Combining wave-equation imaging with traveltime …inversion of wide-aperture crosshole data. The complemen- tary nature of frequency-domain imaging and frequency domain numerical

GEOPHYSICS, VOL. 56. NO. 2 (FEBRUARY IYYI): P. 208-224. 15 FIGS.

Combining wave-equation imaging with traveltime tomography to form high-resolution images from crosshole data

R. Gerhard Pratt* and Fueil R. Goulty*

ABSTRACT

Traveltime tomography is an appropriate method for estimating seismic velocity structure from arrival times. However, tomography fails to resolve discontinuities in the velocities. Wave-equation techniques provide images using the full wave field that complement the results of traveltime tomography. We use the velocity estimates from tomography as a reference model for a numerical propagation of the time reversed data. These “backpropagated” wave fields are used to provide images of the discontinuities in the velocity field.

The combined use of traveltime tomography and wave-equation imaging is particularly suitable for forming high-resolution geologic images from multiple- source/multiple-receiver data acquired in borehole-to- borehole seismic surveying. In the context of crosshole imaging, an effective implementation of wave- equation imaging is obtained by transforming the data and the algorithms into the frequency domain. This transformation allows the use of efficient frequency- domain numerical propagation methods. Experiments

with computer-generated data demonstrate the quality of the images that can be obtained from only a single frequency component of the data. Images of both compressional (V,,) and shear wave (V,s) velocity anomalies can be obtained by applying acoustic wave-equation imaging in two passes. An imaging technique derived from the full elastic wave-equation method yields supe- rior images of both anomalies in a single pass.

To demonstrate the combined use of traveltime tomography and wave-equation imaging, a scale model experiment was carried out to simulate a crosshole seismic survey in the presence of strong velocity contrasts. Following the application of traveltime tomography, wave-equation methods were used to form images from single frequency components of the data. The images were further enhanced by summing the results from several frequency components. The elastic wave-equation method provided slightly better images of the V,] discontinuities than the acoustic wave-equation method. Errors in picking shear-wave arrivals and uncertainties in the source radiation pattern prevented us from obtaining satisfactory images of the V,, discontinuities.

INTRODUCTION

A new method for wave-equation analysis of seismic crosshole data has been developed, based on a finite-difference solution of frequency-domain wave equations. Pre- vious papers (Pratt and Worthington, 1990; Pratt, 1990a, b) developed theoretical and algorithmic aspects of the method for acoustic and elastic wave equations and applied it to synthetic data. In this paper we approach the problem from a more pragmatic viewpoint; the wave-equation method is placed in context as the final step in a processing sequence.

We omit a detailed description of the wave-equation imaging step, since it is contained in the papers cited above.

Traveltime tomography has been used by many researchers to estimate velocity structure from crosshole arrival times (Bois, 1971; Laporte et al, 1973; Dines and Lytle, 1979; McMechan, 1983a; Wong et al., 1983; Ivansson, 1985; Dyer and Worthington, 1988: Bregman et al, 1989). In some cases the results of traveltime tomography have been ambiguous (for example, Lines and Lafehr, 1988). Crosshole seismic data contain complexities such as direct shear waves, guided wave modes, interface waves, various forms of diffractions

Manuscript received by the Editor September 15, 1989; revised manuscript received July I, 1990. *Formerly Imperial College of Science, Technology and Medicine, Department of Technology. London SW7 2BP. UK. Department of Physics, University of Toronto, Toronto, Ont., Canada M5.S IA7. $Department of Geological Sciences, University of Durham, South Road, Durham DH I 3LE, UK. 0 1991 Society of Exploration Geophysicists. All rights reserved.

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Wave-equation Imaging 209

and head waves, as well as reflected and mode converted arrivals. It is potentially advantageous to apply wave-equation techniques, since these use the full information content of the recorded waveform. Moreover, wave-equation methods avoid the asymptotic (high-frequency) ray approxima- tions used in traveltime tomography.

It is important to draw a distinction between the results that can be obtained from traveltime tomography and those from wave-equation methods. The former technique yields quantitative estimates of the smoothly varying components of the velocity distribution, whereas the latter method, as applied in this paper, yields only a qualitative image of the velocity discontinuities. Wave-equation methods are higher order techniques that can lead to a significant improvement in image resolution when compared with traveltime tomography. However, wave-equation methods complement, rather than replace, traveltime tomography.

nite approximation modeling of the frequency-domain wave equation is well suited to the computation of wave fields from a large number of distinct source locations in a single model (Marfurt, 1984; Pratt, 1990b). The application of frequency-domain numerical methods is therefore an ideal approach to use in the modeling, migration, imaging, or inversion of wide-aperture crosshole data. The complemen- tary nature of frequency-domain imaging and frequency domain numerical forward modeling is an important result of Pratt ( 1990b).

To use the full waveform in transmission seismic data, some researchers have proposed a number of techniques that are related to prestack migration. Examples are reverse time migration (McMechan, 1983b; Chang and McMechan, 1986; Hu et al., 1988), nonlinear inversion (Gauthier et al., 1986; Sun and McMechan, 1988; Mora, 1988) and the inverse generalized Radon transform (Beylkin, 1984; Miller et al., 1987). These methods are promising; however, one draw- back is that they are expensive in cases where the number of sources is large. To overcome this, Zhu and McMechan (1988) proposed a stacking technique that reduces the data volume to a more manageable level before migration is attempted.

