Statistical description of seismic reflection wavefields: a step towards quantitative interpretation of deep seismic reflection profiles

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1 Geophys. J. Inr. (1996) 15, Statistical description of seismic reflection wavefields: a step towards quantitative interpretation of deep seismic reflection profiles C. A. Hurich Earth Sciences Department, Memorial University, St. John s, Newfoundland, Canada A1B 3x5 Accepted 1996 January 16. Received 1996 January 15; in original form 1995 March 3 SUMMARY Seismic modelling designed to compare the statistics of a realistic acoustic impedance field with a statistical description of the resulting backscattered wavefield demonstrates a strong correlation between the spatial properties of the impedance field and the resulting wavefield. Experiments in the weak and strongscattering regimes indicate significant and predictable differences in the spatial properties of the wavefield depending upon the scattering mode. Wavefields generated in a mixedscattering regime with concurrent weak and strong scattering are dominated by the strongscattering response. The results of these experiments suggest that the statistical description of reflection wavefields may provide a basis for the quantitative interpretation of the complex reflection wavefields most commonly observed in seismic reflection profiles across crystalline basement rocks, and a tool for extracting more of the information content of the seismic data. Key words: scattering, seismic reflection. INTRODUCTION In the past 0 years, deep seismic reflection studies have provided a wealth of information on the largescale tectonic framework of the continents. However, a significant portion of the information contained in deep seismic profiles is presently not being extracted from the data because of the difficulty of interpreting the complex reflection wavefields that are most often observed in the crystalline crust. Difficulties in interpreting the reflection wavefields centre around an incomplete understanding of the physics of the reflection process and the lack of a methodology to transform the seismic data into useful geological information. The complexity of the reflection wavefield observed from the crystalline crust as compared to the relatively simple response from sedimentary basins arises from the complex geometric distribution of impedance contrasts that characterizes most crustal geology. As a result of the geometric complexity of the geology, the observed reflection wavefield represents the combined result of scattering and specular reflection, which is further complicated by spatial and temporal interference between the various scattered modes (Hurich & Smithson 1987; Gibson & Levander 1988; Raynaud 1988; Gibson 1991; Holliger et ue. 1994; Levander et a!. 1994). An understanding of the relative importance and the effects of the various contributors to the reflected wavefield is fundamental to further progress in the interpretation of deep seismic data. Several workers (e.g. Frankel & Clayton 1986; Levander & Holliger 199; Hestholm, Husebye & Ruud 1994; Holliger & Levander 1994) have demonstrated that seismic models in which the impedance structure of the Earth is treated as a random medium with particular statistical properties are consistent with geological and geophysical observations, and may be a useful approach to understanding the various components of the complex reflection wavefield. The success of statistical approaches to the treatment of the impedance structure of the Earth further suggests that some type of statistical description of the reflected wavefield may be a useful way to relate the complicated reflection wavefield to the impedance structure and ultimately to the geology. Gibson ( 1991) demonstrated theoretically and experimentaily that, for small velocity variations (c. 3 per cent), lateral coherence in the velocity field is correlated with lateral coherence in a zerooffset (coincident source and receiver) seismic section. Holliger, Carbonell & Levander (199), however, in experiments using a velocity model with statistical properties based on analysis of maps and physical properties for an exposed lower crustal section, found no correlation between the lateral correlation length of the velocity field and the lateral correlation length of synthetic seismic data. They attribute the difference in the two results to increased roughness of the more realistic velocity model, which results in more scattering, and to larger velocity contrasts, which result in a larger contribution to the wavefield by higherorder scattering. Treatments of the propagation of waves through heterogeneous media have identified regimes in which the scattering behaviour is dominated by different processes. Depending on the magnitude of the contrasts encountered by the wave and RAS 719

