3-D velocity structure from simultaneous traveltime inversion of in-line seismic data along intersecting profiles

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1 Geophys. 1. nt. (1994) 118, RESEARCH NOTE 3-D velocity structure from simultaneous traveltime inversion of in-line seismic data along intersecting profiles Colin A. Zelt" Geological Suruey of Canada, Continental Geoscience Division, Ottawa, Ontario, Canada, KA OY3 Accepted 1994 February 16. Received 1994 February 16; in original form 1993 August 12 SUMMARY A method for simultaneously inverting in-line seismic-traveltime data recorded along a network of intersecting linear profiles is presented. Consistency of the models at the intersection points is assured and permits 3-D structural and velocity variations to be inferred by interpolation between the profiles. Common model parameters are used at the intersection points, thereby minimizing the number of independent model parameters and maximizing the degree of constraint of each 2-D model. The method has two main applications: (1) to obtain the most constrained set of 2-D models for each profile if subsequent 3-D modelling is not possible, and (2) to provide a 3-D starting model for subsequent 3-D modelling if there is sufficient azimuthal data coverage. Some ideal line geometries are suggested, but the method is applicable to any set of two or more lines with one or more intersection point. A new interpretation of crustal seismic data from the Peace River Arch region of the Western Canada Sedimentary Basin, consisting of four profiles, is presented using the simultaneous-inversion method for both refracted and reflected arrivals. The results are generally consistent with a previous interpretation of the data in which each profile was modelled independently of the others using 2-D forward modelling. However, the models obtained by inversion contain less lateral heterogeneity that more accurately reflects the resolution limits of the data. This illustrates the need to reconsider data that have only been analysed by forward modelling. Key words: 2-D models, 3-D itructure, inversion, seismic, traveltime, velocity NTRODUCTON To acquire 3-D seismic data requires large numbers of shots and/or receivers often beyond the resources available, particularly in land-based experiments. nstead, networks of linear profiles that can be interpreted using 2-D methods are normally obtained and 3-D structure is inferred after independent 2-D modelling of each profile. This paper presents a simple extension of the 2-D traveltime-inversion scheme of Zelt & Smith (1992) that allows simultaneous 2-D traveltime inversion of networks of intersecting profiles and assures consistency at each intersection point. As a result, 3-D velocity and structural variations can be inferred by interpolation between the profiles. This scheme is applicable * Now at: Bullard Laboratories, Department of Earth Sciences, Madingley Rise, Madingley Road, Cambridge CB3 OEZ, UK. to any seismic data recorded along two or more intersecting profiles. deally the intersection points should be sufficiently distant from the endpoints of each profile where the sub-surface is sampled by the seismic data to the maximum depth of interest. This approach has two main applications. First, to obtain the best possible (most constrained) set of 2-D models for each profile if there is an insufficient amount of fan or broadside coverage to allow subsequent 3-D modelling. Second, to obtain a 3-D starting model for the analysis of 3-D data with sufficient azimuthal coverage to apply 3-D modelling methods. Each 2-D model provides only apparent structure in the presence of 3-D (out-of-plane) variations, and thus the resultant 3-D velocity structure obtained by interpolation between profiles is approximate. However, the approximation will be good in regions of weak lateral variations with respect to the resolving power of the data, as 795

2 796 C. A. Zelt is often the case, particularly for most crustal-scale experiments. After describing the simultaneous-inversion method, an application to crustal wide-angle data recorded along four profiles from the Peace River Arch (PRA) region of the Western Canada Sedimentary Basin is presented. The results are compared with the initial interpretation of the data by Zelt & Ellis (1989) using a conventional 2-D forward-modelling approach in which each line was analysed independently of the others. METHOD The simultaneous-inversion method represents one minor modification to the traveltime-inversion algorithm presented by Zelt & Smith (1992); all other aspects of modelling the data along each linear profile for 2-D structure remain unaffected, and so a description of the general procedure is not repeated here. The 2-D models are parametrized in terms of layers in which each layer boundary is defined by an arbitrary number and spacing of depth nodes. The velocity inside each layer is defined by an arbitrary number and spacing of upper- and lower-layer velocity nodes (Zelt & Smith 1992). A layer-stripping approach, that is, modelling