To test the robustness and effectiveness of the combined use of traveltime tomography and frequency-domain wave- equation methods, we generated data in a scale-model simulation of a crosshole seismic survey. The model simulated appropriate dimensions, velocities, and structures by using ultrasonic frequencies and a spatial scale factor of 1:lOOO. While allowing a direct evaluation of the images by compar- ison with the original model, the scale-model approach provides a good simulation of the vagaries of field seismic data. The elastic data from the experiment exhibit direct compressional and direct shear arrivals, mode conversions, trapped wave modes, head waves, and wide-angle reflections. In addition, the data are corrupted by both random and coherent noise. Source directivity effects similar to those exhibited by actual borehole sources are also present in the model data.

DATA PROCESSING PHILOSOPHY

An alternative method for reducing the scale of the problem is to transform the data into the frequency domain and then use only a limited number of frequency components. In addition to the methods used in this paper, Woodward and Rocca (1988) and Petrick et al. (1988) have also proposed frequency-domain inversion algorithms. Decimation of the data in the frequency domain is an appropriate approach in the imaging of crosshole data, since there is theoretical data redundancy in wide-aperture crosshole surveys that allows useful images to be formed from a single frequency component (Pratt and Worthington, 1990). This redundancy has been exploited in synthetic studies (Devaney, 1984; Wu and Toksoz, 1987) and in experiments with scale-model data (Lo et al., 1988; Pratt and Worthington, 1988). The imaging algorithm used in this paper is implemented in the frequency domain in order to take advantage of the redundancy.

In practical applications, the number of source and receiver locations may be limited due to operational considerations (thus reducing the aperture or the spatial sample interval), the signal to noise ratios may be less than ideal, and strong velocity variations may lead to significant wave- field focusing. In these cases, additional frequencies do carry additional information. However, provided the aperture of the survey is large, the use of only a limited number of additional frequency components in several passes of a frequency-domain imaging technique is appropriate.

The wave-equation method used in this paper can be described as an imaging process (see Wu and Toksoz, 1987 for an excellent discussion of the imaging process for seismic and acoustic problems). The imaging process involves focus- sing the scattered data by backpropagating these from the receiver locations, and applying an imaging condition. The imaging condition can be computed by forward propagation of the source function from the source locations. There are three important requirements for the success of the imaging process. First, imaging depends on the accuracy of the model used in the wave propagation. Second, the directly transmitted arrivals satisfy the imaging condition everywhere in the model (Chang and McMechan, 1986) and corrupt the final images. To avoid this corruption, it is desirable to remove the directly transmitted waves from the scattered waves in the data. Finally, the imaging condition requires knowledge of the source time function and of the source radiation pattern. To meet these requirements for the wide-aperture crosshole data, the processing strategy depicted in Figure 1 is used.

The first step in Figure 1 is to use the picked arrival times, along with any a priori knowledge of the structure, such as interface locations, to form velocity tomograms. The velocity field is as important in crosshole imaging as is the velocity field in surface seismic migration. It is the gross velocity structure, rather than the fine details, that is most important for correct wave-equation imaging.

The advantages of frequency-domain imaging are comple- The next step in Figure 1 is preprocessing of the wave mented by using a frequency-domain modeling method for field, to eliminate any directly transmitted arrivals. The forward and backpropagation. The most general techniques correct input is the residual wave field, defined as the for modeling wave propagation through a given medium of differences between the observed wave field and the wave arbitrary complexity are finite approximation methods. Fi- field predicted by forward modeling through the current

210 Pratt and Goulty

estimate of the model. Unfortunately, the residual wave field is extremely difficult to compute. In practice, the residual wave field is dominated by errors introduced by whatever modeling algorithm is chosen.

The scattered wave field is an approximation to the residual wave field that can be computed directly from the data. The scattered wave field is difficult to define rigorously, but can be described as that wave field that reaches the receivers indirectly, having been scattered, reflected, dif- fracted, or otherwise perturbed from the direct wave field by geologic inhomogeneities. If the current model estimate is smoothly varying and describes the arrival times well (as can be expected from traveltime tomography), the scattered wave field will closely approximate the residual wave field. Therefore, the approach used in this paper is to estimate the scattered wave fields in a statistical fashion from the total wave field. The scattered wave fields are used as the input data for wave-equation imaging.

The final input before wave-equation imaging is an estimate of the source time function. This parameter, difficult to measure in most circumstances, is also extracted statistically from the data. The preprocessing step separates the total recorded data into its directly transmitted and scattered components. The direct wave-field estimates are used to provide an estimate of the source time function.

FREQUENCY-DOMAIN WAVE-EQUATION IMAGING

We now review the method of frequency-domain wave- equation imaging by finite differences; see Pratt and Wor-

HIGH RESOLUTION TOMOGRAPHIC IMAGING

a priori constraints picked arrival (interfaces) times

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/ /\ velocity field scattered data source function

1 1 /

Wave theoretical imaging

i high resolution image

(interfaces)

FIG. 1. Flowchart depicting the approach used in this paper to form high resolution images from cross-hole data. Al- though the important aspect of this approach is that it contains a wave equation imaging scheme, this is not proposed as an independent technique. The wave equation method complements, rather than replaces, traveltime tomography.

thington (1990) for a complete treatment of the acoustic method and Pratt (1990a, b) for the elastic wave-equation method.

The imaging technique is derived from Tarantola’s method for wave-equation inversion. Tarantola (1984a) first developed this solution to the inverse problem for the acoustic wave equation and subsequently applied the same method to the elastic wave equation (Tarantola, 1984b). The original papers formulated the problem in the time domain; in Pratt and Worthington (1990) and in Pratt (1990a) the method was reformulated in the frequency domain. The most important aspect of these papers is derivation of a tractable algorithm for calculating the gradient of an objective function with respect to the acoustic (or elastic) parameters of the model. The objective function quantifies the misfit of the current model; many different objective functions can be treated within Tarantola’s formalism.