2 70 C. A. Hurich the characteristic size of the heterogeneities with respect to the radius of the Fresnel zone, the response can be classified into regimes dominated by specular reflection, weak scattering (diffraction) and strong scattering (multiple scattering) (Flatte et ul. 1979; Wu & Aki 1988). In this paper I report on a systematic investigation of the relationship between the lateral (horizontal) properties of the acoustic impedance distribution and synthetic seismograms generated from these distributions over the range of scattering regimes that are normally encountered in the Earth s crystalline crust. The lateral properties are characterized by two estimatorslateral correlation length and mean event length. These investigations are the first step in formulating a methodology for extracting more of the information content from deep seismic data. SEISMIC SCATTER IN G Flatte et al. (1979) present an analysis of the character of the fluctuations in a wave front propagating through isotropic inhomogeneities that can be adapted to provide a general description of the backscattered (reflected) wavefield. They demonstrate that the character of fluctuations in the wavefield is controlled by two parameters representing the strength and relative size of the heterogeneities in the media through which the wave propagates. In the case of reflection seismology, these parameters are represented by fluctuations in the acoustic impedance field. In the adaptation of Flatte et al. (1979) to the seismic reflection problem I have retained the original notation of the authors, even though the notation is not the standard for seismology. In reporting experimental results, however, I have adopted the terminology of previous seismological workers where appropriate. Table 1 provides a summary and crossreference for the terminology. To characterize the size and spatial extent of the heterogeneities, Flatte et al. define A such that where R, is the radius of the first Fresnel zone for a sourcereceiver distance R, and L, is the correlation length of the heterogeneities in a direction orthogonal to the propagation Table 1. Definitions. Flatte et al. (1979) L, Correlation length of impedance field orthogonal to the direction of wave propagation L, Correlation length of impedance field parallel to the direction of wave propagation L, Integral scalelength of impedance field parallel to the direction of wave propagation R, Radius of first Fresnel zone R Distance travelled by wave qo Wavenumber of seismic pulse p RMS fluctuation of acoustic impedance This paper LCL, Horizontal correlation length of impedance fieldin these experiments equivalent to L, LCL, Horizontal correlation length of seismic wavefield defined by a point 50 per cent from the zerolag of the correlation function MEL Mean event length direction. A small value of A (A < 1) corresponds to a backscattered wavefield dominated by specular reflection with minor contributions from diffraction. A large value of A (A > 1) corresponds to a wavefield dominated by diffraction. To characterize strength, they define a that can be approximated by in which q, is the characteristic wavenumber of the seismic pulse, p is the rms fluctuation in acoustic impedance, R is the distance the wave travels, and L, is the integral scalelength. LI can be approximated by 0.4L, where L, is the correlation length in the direction of propagation. Flatte et a/. also suggest that the approximate boundary between the weak and strongscattering regimes, that is between regimes dominated by single and multiple scattering, is defined by A@ = 1 for small values of A, and = 1 for large values of A. Thus, mapping the properties of the acoustic impedance field into A@ space provides an estimate of the relative proportions of the various scattering mechanisms contributing to the reflected wavefield (Fig. la). STOCHASTIC REPRESENTATION OF THE CRYSTALLINE CRUST Owing to the considerable heterogeneity of the crust at a variety of scales, both smaller and larger than the typical seismic wavelength, it is very difficult to produce realistic seismic models of the crust using deterministic models of geology. Deterministic models derived from map data suffer from a dependence on the scale at which the mapping is carried out, the amount of detail that can realistically be included in the map, and the quality of geological exposure. An alternative to using deterministic models is to treat the geology as a stochastic medium with realistic strength (acoustic impedance contrasts) and spatial properties. Acoustic impedance contrasts can be characterized by using physical properties data combined with map information to estimate a probability density function for the acoustic impedance. Spatial properties are more difficult to determine because of the inherent deficiencies of geological maps. The spatial properties of various geological features are an often cited example of a selfsimilar or fractal distribution. Based on the analysis of relatively detailed maps of an exposed lower crustal section in the Ivrea Zone of northern Italy, Holliger, Levander & Goff ( 1993) demonstrated that, at least for the scales included in the maps, the Ivrea Zone geology is essentially selfsimilar with a fractal dimension of.7. A similar analysis on maps of midcrustal Lewisian rocks in Great Britain leads to a similar conclusion, although the fractal dimension varies between.55 and.78 (Levander et al. 1994). Further support for the selfsimilar nature of the Earth s impedance structure comes from modelling studies that demonstrate that selfsimilar impedance distributions produce the best fit to the coda observed in earthquake arrivals. Therefore, it is reasonable to assume that, at least to first order, the acoustic impedance fields of other polydeformed metamorphic terrains are also selfsimilar with comparable fractal dimensions.

