b) ¨ ¸ and W zz O 2P wz wx © wz wx ¹ shown in (a) is highlighted by the dotted circle.
where u, w and ut, wt are respectively the displacement and velocity components along the x- and z-axes, λ, µ and ρ are the elastic 14 constants, and tii and τij refer respectively to the normal and tan14 gential stress components. Equations (12a) and (12b) are differen14 tiated and solved using a 2D staggered finite-difference grid (Virieux 1986; Levander 1988). Boundary conditions are classically defined to model a semi-infinite space, satisfying the freesurface conditions at z = 0. The other boundaries at the grid periphery are coded to satisfy the absorbing conditions of Clayton and Engquist (1977). A spatially localized source is initiated with a second Gaussian derivative function of stress components, using the source insertion principle of Alterman and Karal (1968). Figure 4(a) shows a seismic synthetic shot representing the elastic wavefield generated by a source located at x = 2.25 m, z = 0 m, and recorded at 51 geophones spread over the surface. On this shot, the P-wavefield, consisting of direct and diffracted waves, is difficult to identify because of the presence of surface waves. In order to extract the P-wavefield, we used a Karhunen– Loève (KL) filter to separate dispersive waves from volume waves (Bitri and Grandjean 2004). This method consists of computing the dispersion image of the seismic shot that represents the distribution of the seismic energy in the phase-velocity versus frequency domain (Park et al. 1998). From this image, the dispersion curve can be identified as the set of maximum values (the black line in Fig. 4d). This curve is then used to filter the seismic shot by a KL technique so that surface waves (Fig. 4b) are separated from non-dispersive waves (Fig. 4c). On this filtered data, it is easier to observe not only the direct wave but also the diffracted waves that are related purely to the local heterogeneities of the medium. Finally, the imaging technique described in the previous section can be carried out on the filtered data.
Figure 5 shows the results of both the reconstructed velocity field (Fig. 5a) and the migrated late-arrival signals (Fig. 5b). Compared with the previous acoustic case, the diffraction patterns are not imaged with sufficient accuracy to be correctly interpreted. The reasons that the circular velocity anomaly is unfocused can be related to (i) the presence of multiple diffractions or P–S wave conversions that are not filtered by the KL technique; (ii) damage to the diffracted signals during the filtering processes that introduces phase perturbations. Both these phenomena alter the stacked signals during the imaging step, making it impossible for the migrated section to image local anomalies correctly. We should also mention that a low VP/VS ratio was used in these simulations in order to restrict the generation of higher modes of guided waves and P–S wave conversions. The presence of such complex wavefields could introduce additional complications into our method, particularly when extracting P-waves by filtering. The next section deals with this type of challenge by considering real data recorded on a test site. EXPERIMENT ON A TEST SITE In this section, we apply the above-described methodology to the data set derived from Grandjean and Leparoux (2004). The principle of this experiment was to reproduce, at a reduced scale, the interaction between seismic waves and a void. The geometry and characteristics of the seismic line were thus adapted to the problem considered, i.e. to detect a miniaturized cavity embedded in a host material with a constant P-wave velocity of 200 m/s (Fig. 6). We selected a filling material composed of coarse limestone gravel with a mean diameter of 0.02 m. In order to simulate
© 2006 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2006, 4, 275-283
Imaging subsurface object with seismic tomography
a void in this medium without using masonry, we buried at 0.2 m depth a cylinder made of cellular polystyrene, 0.4 m in diameter and 2 m long. The acquisition system consisted of a series of 32 wide-band accelerometers (0.1–4800 Hz) gathered into a connecting stream and linked to a high-frequency acquisition system. The seismic source was generated by dropping a small (1 cm in diameter) iron ball from a height of 0.2 m on to a small stone anvil laid on the surface of the test site. More details on the site and the acquisition system characteristics can be found in Grandjean and Leparoux (2004). Figure 7 shows an example of a shot for which the low- and high-frequency components have been separated by using bandpass frequency filtering. This separation is efficient enough on this record to determine P-wave first arrivals and diffractions on the high-frequency panel. The same processing as above was applied to the whole data set in order to reject surface waves and retain only volume waves. The first arrivals were manually picked and constituted the input data for the tomographic reconstruction. The initial model was estimated from the direct-arrival slope in the seismic record shot-gathers, and taken as homogeneous. Ten iterations were necessary to converge to a realistic model, explaining 95% of observations. Figure 8 shows the different results obtained after the tomographic and imaging steps. It can be seen that the velocity field, assumed constant at 200 m/s, is affected by some local anomalies ranging from 130 to 220 m/s. These anomalies were interpreted by Grandjean and Leparoux (2004) as local compaction effects related to the building works. The migrated section shows a series of seismic events for which the strongest one is related to the cylinder. Although this event is similar to the cylinder shape and located at the same position, we note that the final reconstruction is not as clear-cut as for the synthetic acoustic cases. We attribute this problem to the poor quality of the diffractions extracted from the seismic records, as shown in Fig. 7(b). As in the elastic synthetic case, the presence of strong surface waves makes it difficult to recover the diffracted signals that, in addition, are intermingled with the first arrivals, and thus are also reduced to zero during the muting process. This example, illustrating our P-wave imaging method applied to the subsurface, indicates the difficulties in filtering surface waves. Surface-wave scattering on the topographic roughness could also generate perturbing signals that are difficult to remove. As mentioned by Snieder (1986), the influence of topography on the propagation and scattering of surface waves produces focusing and defocusing effects making them difficult to filter. Nevertheless, Blonk and Hermann (1994) and Blonk et al. (1995) studied this phenomenon and have proposed some solutions for removing such scattered waves. CONCLUSION A method of P-wave seismic imaging based on traveltime tomography and Kirchhoff migration has been developed. This method is tested to determine its efficiency in tackling the
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FIGURE 6 (a) Schematic representation of the test site and (b) related construction works.