To compute the gradient direction, it is only necessary to be able to solve the forward problem. For data obtained with multiple sources, two forward solutions are required for each source location: the first is the wave field obtained by applying the source function at the source location (forward propagation): the second solution is the response to the timereversed data residuals, applied at the receiver locations (backpropagation). The forward propagated wave fields and backpropagated wave fields are combined to yield estimates of the gradient for the acoustic or elastic parameters of the medium. This final step amounts to application of an imaging condition.

One useful approach in inverse problems is to attempt to find a geologic model that is both consistent with a priori information and can be used to fit the observed data. This can be achieved by minimizing the objection function. The gradient of the objective function is the direction of the steepest descent vector in the model space. Therefore, if the model parameters are varied along this direction, there is some point at which the objective function will be a minimum. The gradient vector can be used in a variety of iterative optimization methods to minimize the objective function (see Tarantola, 1987, for a review), subject to the usual problems of nonuniqueness due to nonlinearities.

Since the gradient vector is a vector in a space with the same number of dimensions as the model space (the space is formally referred to as the “dual” of the model space), the gradient vector can be displayed as an image of the model. The image will be most intense at those spatial locations where the model is most in error. Furthermore, the relative polarities of the image will also be correct; where a parameter needs to be reduced, the image will have the reverse polarity from locations where the parameter needs to be increased. If one begins with a smooth version of the true velocities, the image will depict the high-wavenumber discontinuities in the medium.

The formulation of the problem in terms of the imaging of elastic media properties has a distinct advantage over an approach, such as migration, which attempts to obtain a reflectivity image. In the latter approach, an ad hoc method must be developed to handle the angular dependence of reflectivity, or a further processing step must be used to extract the underlying elastic properties from the angular dependence of the reflectivity. This is especially true in


crosshole seismic surveys where, in the extreme, the reflectivity of an interface “viewed” from below is the negative of the reflectivity when viewed from above. Hu et al. (1988) migrated crosshole data and needed to apply a factor of - 1 where this effect occurred. However, this lacks rigor and does not account for more subtle variations in reflectivity.

ACOUSTIC VERSUS ELASTIC WAVE-EQUATION IMAGING

Shear-wave energy is often present in crosshole experiments. Shear waves originate as source-generated shear waves, from near-field effects such as mode conversion at borehole walls, and from wide-angle mode conversions at interfaces and scattering points. Elastic wave-equation imaging methods have the potential of using all shear-wave energy, including mode-converted waves. However, acoustic methods are easier to develop and cost much less in storage and computer time than elastic wave-equation methods. In spite of the presence of shear wave energy, acoustic wave equation methods can be applied. The acoustic imaging algorithm discriminates against shear waves in the data.

In this section a synthetic study, using a model from Pratt (1990a) (see Figure 2), compares the acoustic and elastic techniques in their application to crosshole data. For the synthetic study, data residuals are computed exactly by running the forward modeling algorithm twice, both with and without the perturbations to the elastic parameters. Initially a full “four component” survey was simulated. The term four component survey is used to refer to a survey where both horizontal force sources (in the radial direction) and vertical force sources (in the axial direction) were used, and where both the horizontal and vertical components of the resulting vector wave field were recorded.

Figures 3a and 3b reproduce the result from Pratt (1990a) in which the elastic wave-equation imaging algorithm was applied to form images of the discontinuities in the compressional velocities (V,) and shear velocities (V,) from all four (motion) components of the data. Both the V,, anomaly and the V, anomaly are well resolved, in spite of the fact that

AV, = 20%

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cl Ap = 20%

FIG. 2. The model used for the synthetic study described in the text, consisting of independent peturbations to p, Vpl, and V,y. Each anomaly is a square with half (compressional) wavelength sides. The background model is a linear velocity gradient from VP = 1600 m/s at the top to V, = 2400 m/s at the bottom of the model. The model is 60 m across and 120 m deep. A crosshole experiment at 50 Hz was simulated using 48 sources at the left edge and 48 receivers on the right edge of the model.

only a single frequency component of the data residuals was used to form these images.

In Figures 3c and 3d images were simulated that might be obtained from a crosshole survey using explosive sources and recorded in hydrophone receivers. Compressional energy in borehole fluids tends to generate horizontal forces at the source borehole wall. Similarly, horizontal particle motion dominates the coupling mechanisms at the receiver borehole. To simulate these mechanisms in a simple manner, only the horizontal data components from the horizontal single force sources were used. In this case the directivity patterns of the sources and receivers restrict the aperture of the P-wave data. Thus the VP anomaly is not as well resolved as in the first result. The V, image is corrupted by a strong X-shaped noise belt due to the very limited effective aperture created by the shear-wave radiation pattern. The two anomalies are still remarkably well resolved, showing that two component receivers are not necessarily required to invert elastic-wave data for VP and V, from wide-aperture crosshole data. This example used the exact source mechanism in the inversion; this parameter is unlikely to be as well known when imaging real data.

The final images from this study use the same elastic data, but the images are formed using two passes of the acoustic wave-equation imaging method. In the first pass the starting velocity model for backpropagation was the linear compressional-wave velocity gradient of the original model. In this case the backpropagation operates on all the data (both compressional and shear waves), but only the backpropagated compressional waves satisfy the imaging condition. The resultant images contain the information from the scattering of compressional waves into compressional waves (P-P scattering), but are corrupted by shear waves of any kind in the data. The result (Figure 4a) is a good image of the VP anomaly. The V, anomaly is hardly visible, showing that the V,y anomaly does not have a large forward scattering cross-section for P-P scattering.