3 Statistical description of rejection wavefieids 71 5 I I 5 10 w lo 5 I 0... A f 5 KM I 0.5 KM ni (4 lo C D KM A I 0.5 KM A Figure 1. (a) Scattering regimes space; (b) scattering field for experiments in the weakscattering regime; (c) scattering field for experiments in the strongscattering regime; (d) scattering field for experiments based on the kinzigite section of the Ivrea Zone. A EXPERIMENTAL APPROACH In order to determine the degree of correlation between the acoustic impedance model and the resulting reflection wavefield and to assess the effects of the various contributors to the wavefield, I have generated and analysed an ensemble of synthetic seismograms. The seismograms were calculated using a secondorder finitedifference solution to the acoustic wave equation in an implementation based on Kelly et al. (1976). Although computationally expensive, the finitedifference solution offers a convenient method both for working with the geometrically complex medium and for determining the complete acoustic wavefield. The experiments are presently confined to the acoustic wavefield. The model space (Fig. ) has lateral and vertical dimensions of 0 and 15 km respectively, with a 5 km thick random zone sandwiched between constantvelocity layers. The grid spacing is 0.05 km. Energy is introduced into the model by designating Gaussian derivative point sources for every other grid point at the top of the model, resulting in a downgoing plane wave. The frequency spectrum of the source wavelet is Gaussian with a 15 Hz centre frequency. Extensive testing of the finitedifference algorithm demonstrates that, for this model space and these model parameters, the solution provides satisfactory accuracy for the problem at hand. The spatial distribution of the acoustic impedance function is characterized by Von Karman statistics (Frankel & Clayton 1986; Holliger & Levander 199) with a fractal dimension of.8 and is selfsimilar over a scale of at least km. The spatial characteristics of the acoustic impedance field are further characterized by the vertical and lateral correlation lengths of the field. The correlation lengths, as determined by the autocorrelation function, designate the level of dependence of a particular point in the discrete impedance field upon its neighbouring values, or, equivalently, they indicate the vertical and lateral repeatability of the field. The vertical and lateral correlation lengths used to generate the acoustic impedance field are designated the theoretical correlation lengths ( LCLT), and are used as a basis of comparison with correlation estimates determined from seismograms. The probability distribution function of the velocity distribution, although originally Gaussian in our algorithm, is converted to a binary distribution for the experiments. The density distribution is correlated to the velocity distribution and determined by the empirical relation p = 0.35( 3000Vp)0~5, (3)

4 7 C. A. Hurich (a> 3km Figure. (a) Example of a single realization of a selfsimilar acoustic impedance field with a fractal dimension of.8. The vertical and lateral correlation lengths are 0.1 and 3.0 km respectively; (b) synthetic seismogram resulting from the impedance field. Estimators of the spatial properties of the wavefield give a LCL, of km and a MEL of 1.64 km. where p isensity and V, is the P wave velocity. The choice of a binary impedance distribution, although not strictly realistic geologically, allows precise and consistent control of the position of the experiments in A 4 space. The choice of a fractal dimension of.8 gives a slightly rougher velocity field than the.7 fractal dimension estimated by Holliger & Levander (199) for the Ivrea Zone. Two separate approaches are taken to determining the statistical description of the synthetic seismograms. The first approach is use of the D autocorrelation to estimate the characteristic lateral correlation length of the seismogram ( LCLs). As with the impedance field, the correlation length is an estimate of the spatial dependence of amplitude and phase at a particular point in the discrete seismogram on the neighbouring values. This is accomplished by taking the Fourier transform of the D power spectrum of the seismogram, windowed to include the response from the entire 0 km by 5 km stochastic zone. The lateral correlation is estimated by a point 50 per cent down from the zerolag value of the autocorrelation function. The choice of the 50 per cent down point to determine the value of the estimator, although somewhat arbitrary, provides a stable estimation, whereas other choices, such as the 80 per cent down point, do not. Fig. 3 (LCL) shows a plot of the theoretical lateral cor relation length for 71 realizations of the impedance field against the correlation length determined by the 50 per cent down estimator. The data show a linear relationship between the theoretical and estimated correlation lengths, demonstrating