Journal of Geophysical Research: Solid Earth

Waveform tomography in 2.5D: Parameterization for crooked-line acquisition geometry


Corresponding author: B. R. Smithyman, Department of Earth, Ocean and Atmospheric Sciences University of British Columbia, Room 2020, Earth Sciences Bldg., 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada. (


[1] A method for 2.5D viscoacoustic waveform tomography that can be applied to generate 2D models of velocity and attenuation from inversion of refraction waveforms on land seismic reflection data acquired along crooked roads is developed. It is particularly useful for typical crustal reflection surveys. First-arrival travel time tomography is applied using a 3D method, but with constraints on the intermediate 3D velocity model; the result is the starting model for the next step. A 2.5D frequency-domain full-waveform inversion stage parameterizes 3D geometry in the seismic source and receiver arrays, with the assumption that the velocity and attenuation models are homogeneous in the out-of-plane direction. This approach results in superior results compared to a strictly 2D approach when the acquisition line is crooked, with a moderate increase in computational cost. A case study using data acquired in the Nechako Basin in south-central British Columbia, Canada, exemplifies and validates the procedure. The velocity model derived from 2.5D waveform tomography is compared with that from a previous study in which 2D waveform tomography was applied to the same data set and with results from 3D travel time tomography. The resolution and accuracy of the velocity model from 2.5D waveform tomography are demonstrated to be greater than those from travel time or 2D waveform tomography. A model of viscoacoustic attenuation, which was not possible in the 2D case, is also generated. These models are interpreted jointly to highlight features of geological interest, such as a sedimentary basin, basement rocks, and faults, from surface to about 3 km depth.

1 Introduction

[2] Waveform tomography is a method for building seismic velocity models through multiple applications of nonlinear inversion of seismic waveform (phase and amplitude) data. An initial stage of inversion is carried out via a conventional ray-tracing tomography approach, typically using an iterative nonlinear inversion framework that seeks to improve a simple starting model by inversion of picked first-arrival travel time data. The model resulting from travel time inversion is then iteratively updated by full-waveform inversion of the seismic data set to better predict the data waveforms. The successful application of the combination of travel time inversion and full-waveform inversion is predicated on the assumption that the model from the initial tomography stage is representative of a low wavenumber version of the ultimate result.

[3] We implement a 2.5D method for the full-waveform inversion stage. It reproduces results equivalent to those of a 3D method with a specially constrained model. This leads to significant computational benefits, in terms of reduced processing time compared to a 3D work flow. The use of a 2.5D method allows for improved accuracy when modeling 3D geometric spreading effects that cannot be reproduced in a 2D work flow. By constructing the equivalent 3D wavefield in the receiver plane, the method accounts for a crooked-line acquisition geometry that would make 2D processing unfeasible. Waveform tomography in 2.5D is thus particularly applicable to most crustal reflection data, which are generally acquired in 2D geometry along existing crooked roads. Waveform tomography cannot be applied to such data without some allowance for the crooked-line acquisition geometry, such as an approximate geometric correction by travel time statics [Smithyman and Clowes, 2012], a modification of the encoded source and receiver locations in 2D to account partially for 3D offset (not considered here), the approach discussed in this paper, or application of full 3D waveform tomography.

[4] The first part of this paper provides a discussion of the 2.5D full-waveform inversion method, including special focus on the extension of the method to account for 3D geometry effects, which we believe represents a new development. We then provide a real-data case study to contrast this approach with a more typical 2D approach based on reprocessing of previous work. Our examples are derived from a data set acquired in south-central British Columbia, Canada.

2 Background

[5] Full-waveform inversion may be implemented using a variety of approximations and simplifications, to suit the computational strategy to the problem at hand. One common choice is the use of the acoustic (or viscoacoustic) wave equation in two spatial dimensions. This limits the size of the problem to a manageable number of model parameters, and the forward-modeling problem can be efficiently formulated in the frequency domain as a boundary-value problem separable over frequencies. This approach is attractive for use with seismic data acquired using 2D seismic acquisition methods. However, the physical world is three-dimensional, and field seismic data always encode some information that cannot be readily synthesized using a two-dimensional numerical code. The use of the acoustic approximation also leads to inaccuracies in modeling (and consequently errors in inversion), which we discuss in section 3.4.

[6] Active-source waveform tomography and full-waveform inversion have become trusted tools in research [Takougang and Calvert, 2011; Malinowski and Operto, 2011; Kamei et al., 2012] and are being increasingly used in industry settings as well. A number of research groups and companies have developed practical massively parallel full-waveform inversion programs that operate in 3D (e.g., Ben-Hadj-Ali et al.[2008]; Plessix [2009]). The researchers responsible for these efforts aim to tackle not only the challenging problem of 3D full-waveform inversion but also the significant computer science problems (and computational costs) relating to the scale of the science. These computational issues are in most cases inseparable from the geophysical problem. There exists, however, a middle ground between costly 3D processing and comparatively simplistic 2D processing: a two and one-half dimensional method solves a simplified 3D problem in such a way that the overall solution is formed from a series of solutions to smaller problems [Bleistein, 1986]. Each of these smaller problems is a 2D problem of much lower complexity than the full 3D problem. For certain classes of problems, the use of a 2.5D method can convey some of the benefits of 3D processing, while reducing significantly the computational demands in comparison to a 3D approach. We extend the work of other researchers in the field [Song and Williamson, 1995; Song et al., 1995; Pratt et al., 1996; Zhou and Greenhalgh, 2006] to include the ability to handle 3D geometry.