In Figure 4b, the acoustic method was used again, but the background velocities were the V, ones. In this case the V, anomaly is resolved, although there is some corruption of the image due to compressional waves in the data (the coherent noise). The shear radiation pattern from the single- force sources used to model the data creates upgoing shear waves that are 180” out of phase with the downgoing shear waves. In the imaging, a uniform source radiation pattern is assumed. This leads to the contributions to the image from the top half of the receiver array having the reverse polarity from the contributions from the lower half. The errors in modeling the source mechanism lead to destructive interfer- ence and an unsatisfactory V, results. The VP anomaly is poorly resolved due to its small S-S forward scattering cross-section.

The results of this synthetic study provide confidence in the imaging of single-component elastic-wave data. The images will likely be best if an elastic wave-equation algorithm is used, but useful results can be obtained even by applying the acoustic wave-equation method. The assumption of a uniform radiation pattern can still yield useful images, although the V,, results are better than the V, results with this assumption. The remainder of the paper deals with the application of the same methods to scale-model data.


FIG. 3. Elastic wave equation monofrequency images formed from synthetic elastic wave data generated using the model shown in Figure 2. (a) VP image and (b) V, image formed using all four (motion) Fomponents of the data. (c) V, and (d) VP formed using only the horizontal receiver component data from the honzontal force sources.

Waveequation Imaging 213

SCALEMODEL DATA ACQUISITION

To evaluate the wave-equation methods in their application to crosshole seismic data, a survey was simulated in the ultrasonic seismic modeling system at the University of Durham (Sharp et al., 1985). This system has been used in the past to simulate crosshole seismic surveys (East et al., 1988; Pratt and Worthington, 1988), but previous models consisted of simple anomalies embedded in homogeneous media. In this case the epoxy-resin model used (Figure 5) consisted of a plane layered structure with increasing velocities from top to bottom. Nominal velocities varied from 2.107 km/s at the top of the model to 2.92 km/s at the bottom. Two low-velocity layers were inserted between higher velocity layers, in order to simulate reservoir rocks juxtaposed against impermeable ones. Two further geologic anomalies were simulated. The first modeled a dipping layer with a vertical discontinuity, such as might be caused by a fault. The second anomaly was a semicircular arc that cut into one of the layers, much as a buried river channel might do.

The model was submerged in water and a transmission survey was carried out by positioning ultrasonic piezoelectric transducers at the left- and right-hand edges of the model. The source transducer was circular in cross-section, with a diameter of 7 mm. The transducer was positioned with a 1.25 mm clearance between it and the edge of the epoxy- resin model. The receiver active element had a cross-section of roughly 1 mm and was located 2 mm from the edge of the model.

Waveforms from 51 source positions were recorded at each of 5 1 receiver locations, giving a total of 2601 waveform recordings. A 16-fold stack was used at each source-receiver position to enhance the data quality. In order to treat the source element as a point source, a static correction was made to the recorded waveforms to allow for the circular wavefronts being 3.5 mm from the center of the source at firing.

The distance between the source line and the receiver line was 54.5 mm (measured from the center points of the active elements), and the spatial sampling interval of the sources and receivers was 2.5 mm. The source frequencies ranged between approximately 200 and 500 kHz, giving a spatial sample interval that is everywhere less than half a compressional wavelength at the dominant source frequency. The two edges of the model are separated by approximately seven wavelengths.

The model can be scaled up to realistic dimensions by multiplying the distances and the times by a factor of 1000, so that millimeters become meters, microseconds become milliseconds and kHz become Hz. The simulated borehole separation was thus just over 50 m and scaled frequencies were at most 500 Hz. In field experiments higher frequencies could likely be recorded (see Worthington et al., 1989, for an example of real crosshole data with kHz frequencies at similar well separations). If these data can be processed successfully, still better results could be obtained from higher frequency data. However, the spatial and temporal

FIG. 4. Acoustic wave equation monofrequency images formed in two passes from the same synthetic elastic wave data as used in Figure 3c and 3d. (a) The acoustic velocity image formed using the VP velocities for the a priori model, and (b) the acoustic velocity image formed when the V, velocities are used.


sampling would need to be correspondingly denser for higher frequency data.

normally delayed arrival times, but the edges of the layer are obscured by the presence of head waves arriving from the faster layers on either side. The traces from source positions that are also within the layer (sources 32 to 36) show evidence of high-amplitude, guided-wave modes that persist up to late delay times.

Qualitative data evaluation

Some of the waveform data from the experiment are shown in Figure 6. Traces from all 51 source positions are shown in each of these three common-receiver panels. In the first panel, the data are from a receiver position close to the bottom of the model (receiver 7). The deepest source position is at the left-hand side, the shallowest is at the right. In addition to the direct compressional arrival, the direct shear arrival and some reflected waves can be identified. In the second panel, the receiver was located just above the bottom interface (receiver 22). There is a clear traveltime effect due to the large velocity change above. The reflection generated at this interface can also be seen. The third panel is from receiver position 34, located exactly at the center of the thicker low-velocity layer. The slower velocities cause ab-

2.107 km/s

2.81 kmJs

2.123 km/s c No. 34

2.81 km/s

125.0

No. 22

FIG. 5. A schematic of the epoxy resin model used to generate the data in the ultrasonic scale model experiment. A total of 51 source positions were used and waveforms were recorded at 51 receiver positions. The model extended vertically beyond the limits shown here by 37 mm in either direction, and by 87 mm in both directions out of the plane.

PREPROCESSING

Traveltime tomography

The first step in preprocessing the crosshole data from the experiment was to digitize the 2601 direct arrival times and perform traveltime tomography. Many algorithms solve the nonlinear problem of incorporating ray bending due to velocity structure into traveltime tomography (Bois et al., 1972; Lytle and Dines, 1980; Dyer and Worthington, 1988; Bregman et al., 1989). To simulate the optimal performance of a curved ray tomography algorithm, bent raypaths were computed using the model specifications of Figure 5.