that the estimator is stable for the model space over the range of correlation lengths considered. Linear regression estimates a standard error of 0.31 km for the determination of the theoretical lateral correlation length (LCL,) from the measured lateral correlation length of the impedance field (LCL,). The data also demonstrate an increase in the variance of the estimator with increasing correlation length, suggesting that the variance of the estimator is lagdependent. These observations indicate that, when using the autocorrelation to compare correlation lengths in the synthetic seismic data with the correlation lengths in the associated impedance fields, an increase in the scatter of observations at larger correlation lengths is expected, but deviation from a linear trend should be attributable to the mechanics of the reflection process and not to the estimator. The second approach to estimating the statistics of the seismograms involves automatic picking of events on the seismogram and derivation of descriptive statistics for the event population. Determination of the properties of the estimators derived in this fashion is more difficult than for the correlation

5 Statistical description of reflection wavefields 73 Figure 3. Characteristics of the lateral correlation length (LCL) and mean event length (MEL) estimators determined by comparison with the theoretical LCL of the impedance field. The slight nonlinearity of the MEL estimator suggests that the two estimators reflect different but related aspects of the wavefield. length. The problem centres on the lack ofa theoretical standard for the estimator and in the detection of bias introduced by the algorithm used for event picking. The results of using mean event length (MEL) as an estimator of the lateral properties of the impedance field are also shown in Fig. 3. In this case, the eventpicking algorithm was set to detect events that had no more than 50 m of topography (c. 1/8 of a wavelength) and would qualify as surfaces suitable to produce specular reflection (Sheriff & Geldart 198). Unlike the LCL estimator, the response of the MEL estimator with respect to the LCL, is slightly nonlinear. The nonlinearity of the MEL with respect to the LCL, and the difference in the absolute values of the spatial properties returned by the two estimators is not surprising, as the two estimators provide information on somewhat different aspects of the impedance field. As with the LCL estimator, the significant deviation of thrmel estimator from the trend demonstrated in Fig. 3 when the estimator is applied to the seismic wavefield should be associated with the mechanics of the backscattered wavefield and not with the estimator. The advantage of the eventpicking approach to estimating the statistics of the seismograms is that a number of different statistics can be estimated for the event population, and event subpopulations based on a number of appropriate discriminators can be extracted and analysed. EXPERIMENTAL RESULTS Weak scattering Experiments in the weak scattering regime (Fig. lb) were carried out with an impedance field which varied between 16.3 and 17.73, producing a modest absolute reflection coefficient of A reflection coefficient of this magnitude might represent lithological variations between a granite and a granodiorite or between a plagioclaserich and a plagioclasepoor gabbro. Impedance contrasts of this magnitude are common in the crystalline crust (Hurich & Smithson 1987), but it is not yet clear how often reflections from such modest contrasts are detected in standard deep seismic profiles. The results of 50 different realizations of the impedance field and the analysis of the resulting synthetic seismograms using the LCL estimator are shown in Fig. 4. These data demonstrate a strong linear correlation between the LCL, of the impedance field and the measured lateral correlation length of the seismogram (LCL,). Linear regression of the data set gives a standard error of 0.1 km for the estimation of LCL, based on LCL,. The similarity of the error estimation to that for the impedance field suggests that the spread in the LCL, estimates is associated with the variance of the estimator coupled with variations in the individual realizations of the acoustic impedance field and not with phenomena associated with wave propagation and scattering. Fig. 5 shows a comparison between the response of the MEL estimator applied to 141 synthetic seismograms from experiments in the weakscattering regime and the response of the MEL estimator applied to the associated impedance fields (as previously shown in Fig. 3). The MEL of the seismograms is determined for 100 per cent of detected events above the background noise level. The correlation between the MEL of the seismic wavefield and the LCL, is nonlinear, as expected, but is considerably more nonlinear than the relation determined from the impedance field. It is presently not