shallow layers first and fixing their structure as progressively deeper layers are modelled, remains the most efficient way to model the traveltimes of typical wide-angle data. However, with the simultaneous-inversion method, the same layer, or set of layers from each profile is modelled simultaneously before proceeding to the next deepest layer in each model. Ray tracing through each model generates a partial derivative matrix, A, with elements uj, where i (row number) corresponds to traveltime and j (column number) corresponds to the model parameter (Zelt & Smith 1992). A super matrix, A, is constructed from the individual A matrix for each line in which a parameter from one model that is considered common with a parameter from another model, i.e. a velocity or boundary node at the intersection point of the lines, are treated as one parameter in A by placing the derivatives for these parameters in the same column. A damped least-squares inversion is then applied to A and all parameters for the current layer(s) in each model are updated before ray tracing each model again in the next iteration. The algorithm allows any single parameter from two or more models to be treated as one parameter in the inversion so that its value in each model remains the same after inversion. f the parameter starts with different values in each model, the absolute or relative difference between the values can be maintained during the inversion process; for velocity nodes, this would allow azimuthal anisotropy to be considered. Any number and configuration of in-line seismic profiles can be considered. The more intersection points within the line network, the greater the degree of constraint of each 2-D model, since for the same amount of data, the number of independent model parameters is a minimum when using the simultaneous-inversion method (no increase in the degree of constraint is achieved if the intersection points are spaced within the lateral resolution of the data; see below). t is desirable for the ends of each profile to extend beyond the intersection points so that the maximum depth of Figure 1. Line geometries for three and four profiles. (a) Three-, and (b) four-profile geometries with the maximum number of intersection points; (c) four-profile geometry using a rectangular grid; (d) geometry used by Kanasewich & Chiu (1985). sampling by the wide-angle data occurs beneath as many intersection points as possible. Figs l(a)-(b) show line geometries for three and four profiles with the maximum number of intersection points in each case; the maximum equals fn(n-1) where n is the number of profiles. Rectangular-grid geometries, like that shown in Fig. l(c), have less than the maximum number of intersection points equal to n1n2 where n1 and n2 are the number of profiles in each direction and n1 + n2 = n. The maximum-intersectionpoint geometry provides the greatest degree of constraint for each 2-D model. The advantage of the rectangular-grid geometry is more uniform sampling, and hence less prior knowledge of the target is required. Figure l(d) shows the line geometry used by Kanasewich & Chiu (1985) in which they acquired in-line data along the sides of the triangle, as well as broadside data across the interior of the triangle, to allow 3-D modelling of the structure. The advantage of using the geometry in Fig. l(d) as compared to that in Fig. l(a) is that the area enclosed for 3-D sampling is larger. The disadvantage is that the 2-D models for each line will be more poorly constrained since there is no sampling at depth beneath the intersection points. Although each model layer should have a velocity and boundary node at each intersection point with another line, there is one other consideration to be made regarding node placement: another node should not be placed within the estimated lateral resolution of the data at this depth. This is sensible since it avoids a poorly resolved model, but there is another equally important reason. Consistency between the models will be maintained at the intersection point, but could be artificial in the sense that the velocity or boundary depth will be able to change over a short distance from the point, and hence defeat the purpose of the simultaneousinversion approach by decoupling the models at the

3 3-0 velocity structure 797 intersection point. f two or more line-intersection points occur within the lateral resolution of the data at a particular depth, a single centrally positioned node should be used that is common in the models of each intersecting line. PEACE RVER ARCH DATA The PRA wide-angle data were recorded along four -300 km long lines called A, B, C and D (Fig. 2); line B is approximately coincident with the Devonian axis of the PRA. There are four intersection points, the closest of these to the end of a line is the east end of line B, 57 km. Lines A and B were end and centre shot, with a receiver spacing of -3 km. Lines C and D were only end shot with a receiver spacing of -5 km. A total of nearly 900 vertical-component seismograms of in-line data were acquired. Further