In addition to computing and using the direct raypaths, raypaths were also computed for all possible head waves. The detrimental effect that head-wave arrivals can have on traveltime tomography has been recognized by several researchers (Wielandt, 1987; Worthington et al., 1989). Where head waves arrived earlier than the direct waves, the head- wave raypaths were used in the tomographic inversions. Since the first-arrival times were always used, rather than the (sometimes delayed) higher amplitude direct arrivals, this is an appropriate strategy for correctly handling head- wave arrivals that does not require repicking the traveltimes. A subset of the raypaths used is shown in Figure 7.

The raypaths described above were used to form the tomogram shown in Figure 8, with 15 iterations of the simultaneous iterative reconstruction technique (SIRT) (Dines and Lytle, 1979; Ivansson, 1985). By using the true model to compute the raypaths, an optimal tomography result has been obtained. The a priori knowledge of the raypaths allows the layer boundaries to be extremely well resolved.

Estimating the direct wave field

The estimation and removal of the direct wave field is related to the problem in the preprocessing of VSP data of separating the downgoing (direct) waves from the upgoing (reflected, or scattered) waves. There is a large body of published work on this problem (Seeman and Horowitz, 1983; Stainsby and Worthington, 1986; Hardage, 1983; Aminzadeh, 1986; Moon et al., 1986).

In crosshole applications the problem is somewhat more complicated, since the time moveout of the direct wave is not always different from that of the scattered wave, as seen from Figure 9a. The moveout of the reflected wave on a common-source gather is in some circumstances very similar to the time moveout of the direct wave. This occurs when both the source and the receiver are on the same side of the reflector and the receiver is farther away from the reflector than the source is. This effect was recognized as problematic by Iverson (1988), who pointed out that wave-field separation techniques that rely on differential moveout will not work in these regions of common-source data.


When the data are regathered into common ray angles (Figure 9b), the moveout for the same events on the same trace differs from the direct wave moveout. This simple way of reorganizing the data allows scattered and direct arrivals to be discriminated on the basis of time moveout. A similar discrimination is possible if the data are reassembled as common-receiver gathers. As both, sources and receivers approach the reflector, the two wave types become difficult to discriminate, no matter how the data are organized.

The direct wave was extracted from the experimental data in the following way: for each trace that was to be operated on, 11 common ray-angle gathers were selected. These were the gathers from the given ray angle and from the five adjacent ray angles on either side. From each of these gathers, five traces were selected: the traces from the given receiver location and two traces on either side. Thus a total of 55 input traces were used for each output trace (except at the edges of the survey).

The next step was to align the 55 traces according to their arrival times and average them. This averaging process is implemented as a running mean filter, as described in Hard- age (1983) for VSP processing. The filtering operation en- hances the direct arrivals at the expense of the scattered waves. The process was repeated for all 2601 traces in the survey. The results for one of the common-receiver gathers are shown in Figure 10a. The actual number of input traces used in the mean filter was determined by interactively processing single traces and evaluating the filter performance qualitatively.

Source signature estimation and deconvolution

Directional variation of the source behavior is likely to be important in borehole seismic investigations, since the pres-

ence of the borehole wall and the source and receiver coupling mechanisms introduce strong directional effects. In the scale-model data directional effects are introduced by coupling mechanisms between the piezoelectric transducers and the wall of the epoxy resin model, and by inherent directivities in the transducers themselves. As well as a dependence on the direction of propagation in the amplitudes, there may also be directional dependencies in the source signature (time dependence). It is easier to compen- sate for the latter directional effects than to attempt to determine them and include them in the modeling. The direct wave estimates (Figure 10a) provide a robust way of decon- volving the data. Each trace of the direct wave field is an estimate of the far-field source signature, time shifted by the arrival time for that trace. Since the traces were only filtered with other traces that had approximately the same ray angle, the directional variation of the source signature is reproduced in the direct-wave field.

A deterministic deconvolution was performed by computing a least-squares filter at each trace to shape the direct wave estimate to a minimum-phase wavelet (the “desired output”) with the same bandwidth. Once the filter was determined for that trace, the filter was applied to the corresponding trace in the original data. By using a high level of prewhitening, the data were also barid-pass filtered by the process (Hatton et al., 1986). Figure lob shows the deconvolved common-receiver gathers. The tightening of the source wavelet makes the data easier to interpret qualitatively, and the source wavelet is now consistent throughout the survey. The desired output wavelet is a reasonable estimate of the new source wavelet contained within the deconvolved data.

FIG. 6. Full waveform common receiver data from the ultrasonic tank experiment. Each of these three panels is a common receiver gather containing data from all 51 source positions (see Figure 5).


Estimating the scattered wave field used, these arrivals are as important to remove as are the direct compressional arrivals. Exactly the same approach

The final preprocessing stage for the ultrasonic tank data was used to remove them: the direct shear arrival times were was to use the deconvolved total wave field (Figure lob), and digitized, and a multitrace mean filter was applied to enhance apply exactly the same process of estimating the direct wave them. This direct shear-wave field was then subtracted from field described earlier. Once this new direct wave field was the original estimate of the scattered wave field. The final

estimate of the scattered wave field is shown on Figure 10d. computed,it was subtracted from the deconvolved total A great deal of information is contained within these wave field to produce an estimate of the scattered wave field. waveforms; the challenge is successfully extracting it. The Results are shown in Figure 10~. highest amplitude events on the scattered wave-field esti-

Direct shear arrivals are still present in these data, be- mates (Figure 1Od) are the guided wave modes, discussed in cause the direct shear arrivals do not have the same moveout the section “Qualitative data evaluation.” These events will as the direct compressional arrivals (no matter how the data have an important effect on the final images. Provided the are gathered). They are therefore poorly represented in the imaging algorithm properly accounts for these wave types, direct-wave estimate. If the elastic-wave equation method is they will contribute constructively to the image.