clear exactly why the difference in the two results occurs. As indicated previously, the departure of the estimator applied to the seismic wavefield from the results of the same estimator applied to the impedance field is probably the result of the mechanisms operating to produce the wavefield rather than a property of the estimator. A significant change in the curvature of the powerlaw fit to the experimental data, which correlates with the average radius of the Fresnel zone, suggests that the MEL estimator may be responding to a difference in wavefields dominated by diffraction and wavefields dominated by specular reflection. A comparison between the response of the MEL applied to 100 per cent of the detected events and a subpopulation of the top 5 per cent of these events, derived by applying a discriminator based on the product of event length and mean event amplitude, is shown in Fig. 6. The subpopulation includes the strongest and longest events in the population, representing the reflections most likely to be detected in a seismic experi

6 74 C. A. Hurich 1.4 F Q LCL Impedance Field (km) Figure 4. Theoretical LCL versus measured LCL for experiments in the weakscattering regime. These experiments demonstrate a clear correlation between the lateral spatial properties of the impedance field and the seismic wavefield. 1.5 E 4 _... 0 SEISMIC WAVEFIELD 100% 1 m 1mEDmCEFIELD Q I LCL Impedance Field (km) FigugeA Comparison of the MEL estimator applied to 141 realizations of the impedance field and the associated seismograms. z E YI s E 1. h E 5 s W s VI hl B h Y, 0 Top5Yo Fi I 1 E el LCL IMPEDANCE FIELD (km) Figure 6. MEL for 100 per cent and the top 5 per cent of the event populations derived from synthetic seismograms for experiments in the weakscattering regime. Although the absolute values of MEL are different, the similarity in the response of two event subpopulations suggests that the geometric properties of the wavefield are independent of the detection threshold of the seismogram. F, marks the average radius of the Fresnel zone.

7 Statistical description of rejection wavejields 75 ment. Although the absolute MELs are quite different for the entire population and the subpopulation containing the longest and strongest events, the curvatures (exponents) of the bestfitting power laws are very similar. The similarity suggests that the nonlinear response of the MEL estimator with respect to the LCL, of the impedance field reflects a fundamental property of the wavefield that is independent of the threshold of detectability in a particular seismic experiment, and suggests that the seismic wavefield may retain the selfsimilar properties of the impedance field. Fig. 7 (weak) shows the relationship between the MEL for the 78 realizations of the impedance field and the MEL of the resulting seismograms. There is considerable scatter in the data, probably due to the combined variance of applying the MEL estimator to both the impedance field and the seismic data. Owing to the lack of an evident argument to the contrary, I presently take the simplest model and view the relationship as linear. Linear regression on the data estimates a standard error of 0.48 km for the prediction of the MEL of the impedance field based on the MEL of the seismic data in the weakscattering regime. Strong scattering A set of experiments with a binary acoustic impedance distribution that varies between and 3.40 maps dominantly into the strongscattering regime in A@ space, although for higher correlation lengths scattering for frequencies less than about 15 Hz should be in the weak regime while scattering for higher frequencies should occur in the strong regime (Fig. lc). The acoustic impedance contrast in this set of experiments gives an absolute reflection coefficient of 0.18 and represents the contrast between a lithology with a Pwave velocity of 6.0 km s' and density of.7 x lo3 kg m3, and a lithology with a 8.0 km sl velocity and a.95 x lo3 kg m3 density. This reflection coefficient might represent the contrast between a granite and an amphibolite, mafic granulite or feldsparrich peridotite, and would be about the highest contrast expected 9'!1 the crystalline continental crust. Fig. 8 shows the LCL, of synthetic seismograms associated with 60 realizations of the acoustic impedance field in comparison with the data already discussed for the weakscattering regime. As with the results for the weakscattering regime, LCL, and LCL, are correlated and the scatter of LCL, increases with increasing LCLT. Unlike the results for weak scattering, however, the relationship is nonlinear, indicating that multiple scattering decreases LCLs, particularly for longer LCL,. The dominance of the effect at longer LCL, suggests that multiple scattering in the form of specular or quasispecular intrabed multiples is more important to the response in the strongscattering regime than multipathing due to closely spaced