FIGURE 7 Example of a shot for which surface waves are separated from volume waves by a simple band-pass filter. 1: first arrivals, 2: surface wave, 3: P-wave diffraction on the heterogeneity.
FIGURE 8 (a) Background velocity field inverted from the first-arrival traveltimes. (b) Anomaly reconstructed from the late arrivals using the velocity field shown in (a). The black dotted circle indicates the real position of the heterogeneity.
© 2006 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2006, 4, 275-283
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problem of detecting and imaging buried heterogeneities in highly contrasted subsurface media. We implement previously known tomography and migration concepts to develop a twostep imaging technique using algorithms adapted for imaging highly heterogeneous media. This technique consists of processing first arrivals as well as later arrivals, and using them both in tomography and migration processes, sequentially. Tomography is implemented here as an iterative technique based on probabilistic SIRT for reconstructing the background velocity field from the first-arrival traveltimes. In this formulation, Fresnel wavepaths constitute the forward step of traveltime inversion. Then, from the inverted velocity field, the seismic signals are migrated by a Kirchhoff method implemented in the space domain. In particular, we studied solutions for improving the quality of the seismic imaging of high-contrast media: the use of Fresnel wavepaths instead of classical seismic rays to take into account the seismic resolution; a migration based on Kirchhoff summation in the space domain, which is equivalent to a prestack depth migration; the identification of an adapted filter based on the Karhunen–Loève transform for removing the P-wave signal-tonoise ratio. The proposed methodology was first applied to a synthetic acoustic case, where both the background velocity field and local impedance contrasts were reconstructed as defined in the predicted model. To examine a more complex situation, a second synthetic case was considered in the elastic approximation. In spite of using adapted filtering for removing them, the surface waves reduce the diffracted signals to a poor quality. Consequently, the reconstructed model shows the anomaly with an incomplete and unfocused shape. Finally, an experimental case specifically designed for the purpose was considered, to test the algorithms under real conditions. The results are in agreement with reality, even if the surface waves are again a problem. This work demonstrated the theoretical suitability of the proposed method. The tests carried out on synthetic and real data sets highlighted the restrictions to be considered, particularly suitable data preprocessing to remove surface waves that could mask P-wave signals. REFERENCES Alterman Z. and Karal F.C. 1968. Propagation of elastic waves in layered media by finite-difference methods. Bulletin of the Seismological Society of America 58, 367–398. Berkhout A.J. 1987. Applied Seismic Wave Theory. Elsevier Science Publishing Co. Bitri A. and Grandjean G. 2004. Suppression of guided waves using the Karhunen-Loève transform. First Break 22(5), 45–48. Blonk B. and Hermann G.C. 1994. Inverse scattering of surface waves: a new look at surface consistency. Geophysics 59, 963–972. Blonk B., Hermann G.C. and Drijkoningen G. 1995. An elastodynamic inverse method for removing scattered surface waves from field data. Geophysics 60, 1897–1905. Cao S. and Greenhalgh S. 1994. Finite-difference solution of the eikonal equation using an efficient, first-arrival, wavefront tracking scheme. Geophysics 59, 632–643.
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Pratt T.L., Dolanz J.F., Odum J.K., Stephenson W.J., Williams R.A. and Templeton M.E. 1998. Multiscale seismic imaging of active fault zones for hazard assessment: A case study of the Santa Monica fault zone, Los Angeles, California. Geophysics 63, 479–489. Reynolds A.C. 1978. Boundary conditions for the numerical solution of wave propagation problems. Geophysics 43, 1099–1110. Robein E.,1999. Vitesses et Techniques d’Imagerie en Sismique Réflexion. Principes et Méthodes. Tec & Doc Ed., Paris. Sethian J.A. and Popovici A.M. 1999. 3-D traveltime computation using the fast marching method. Geophysics 64, 516–523. Snieder R. 1986. The influence of topography on the propagation and scattering of surface waves. Physics of the Earth and Planetary Interiors 44, 226–241. Virieux J. 1986. SH-wave propagation in heterogeneous media: Velocitystress finite-difference method. Geophysics 49, 1933–1957. Watanabe T., Matsuoka T. and Ashida Y. 1999. Seismic traveltime tomography using Fresnel volume approach. 69th SEG Meeting, Houston, USA, Expanded Abstracts, 1402–1405.
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