details of the experiment and plots of the data are provided by Zelt & Ellis (1989). Their analysis of the data was through conventional ray-trace forward modelling of traveltimes and amplitudes to obtain 2-D P-wave velocity models to upper mantle depth for each line. The main features of their models are: (1) weak-to-moderate lateral variations in structure, with no evidence for significant velocity discontinuities or thick low-velocity zones within the crust; (2) an average crustal velocity of 6.6 km s-', average crustal thickness of 40 km, and average upper mantle velocity of 8.25 km s-'. n addition, a number of subtle features of the models may be associated with the approximately east-west-trending Devonian axis of the PRA, suggesting a Palaeozoic thermo-extensional origin for its anomalous 58" 56' 54" ' km shot point - receiver site 120" 118" 116" Figure 2. Location map for the Peace River Arch seismic experiment. The 10 shot points and 336 receiver sites are indicated for each of the four profiles. The shot points labelled Al, A2 and A3 identify line A, and similarly for lines B, C and D. The Devonian axis of the Peace River Arch is approximately coincident with line B. vertical movements (Stephenson et al. 1989). These features include: (1) a relatively high-velocity lower crust beneath the eastern end of the PRA axis (line B); and (2) a slight crustal thickening beneath the arch axis. Halchuk & Mereu (1990) present an interpretation of the PRA data by forward modelling that is largely consistent with the models of Zelt & Ellis (1989). The P-wave phases considered in this study are refracted waves through the upper crust (P,), lower crust (Pc) and uppermost mantle (P,,), and reflected waves from the Moho (P,,,P). The P, phase is a later amval after P, and its presence as a coherent branch on most record sections is an unusual feature of the PRA data. t provides constraint on lower crustal velocity not normally possible. The same picks of these phases as used by Zelt & Ellis (1989) were used in this study, although -10 per cent of the picks, particularly P, and P,P arrivals at far offset where these phases converge, were reassessed and subsequently relabelled as another phase or omitted in cases of doubt. The uncertainties assigned to the picks and used in the traveltime-inversion algorithm (Zelt & Smith 1992) are 100ms for Pn, 150ms for P,P and a linear offsetdependent uncertainty for Pg and P, such that a value of 50ms was assigned at Okm and 100ms at 180km offset, roughly the P,-P, crossover point. These uncertainties are based on recent and more densely spaced crustal data in which a detailed determination of the appropriate uncertainties to use for traveltime inversion was made (Zelt & Forsyth 1994). The shot-receiver geometry for each profile is close to linear for such a large-scale experiment so that 2-D modelling of each line is considered reasonable. The reciprocity of traveltimes for each phase along the four profiles was checked and in all cases falls within the assigned picking uncertainties. The shallow sedimentary velocity structure was not modelled, but obtained from sonic logs along the profiles (Zelt & Ellis 1989). A laterally varying two-layer sedimentary model was obtained for each profile and incorporated into each crustal model before the simultaneous-inversion procedure was applied to obtain the subsedimentary velocity structure. The velocity and thickness of the sediments vary between km s-l and 2-5 km. The data were corrected to a datum level equal to the average elevation of the survey region, 0.65 km above sea level, and depths within the models are measured from this level. APPLCATON AND RESULTS Estimates of the vertical velocity gradients throughout the crust were required before proceeding with the inversion since traveltimes alone do not generally provide sufficient constraint on this model feature. The traveltime and amplitude analysis of Zelt & Ellis (1989) showed that the crust throughout the PRA region is divisible into three layers based on changes in the velocity gradient with depth: moderate gradients of s-' in the upper 15 km, small gradients of s-l between km depth, and large gradients of s-l below 30 km. The average upper mantle gradient is -0.01s-'. Thus, a three-layer crust and single upper mantle layer with these gradients were used in the inversion of the PRA data. The crustal

4 798 C. A. Zelt 0 50 DSTANCE (km) to- n rn W *. *- \ 0 L h(- south nortt 0. to- n v) W b *- \ 0 - q- A2 to- n rn W b *- \ Q - m DSTANCE (km) Figure 3. Ray diagram and traveltime comparison for the final line-a model. Ray paths for the P'., Pc, P,P and P, phases for shots Al, A2 and A3 connecting all source-receiver pairs are shown (see text for description of phases). Sedimentary-, crustal- and Moho-layer boundaries indicated by dashed lines. Observed traveltimes indicated by vertical bars with heights representing pick uncertainty; calculated data indicated by dots; shot-point locations indicated by large dots along distance axis. A 7 km s-' reducing velocity has been applied.