THE APPLICATION OF WAVE-EQUATION Ih4AGING

The three essential requirements for wave-equation imaging depicted on Figure 1 (the velocity field, the scattered wave field, and the estimate of the source function) have been met by the preprocessing steps described in the previous section. The application of wave-equation imaging using

FIG. 7. Ray paths for a single receiver position. Ray paths FIG. 8. Curved ray tomogram formed using the ray paths were computed using the nominal model velocities from traced through the “true” model of Figure 5. This represents Figure 5. Both direct body wave ray paths and head wave the optimum result obtainable through ray theoretical travel ray paths were computed, only the first arrival ray paths are time tomography. The interfaces from the true model are shown. superimposed.

Wave-equation Imaging

these results is described in this section. The acoustic wave-equation method (Pratt and Worthington, 1990) is used first, beginning with a single frequency component.

217

differences was used to provide solutions to the frequency- domain acoustic wave equation. These solutions were used to forward propagate the appropriate Fourier component of the source function estimate, and to backpropagate the Fourier components of the scattered data estimates. These two wave fields from each source location are combined by first applying an adjoint operator (Mora, 1987) (the form of which is dependent upon the parameter being estimated) and then simply multiplying the results at each image location. This yields an estimate of the gradient (with respect to the parameter being imaged) of the objective function for each source location. The final step is to sum the results from all source locations. As described in the section “Frequency- domain wave-equation imaging,” the gradient can be displayed as an image.

Acoustic wave-equation imaging

The compressional velocity tomogram of Figure 8 was used to define compressional velocities on a regular grid. The velocities at the grid points were obtained from the tomogram by linear interpolation. These velocities were then used to compute bulk moduli and densities in the reference model by using Gardner’s equation (Gardner et al., 1974). This model was used in the forward and backpropagation steps of the acoustic wave-equation imaging scheme.

As described in the Introduction, the method of finite

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FIG. 9. (a) Where sources and receivers are both on the same side of a reflector and the receiver is farther from the reflector than the source, time moveouts of the direct arrival and the reflected arrival on common source gathers are similar. (b) If the data are regathered into common ray angle gathers, the moveouts of the direct and reflected arrivals for the same source-receiver pair are different. The reflected events A’ and B’ correspond to reflections from horizontal layers A and B.

Time-domain windowing was applied to the scattered data estimates before Fourier transformation. A mute was applied to the scattered field traces from zero time up to the first arrivals. A second mute was applied 32 ps following the first arrival. The windowing was used to remove the energy scattered from the direct shear waves, since this is not treated correctly by the acoustic wave-equation method.

A monofrequency acoustic wave-equation image of the velocity anomalies in the ultrasonic tank model is shown in Figure lla. The imaging frequency used to generate this image was 380 kHz. As a structural or stratigraphic interpre- tation tool, the wave-equation image is a significant improvement on the straight-ray tomograms. Layer resolution is improved, and the presence and the location of the buried channel and the dipping layer are now clear. Although the acoustic method allows computation of both velocity and bulk density images, synthetic studies have shown that the density is poorly resolved by crosshole data (Pratt and Worthington, 1990; Pratt, 1990a). The density images obtained from these data (not shown) were uninformative.

The wave-equation images of this paper can only be interpreted qualitatively. In nonlinear inverse theory the gradient vectors these images represent would be used to update the model quantitatively, a new residual wave field would be computed, and the next iteration would be based on the updated model. It is not possible to proceed in this manner because the finite-difference modeling method used is two-dimensional. Amplitudes are therefore modeled incor- rectly, implying that the gradient calculation will be in error, and that one cannot calculate a new residual field directly from any given model. Because the true residual field cannot be calculated directly, the scattered wave field is used instead. Iterative inversion is therefore not possible. How- ever, because of the focusing process inherent to the algorithm, the use of the scattered wave field results in a high-resolution image of the geologic discontinuities. The two roles of velocity determination and imaging complement each other here in much the same way as do tomography and migration in surface seismic data processing (Bording et al., 1987).

Multifrequency imaging

As described in the Introduction, wide-aperture crosshole surveys are theoretically redundant. However, in this data set, there is a finite aperture and a finite signal to noise level.


Furthermore, there are strong velocity contrasts in the model that distort the wave paths. These conditions mean that additional frequencies will contain additional information.

Figure 11 b shows the result of stacking the wave-equation results from five separate frequency components of the data, ranging from 290 kHz to 410 kHz. This is an effective way of stabilizing the image and has the additional effect of spatially low-pass filtering the image. This is desirable, since single frequency images contain a sharp wavenumber cutoff at 2k,, where k. is the dominant wave number in the model (Dev- aney, 1984). This cutoff leads to a spatial reverberation in the x-z domain, due to Gibb’s phenomenon. Stacking several results with varying k. values inhibits this reverberation by tapering the K-space coverage.

Sensitivity tests

To better define the regime of application of the technique, several tests were carried out. The images were perturbed by altering (1) the grid interval, (2) the spatial sampling along the source and receiver arrays, and (3) the length of the source and receiver arrays (the survey aperture). The results are shown in Figure 12.

Figure 12a reproduces the monofrequency image from Figure 1 la. In Figure 12b the grid interval has been changed from 0.5 m to 0.833 m. There are now fewer than seven grid points per wavelength rather than 11 (at 380 kHz and in the

lowest velocity zone). The image quality is not severely degraded, suggesting that for the imaging problem the criterion of 10 grid points per wavelength can be relaxed.