diffractors. Together, the results for the weak and strongscattering regimes define an envelope that constrains the expected results for the majority of possible impedance contrasts in the continental crust. The results of applying the MEL estimator to both the impedance field and seismic data in the strongscattering regime (Fig. 7, strong) demonstrate that strong scattering tends to reduce the length of coherent events in the wavefield. As in the weakscattering experiments, the relation between the MEL of the impedance field and the MEL of the wavefield appears to be linear, at least to first order. Ivrea lower crust Physical properties data from the Ivrea Zone of northern Italy, an exhumed continental lowercrustal section, form the basis of a set of experiments designed to investigate the seismic response of an impedance field with contrasts that may commonly occur in the lower crust (Fountain 1976; Burke & Fountain 199; Holliger et al. 1993). A binary impedance field varying between and 19.3 (reflection coefficient = 0.11) represents the contrast between amphibolitegrade pelitic paragneisses and metabasites in the kinzigite portion of the Ivrea Zone. The contrast is consistent with reflection coefficients of 0.1 estimated by Warner (1990) from lower crustal reflections in the area around Great Britain. One of the interesting aspects of this set of experiments is that mapping in A@ space (Fig. Id) shows that the seismic.response of an impedance field with contrasts similar to those in the Ivrea MEL Impedance (km) Figure 7. MEL measured on the impedance field versus MEL measured on the resulting synthetic seismograms for experiments in the weak and strongscattering regimes.

8 76 C. A. Hurich 1.6 I L 8 n E & d LCL Impedance Field (km) Figure 8. LCL, versus LCL, for synthetic seismograms for experiments in the weak and strongscattering regimes. Zone spans the boundary of the weak and strongscattering regimes. Weak scattering dominates for frequencies less than 9 Hz, and strong scattering dominates for higher frequencies. Mapping in A@ space also suggests that for LCL,s longer than approximately 3 km the seismic response should be almost entirely in the weakscattering regime. The results of the LCL estimator for the Ivrea experiments along with the results for the experiments in the weak and strongscattering regimes are shown in Fig. 9. The Ivrea experiments are essentially identical to those of the strongscattering experiments for LCL,s of 4 km and shorter. Even though the A 4 space mapping for the Ivrea experiments implies a mixedscattering response, the strong scattering of the higher frequencies apparently dominates the correlation properties of the backscattered wavefield. In accordance with the A 4 mapping, for LCL,s greater than 4 km the Ivrea data show a distinct shift towards the weakscattering response. DISCUSSION The stochastic description of the acoustic impedance fields associated with geology typical of the continental crust provides a basis for the application of seismic modelling to understanding the complex reflection wavefields commonly observed in deep seismic profiles. The impedance fields have selfsimilar spatial properties and consequently include impedance variations that span the range from significantly smaller, to much larger, than the seismic wavelength. As a result, the seismic response of these impedance fields provides an accurate estimate of the characteristics of the reflected wavefields observed in deep seismic profiles. The success of the stochastic description of the impedance fields suggests that a statistical description of the seismic wavefields may provide insight into the relative importance of the various contributors to the wavefield and perhaps provide a basis for a quantitative interpretation of the wavefield. The results of these experiments clearly demonstrate that the spatial properties of the backscattered wavefield are correlated to the spatial properties of the acoustic impedance field. This is in contradiction to the conclusions of Holliger et ul. (199), which were derived from a limited number of experiments in the strongscattering regime. It is probable that the considerably larger size of the sample space, combined with the ensemble approach in the present experiments, provides more stable and reliable statistics for the wavefields. These experiments also demonstrate a clear difference in the spatial properties of wavefields generated in the weak and strongscattering regimes. Wavefields associated with strong scattering show consistently shorter lateral correlation lengths (LCL,) and mean event lengths (MEL) as well as a larger number of total events detected than wavefields associated with weak scattering. The results for the LCLs estimator applied to wavefields generated in the mixedscattering regime indicate that, when both weak and strong scattering occur concurrently, the spatial properties of the wavefield are dominated by the