5 3-0 velocity structure 799 boundaries at 15 km and 30 km were fixed since intracrustal reflected phases were not considered in this analysis and the velocity discontinuities across these interfaces are relatively small, less than 0.3 km sc1, and refracted arrivals provide little constraint on interfaces that represent small discontinuities. The starting model for each layer was laterally homogeneous representing an average of the four 2-D models obtained by Zelt & Ellis (1989). The inversion of the traveltimes was performed in two steps: (1) Pg and P, arrivals were modelled simultaneously to determine the velocity of the three crustal layers, and (2) P,P and P, were modelled simultaneously to determine the Moho boundary and upper mantle velocity. The crustal velocities were fixed during step 2, although the final few iterations also included P, and the lower crustal velocities were allowed to vary since both P, and the furthest-offset P, P arrivals constrain these velocities. Figure 3 shows the ray coverage provided by all shots and phases for line A that is typical for each profile. Note the velocity between km depth is poorly constrained due to a lack of refracted rays with turning points in this depth range. Also, the ray coverage at the Moho is offset horizontally -50 km from either end shot. Fig. 3 also shows the comparison between the observed and calculated traveltimes for each shot and phase for line A. The observed data most poorly fit by the calculated data (e.g. P, at 275 km for shot Al, P,P for shot A2 and P,, at 100 km for shot A3; Fig. 3), were similarly misfit by Zelt & Ellis (1989). An inversion of the line A data alone also yielded the same poor fits. This suggests these features of the data: (1) would require a significant and unrealistic velocity anomaly to provide a fit within the pick uncertainties since they were misfit by a forward- and inverse-modelling approach; (2) are picking errors, and/or (3) are due to out-of-plane (3-D) structure. Nevertheless, the overall traveltime fit of the PRA data is within the assigned pick uncertainties, that is, a normalized x2 value of -1 (Table 1). Figure 4 presents the 3-D structure of the PRA region in the form of contour plots of four model features obtained by simple interpolation and extrapolation of the velocity structure defined by the 2-D models of the four lines. The number of velocity and boundary nodes used in each layer of the models, in addition to those at the four intersection points, is the minimum required to fit the data within the picking errors. For each layer and parameter type, two nodes were placed some distance in from the ends of the line to bracket either side of the ray sampling at that level in the model: 0, 25 and -50 km for the upper crust, middle crust and lower crust/upper mantle, respectively (Fig. 3). One exception was at the eastern end of line B in the lower crust where the intersection point with line A is roughly 50 km from the end of the line (Fig. 4). For the basement and upper mantle velocities, the 12 and 11 intersection- and end-point nodes were sufficient to achieve satisfactory fits to the data (Fig. 4). For the lower crustal velocity and Moho depth, 2 and 7 additional nodes were added to achieve the final fit to the data. There are 63 independent velocity and boundary nodes in the four 2-D models. The average crustal and upper mantle velocities (6.56 and 8.21 km s-l) and crustal thickness (39.8 km) agree to within 1 per cent with the values obtained by Zelt & Ellis (1989), although the average basement velocity (6.04 km s-l) is 0.09 km s-l faster, and the average lower crustal velocity (7.17 km sc1) is 0.18 km s-l slower than the Zelt & Ellis values. The major regional variations presented in Fig. 4 generally agree with the results of Zelt & Ellis, including decreasing basement velocity to the north-west, increased lower crustal velocity to the north-east and west, increased upper mantle velocity to the north-east, and crustal thickening and thinning in the north-east and north-west, respectively. However, these variations are generally slightly larger in magnitude in the models of Zelt & Ellis, and most small-scale anomalies, i.e. -50 km or less in lateral extent, in the Zelt & Ellis models have no counterpart in the models of this study. This is not surprising given the relatively poor lateral resolution of these data (cf. White & Boland 1992; Zelt & Forsyth 1994) and that the inverse approach is designed to yield a minimum-structure model (Zelt & Forsyth 1994), whereas when forward modelling it is generally a difficult task to construct a single model that adequately satisfies the data. n other words, most of the small-scale anomalies in the forward models are artefacts of the forward-modelling procedure and reflect the nonuniqueness inherent in the data. DSCUSSON Networks of linear profiles with a maximum number of intersection points and sampling to the maximum depth of interest beneath as many of these points as possible is the optimal manner in which to acquire seismic data when 3-D imaging is required but sufficient 3-D coverage to allow 3-D Phase p, > pc Number of Average RMS misfit Normalized Model parameters constrained observations uncertainty (ms) (ms) X2 velocity nodes boundary nodes P3 prl Total