In Figure 12~ the image was formed using alternate source and receiver positions. In this case the Nyquist spatial sampling criterion is violated, since source and receiver intervals are now 5.0 mm (almost one wavelength in the lowest velocity zone). The image quality is slightly degraded, but this is more likely to be due to the lower level of redundancy. This suggests that the sampling criteria can be relaxed beyond the Nyquist criteria, consistent with a result in Pratt and Worthington (1990) in which synthetic data were used.

In Figure 12d only the top 30 receivers and the top 30 sources were used, to simulate the image that could have been obtained in a situation in which the boreholes did not penetrate much deeper than the objective. It can be seen that the buried channel can still be resolved, which is an important result, since this restriction is often met in practice.

Elastic wave-equation imaging

To form elastic wave-equation images, the same V, velocities (from the curved-ray tomograms) were used as in the acoustic wave imaging. However, the application of the elastic wave-equation imaging technique required further a priori information. The shear velocities are required in order to propagate shear waves correctly and a knowledge of the

(4 (b) (cl (:I FIG. 10. Preprocessing of the data from the ultrasonic tank. The common receiver data shown here are from receiver position 7 (see Figure 6): (a) the direct wave estimates formed using the mean stacking procedure, (b) the original data following trace by trace deterministic deconvolution using the estimates from (a), (c) the deconvolved scattered field estimates following the removal of the direct wave estimates; and (d) the scattered wave estimates following the further removal of the direct shear wave estimates. The reverberations at early times on (c) and (d) are due to the convolution of the wave shape filter with the spike that occurs at zero time on the raw data.


source mechanism is required. Furthermore, if the source mechanism (the spatial distribution and orientation of force vectors) is not correctly modeled, the radiation pattern will be in error, and an incorrect imaging condition will be applied. In essence this is what happens when the acoustic method (which assumes point sources with no directivity) is used to image vector data, as in Figure 4.

The a priori shear velocities can be obtained by perform- ing traveltime tomography using the picked shear arrival times. It was extremely dil5cult to pick these arrivals con- sistently, especially at low ray angles where little direct shear-wave energy exists. The resultant tomogram (Figure 13) is a poor image of the structure of the model, although the magnitudes of the resulting shear velocities are reasonable. This tomogram was formed using 15 iterations of SIRT, assuming straight rays since the shear arrival times are not of

sufficient quality to permit a full curved-ray tomographic inversion.

Unfortunately, the source mechanism is not as straightforward to obtain. The sources and receivers in this experiment have a complex directivity pattern from the interaction of compressional wavefronts in the water with the vertical edge of the model. In the absence of an exact method for extracting the source mechanism from the data, trials were made using different artificial mechanisms. In the first trial, the source mechanism used was a single horizontal force. Similarly, the data were backpropagated by multiplying the complex conjugates of the data residuals by the numerically computed Green’s tensor for a horizontal single-force source. This is equivalent to assuming that the data were recorded only at horizontal component geophones. The absence of direct shear waves at zero ray angles (see Figure

(a) (b) FIG. Il. (a) Acoustic wave equation image from the ultrasonic tank data. A single frequency component of the data at 380 kHz was used to form this image. The a priori velocities were obtained from the tomographic image of Figure 8. The interfaces from the true model are superimposed. (b) Acoustic wave equation image formed by stacking the images from five frequency components of the data from 290 kHz to 410 kHz.


lob or 1Oc) is reproduced by this trial form of the source mechanism.

The V, image is of poor quality. The assumed source and receiver mechanisms result in radiation patterns that are strongly biased against horizontally traveling shear waves. Although at exactly horizontal ray angles no direct shear energy is observed, direct shear energy is present over a wide range of angles. The horizontal single force mechanism has not used much of this data, and the X-shaped noise belt (see Figure 3d) dominates the image.

The forward propagated wave fields and backpropagated wave fields were combined to estimate the gradient with respect to the densities and the two Lame parameters. The gradient was then recomputed in terms of the compressional and shear velocity anomalies (as in Mora, 1987, and Pratt, 199Oa).

For the elastic wave-equation imaging, a different time- In addition to the problems created by using an inappro- domain windowing of the scattered field was used prior to priate source mechanism, there are other reasons for the computing the Fourier components. Rather than excluding poor quality of the V, image. The shear waves are of lower the scattered waves that originated as direct shear energy, amplitude and are less coherent than the compressional these are deliberately included. P-P, S-S, S-P, and P-S waves. As a result, the shear-wave tomogram is of lower scattering are all modeled correctly by the elastic-wave quality. These two problems combine to result in poor equation. Hence the scattered wave field from a 64~s focusing of lower signal-to-noise ratio data. Furthermore, window following the first arrival was used (twice the length the shear waves are spatially undersampled by both the used in the acoustic wave equation imaging). Figure 14a receiver spacing and the finite-difference model. Due to disk depicts the image for V, and Figure 14b shows the image for storage limitations the grid interval was left to 0.5 mm, in V, obtained from the elastic wave-equation technique. spite of the fact that the shorter wavelengths of the shear Again, images from five frequencies between 290 kHz and waves dictate a correspondingly tighter grid. The justifica- 410 kHz were stacked to form these images. Some improve- tion for this violation of the principles of numerical modeling ment can be observed in the V, image when compared to the is the sensitivity study described above, in which a grid acoustic wave equation results (Figure llb). This is most spacing of seven grid points per wavelength was found to be clear at the top interface and in the regions closest to the adequate. However, elastic shear-wave imaging may be source and receiver arrays. more sensitive to undersampling than acoustic compression-

FIG. 12. Sensitivity tests using the acoustic wave equation imaging technique. (a) Image from the single frequency component at 380 kHz shown reproduced from Figure 1 la. (b) Image formed using a finite difference grid interval of 0.833 mm rather than 0.5 mm, as used in (a). (c) Image formed as in (a), but using every alternate source and every alternate receiver position. (d) Image formed as in (a), but using only the top 30 source positions and the top 30 receiver positions.


al-wave imaging. Finally, the source time function we use has been extracted from the direct compressional waves and may not adequately model the transient behavior of the shear waves. Some of these problems would be alleviated by the use of vector (geophone) data, in which case the additional information would better constrain the backpropagation of the data residuals.