strongscattering response. The difference in the spatial properties of wavefields dominated by weak and strong scattering highlights the question of the relative importance of the two scattering regimes in deep seismic data. The relative contribution of the two scattering modes depends upon the magnitude of the impedance contrasts and the frequency of the seismic source. A rough estimate of the possible reflection coefficients in the continental crust (Hurich & Smithson 1987) suggests that about 80 per cent of the possible reflection coefficients are less than 0.1, and about 45 per cent are less than The range of most common reflection coefficients spans the range of impedance contrasts represented by our experiments in the weakscattering regime and the Ivrea lower crust. For the lower end of the typical seismic spectrum (55 Hz) and the most common impedance contrasts, the backscattered wavefield is dominated by weak scattering. For higher frequencies, however, the contribution from strong scattering is increasingly important even in the case of small impedance contrasts. Furthermore, as large impedance contrasts are more likely than small contrasts to produce reflections that are detectable in deep seismic profiles, it seems likely that most of the obserwd reflection wavefields containing frequencies higher than about 5 Hz are dominated by strong scattering. The experiments also suggest that the spatial properties of

9 Statistical description of rejection wavefields Weak w Strong I LCL.,. Impedance Field (km) Figure 9. Theoretical LCL versus measured LCL for experiments in the weak, mixed (Ivrea) and strongscattering regimes. The Ivrea experiments are dominated by the strongscattering component of the wavefield for LCLs of 4.0 km and less, but the weakscattering component becomes more important for LCLs greater than 4.0 km the wavefield may provide information on the relative importance of specular reflection and diffraction in the reflection wavefield. In particular, the experiments suggest that the MEL estimator may be sensitive to the different processes operating in wavefields dominated by diffraction as opposed to specular reflection. The LCLs estimator gives no indication of this possibility. The difference in the response of the two estimators, with respect to LCLT, is consistent with the notion that the two estimators respond to different aspects of the wavefield. It is encouraging that the experimental results suggest that a statistical approach to the description of the spatial properties of the backscattered wavefield may provide a viable approach to quantitative interpretation of complex reflection wavefields. The results presented for the MEL estimator (Fig. 7) and the LCL estimator (Figs 8 and 9) provide envelopes of the expected response for the extremes of acoustic impedance contrast likely to occur in the continental crust. It is expected that these envelopes will form the basis for comparison with further experiments and more importantly for comparisons with actual seismic data. There is an important limitation on the present set of experiments that requires consideration and that places bounds on the applicability of the results. It is probable that the spatial properties of the reflection wavefield are in some way related to the relative abundance of impedance contrasts as well as to the correlation lengths of the impedance field. That is, the spatial properties of the wavefield depend upon the number of reflectors as well as upon the spatial distribution of reflectors. A set of experiments that explores the role of the abundance of impedance contrasts, and will be discussed elsewhere, indicates that the general trends of the results presented herein are independent of the abundance of contrasts, but under some circumstances the absolute values of the estimators are dependent on the number of impedance contrasts as well as on their spatial distribution. ACKNOWLEDGMENTS The author would like to thank T. Kocurko for computing support and K. Holliger for very useful discussions and software. The research was funded by NSERC under the Individual Research Grant program. REFERENCES Burke, M.M. & Fountain, D.M., Seismic properties of rocks from an exposure of extended continental crustnew laboratory measurements from the Ivrea Zone, Tectonophysics, 18, Flatte, S.M., Dashen, R., Munk, W.H., Watson, K.M. & Zachariasen, F., Sound Transmission Through a Fluctuating Ocean, Cambridge University Press, New York, NY. Fountain, D.M., The IvreaVerbano and StronaCeneri zones, northern Italy: a cross section of the continental crustnew evidence from seismic velocities of rock samples, Tectonophysics, 33, Frankel, A. & Clayton, R.W., Finite difference simulations of seismic scattering: implications for the propagation of shortperiod seismic waves in the crust and models of crustal heterogeneity, J. geophys. Res., 91,

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