6 800 C. A. Zelt 57 56' 55" 120" 118" 116" Figure 4. Basement, lower crustal and upper mantle velocity and crustal thickness in the Peace River Arch region as determined from the 2-D models for each line; basement refers to the top of the upper crustal layer beneath the sedimentary layers. Contour intervals are 0.1 km s-' for velocity and 2 km for thickness. The straight lines and grey squares indicate the four seismic profiles and shot points (Fig. 2). The dots indicate the position of the nodes within the 2-D models representing the particular contoured parameter; note that for the basement velocity, there is a node coincident with the position of each end shot. Only the region enclosed by the dashed line is considered constrained by the seismic data. modelling is not possible. t is for this type of experimental geometry that the simultaneous-inversion method is intended. Although the application to the PRA data consisted of four profiles, there is in theory no limit on the number of profiles that could be involved, only the normal considerations when dealing with large numbers of model parameters and data in an inverse problem. f there is a sufficient number of profiles to allow 3-D modelling, or azimuthal as well as in-line data is recorded, the simultaneous inversion method could be applied to obtain the best possible 3-D starting model. The need for very 'close' starting models when using a variety of modelling techniques has been widely recognized, for example fullwavefield inversion (Sun & McMechan 1992) and in the case of traveltime inversion of relatively sparse 3-D data (B. C. Zelt, personal communication, 1993). Another application

7 3-0 velocity structure 801 of the method is the reassessment of data that have only been analysed by 2-D forward modelling; the application to the PRA data presented here demonstrates the potential for uncovering modelling heterogeneity that is not required to fit the data. Consistency of the 2-D models at their intersection points is automatically obtained and eliminates the ambiguity when presenting the regional variations from forward modelling since consistency at the intersection points is often not achieved. f azimuthal anisotropy is present in the survey region, consistency of velocities at the intersection points would not be expected. However, if the data can be adequately fit without considering anisotropy, or there is no independent information suggesting its presence, then assuming isotropy and obtaining consistency of velocities at intersection points is preferable since this represents the simplest-model approach. f necessary, a constant or relative difference between the value of two or more nodes at an intersection point can be maintained throughout the inversion. n this way azimuthal anisotropy can be included based on evidence from other data or determined by prior analysis of the wide-angle data. Alternatively, the amount of azimuthal anisotropy may be determined by trial and error using the simultaneous-inversion method by testing different levels of anisotropy to determine which provides the best fit to the data. Another advantage of the simultaneous-inversion method is that it is possible to constrain a particular depth range of a model that has no ray coverage if an intersecting profile has ray coverage at the same depth. The simultaneous-inversion method presented here has made use of the 2-D inversion algorithm of Zelt & Smith (1992). Since the model parametrization of this approach permits irregular node spacing, it is always possible to position nodes at the intersection points of lines. Other inversion algorithms that use regular-grid parametrizations may or may not run into problems in this regard, depending on the particular network geometry of the profiles. This point aside, the simultaneous-inversion method could be applied using any 2-D traveltime-inversion scheme since it is only the construction of the super partial derivative matrix A, that is necessary. The method may be applicable to the analysis of any geophysical data using a discretely parametrized 2-D inversion algorithm. The method of simultaneous inversion of multiple in-line profiles has produced a new set of 2-D models for the PRA data from which the 3-D regional velocity structure can be inferred. The quality of the traveltime fit is as good as the fit obtained by forward modelling, but the final models obtained by inversion contain less lateral heterogeneity that more accurately reflects the resolution limits of the data. The main features and trends of the two sets of models are similar and, in particular, the two features that led to the suggestion of a Palaeozoic thermo-extensional origin for the vertical movements of the PRA, are present in the inverse models: relatively high velocities in the lower crust and slight crustal thickening beneath the arch axis. A new feature of the inverse models that is not present in the models of Zelt & Ellis (1989) is the suggestion of crustal thickening toward the Rocky Mountains in the south-west (Figs 2 and 4), although the 44 km contour is extrapolated from the constrained region. ACKNOWLEDGMENTS Reviews of earlier versions of the paper by R. Ellis, B. Milkereit, J. Wu and B. Zelt improved its clarity. The submitted manuscript was reviewed by D. Caress and an anonymous reviewer. This work was done while the author was a Canadian Government Laboratory Visiting Fellow at the GSC. Geological Survey of Canada Contribution REFERENCES Halchuk, S.C. & Mereu, R.F., A seismic investigation of the crust and Moho underlying the Peace River Arch, Canada, Tectonophysics, 185, Kanasewich, E.R. & Chiu, S.K.L., Least-squares inversion of spatial seismic refraction data, Bull. seism. SOC. Am., 75, Stephenson, R.A., Zelt, C.A., Ellis, R.M., Hajnal, Z., Morel-8- Huissier, P., Mereu, R.F., Northey, D.J., West, G.F. & Kanasewich, E.R., Crust and upper mantle structure and origin of the Peace River Arch, Bull. Can. Perrot. Geol., 37, Sun, R. & McMechan, G.A., D full-wavefield inversion or wide-aperture, elastic, seismic data, Geophys. J. nt., 111, White, D.J. & Boland, A.V., A comparison of forward modeling and inversion of seismic first arrivals over the Kapuskasing Uplift, Bull. seism. SOC. Am., 82, Zelt, C.A. & Ellis, R.M., Seismic structure of the crust and upper mantle in the Peace River Arch region, Canada, J. geophys. Res., 94, Zelt, C.A. & Forsyth, D.A., Modeling wide-angle seismic data for crustal structure: southeastern Grenville Province, J. geophys. Res., in press. Zelt, C.A. & Smith, R.B., Seismic traveltime inversion for 2-D crustal velocity structure, Geophys. J. nr., 108,