To further investigate the problems in forming a V, image, an image was formed by using a source mechanism that radiated shear and compressional waves uniformly in all directions. Although it is straightforward to produce such a source mechanism in the finite-difference formulation, this is of course a totally artificial source that bears no relation to the true source mechanism. However, this mechanism has the advantage of using data from ah directions.

The result of applying the elastic wave-equation imaging method using the uniformly radiating source mechanism is shown in Figures 14c and 14d. The layered structure of the model can now be seen on the V, image, although the image is still unsatisfactory. The use of this source mechanism results in a slightly noisier VP wave image.

Figure 15 shows the vertical derivative of the VP elastic wave-equation multifrequency image (Figure 14b), displayed using the more conventional variable area representation. The image clearly shows the locations of all of the interfaces

FIG. 13. Straight ray tomogram formed using the direct shear wave arrival times.

in the model. The location, shape, and size of the buried channel are well reproduced. Since guided wave modes dominate the scattered data, it is reasonable to conclude that these wave modes have been correctly handled by the imaging algorithms and have contributed to the images of the two low-velocity zones.

The position of the vertical fault plane is somewhat blurred. Vertical features are difficult to detect using tomographic techniques of any kind. A “smile” artifact can be clearly seen at the fault position. This feature is interpreted as being the expression of the migration-like behavior of the algorithm, which collapses the diffractions in the data at their location of origin in the image.

Computational considerations

The use of data from all 51 source positions meant that a total of 102 forward modeling runs of either the acoustic or elastic wave equation were required to form a single image. The advantage of frequency-domain finite differences is that these can all be carried out by factorizing a single “finite- difference matrix” only once, and solving for the additional source positions using forward reduction and backsubstitu- tion. In applying the acoustic method, a grid interval of 0.5 mm was used, chosen to respect the criterion of 10 grid points per wavelength generally used in finite-difference modeling. A 122 by 263 point grid was used (frequency- domain modeling also allows the grid size to be optimally chosen for each imaging frequency, although this was not done).

The storage space required for the finite-difference matrix in the acoustic wave-equation simulation was approximately 15 megawords. The FPS-164/MAX computer used was equipped with 3 megawords of main memory, 155 megawords of disk storage, and two matrix accelerator boards. In this environment, acoustic wave-equation images could be obtained in about 30 minutes per frequency component.

Due to storage and CPU time limitations, the same grid spacing was used in both the acoustic and elastic wave- equation modeling. The storage required for the elastic wave-equation finite-difference matrix resulting from the same 122 x 263 point grid was approximately 60 megawords (four times that required by the acoustic wave-equation method). The elastic wave-equation images took eight times as long to compute as the acoustic wave-equation images, or roughly four hours per frequency component. Normally, elastic wave-equation modeling requires more grid points than the acoustic wave equation. Taking this into account, the true increase in cost of using elastic wave-equation imaging over acoustic wave-equation imaging is consider- ably more than the figures above would indicate.

CONCLUSIONS

A complete strategy for data processing full waveform crosshole data has been developed and applied. A significant aspect of this strategy is that wave-equation methods are complemented by traveltime tomography, since they depend upon accurate velocities for optimal performance. The use of a geologically plausible scale model has demonstrated the potential of the method. These data exhibit the realistic problems of both coherent and random noise, along with

(4 (b)

(cl (d)

FIG. 14. (a) V, and (b) VP elastic wave equation images formed from five frequency components of the data using the assumption of a single force source. (c) V, and (d) V, images formed using the artificial uniformly radiating source.


azimuthally variable source behavior; they contain many of the wave types that have been problematic in the past. Specifically, guided wave modes are present. These modes are correctly handled by the finite-difference modeling of the wave equation, and they contribute significantly to the final results. The source time function for compressional waves can be reliably extracted from the data; however, more research will be required to determine the best method of extracting the source mechanism and using it in full elastic wave-equation imaging.

Wave-equation imaging is likely to provide a much improved way of imaging underground structure between boreholes. Both acoustic wave-equation imaging and elastic wave-equation imaging can be applied to multisource data using frequency-domain finite-difference modeling. The elastic wave-equation technique provides slightly better results and provides both VP and V, images, although a method for estimating the source mechanism is required before optimal V, images will be obtained. Useful images can be obtained from single frequency components of wide-aperture cross-

FIG. 15. Wiggle display of the vertical derivative of the five frequency VP image from the elastic wave equation method (Figure 14b). The image is displayed in the reverse polarity.

hole data, and further enhancement can be obtained by stacking the results from several frequency components.

The quality of the data used in this study was better than the quality of most field crosshole experiments. However, the scaled frequencies were also lower than might be expected from field data. With the current interest in develop- ing acquisition equipment, it is reasonable to expect that field data of high quality will become more widely available. A crucial experiment that needs to be performed is the application of these and other methods to real crosshole data. This study has demonstrated that the combined use of frequency-domain wave-equation imaging and traveltime tomography will perform well in such an experiment.

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