[7] The use of a 2.5D processing methodology for full-waveform inversion is not, in and of itself, a new idea. The 2.5D approach to generating 3D-equivalent wavefields is used commonly in the field of seismic migration. Song and Williamson [1995] discuss the use of a 2.5D method for forward modeling of acoustic waveforms, and Song and Williamson [1995] extend the method to full-waveform inversion in a companion paper. More recently, Zhou and Greenhalgh [2006] present a methodology for efficiently sampling the transverse wavenumber component when synthesizing seismic data. In the past decade, much research in this field has been toward full-waveform inversion in 3D [Ben-Hadj-Ali et al., 2008; Plessix, 2009] and improved accounting for elastic wave propagation [Sears and Barton, 2010]. Because of the increased cost when compared to 2D processing (significant at the time), the 2.5D method of Song and Williamson [1995] did not enter into common use. However, there exists a plethora of seismic data collected along 2D acquisition profiles that do not contain cross-line information necessary to solve for a 3D model, but do contain 3D geometry that is irreconcilable with a 2D approach. Furthermore, while the computational cost of 2.5D forward modeling (and inversion) is somewhat higher than the equivalent 2D approach, the cost is still on a similar computational order; 3D processing is two computational orders more expensive than 2D processing. Significant savings can be made by choosing 2.5D processing over a full 3D implementation. This is often a controlling factor in academia, or for small companies working in the geophysics or engineering sectors.

3 Methodology

[8] Waveform tomography comprises a series of processing and inversion steps designed to produce a model of velocity, and possibly viscoacoustic attenuation [Song et al., 1995; Brenders and Pratt, 2007], that optimally predicts a set of field seismic data. An initial 1D model is constructed of velocity variation with depth, based on txinterpretation or known average properties. This model is then updated through travel time inversion (section 3.1) to generate a gridded model that optimally predicts the seismic first-arrival data. Full-waveform inversion is applied (section 3.3) to update the intermediate model from travel time inversion, imaging structures that are illuminated by wave arrivals and that are not well predicted in the tomography stage (which is limited by asymptotic ray theory). Because of the use of the acoustic wave equation, isotropy assumptions, simple representation of the source and receiver arrays, and a variety of other approximations implicit in the method, some heuristic or empirical corrections must be applied before and after full-waveform inversion (section 3.4). The result is assessed based on a priori geological knowledge and comparison of synthetic data waveforms with the true data waveforms, for purposes of quality control.

3.1 Travel Time Inversion

[9] Because full-waveform inversion is a highly nonlinear process, it is necessary to approach the inversion with a reasonably good estimate of the true model. This model (called the starting model or initial model) is iteratively updated by the full-waveform inversion process to produce a model that better predicts the data waveforms. In the method of waveform tomography, the starting model for full-waveform inversion is typically built by a more conventional velocity analysis stage. This is less computationally intensive than the later full-waveform inversion stage, but no less important to the final result.

[10] We use the program FAST (First Arrival Seismic Tomography) [Zelt and Barton, 1998] to produce a starting model for full-waveform inversion. A first-arrival travel time datum is picked (typically by hand) for each of the seismic traces to be considered in the later full-waveform inversion. This travel time represents the arrival of the earliest waveform to reach a given part of the receiver array from the selected source. FAST is initialized with a simple (near-1D) model of the estimated velocity structure of the site and models these travel times using an eikonal equation method (i.e., fast marching) [Vidale, 1990; Hole and Zelt, 1995]. The predicted travel time for each trace is compared to the measured travel time (i.e., the pick), and the residual is weighted based on the inverse of the expected picking error. The weighted residuals are spread across the (smoothed) numerical raypaths by iterative backprojection [Hole, 1992]) to form a descent direction at each iteration (equivalent to a gradient method of inversion). The descent vector is regularized by the addition of scaled vectors representing the roughness in the vertical and horizontal directions, which encourages a smooth model. The initial model is perturbed with a series of modified versions of the regularized gradient, scaled by a step length. The new model that optimally reduces the data misfit is found, and the process is repeated using the new model as the starting model for the subsequent iteration. A conjugate gradient scheme is followed until the data are fit within the expected picking error.

[11] In order to produce a model compatible with the application of 2.5D full-waveform inversion, it is necessary to extract a 2D slice from the 3D volume representing the travel time tomography model. To simplify this process, we implement two additions to the method of Zelt andBarton [1998]: (1) the local grid of the model is oriented such that the x and z directions match those of the consequent 2D model, and (2) the 3D model from FAST is strongly smoothed in the ydirection at each iteration, to ensure that the model heterogeneity is nominally 2D. The resulting volume is then sliced along the x-z plane and possibly re-sampled for use in full-waveform inversion.

3.2 Full-Waveform Forward Modeling

[12] The numerical forward modeling in this study is carried out by the method of Pratt [1999], with modifications discussed in the following text. We proceed in a similar fashion to the discussion in Pratt and Song [1996], for the case when y is not necessarily zero. The 3D acoustic wave equation (equation ((1))) describes the propagation of acoustic waves in space and time, through a region of acoustic velocity c(x,y,z).

display math(1)

However, the method of frequency-domain full-waveform synthesis solves the Helmholtz equation (equation ((2))), which represents the time-independent form after applying the method of separation of variables.

display math(2)

[13] A spatial Fourier transform is applied to equation ((2)) that separates the result into a number of 2D wave equations as presented in equation ((3)), with the additional assumption that the model does not vary in the direction described by the y coordinate (i.e., c(x,y,z)=c(x,z)). This represents a decomposition over plane waves in the y coordinate and therefore leads directly to the required assumption of infinite support in that axis (i.e., model homogeneity).

display math(3)

[14] In equation ((3)), kyis set separately for each Fourier component; this means that each solution for a different value of ky is independent. In order to reconstruct the wavefield equivalent to the solution of the 3D wave equation, an inverse spatial Fourier transform is applied, i.e.,

display math(4)

[15] It is convenient to solve the 2D Helmholtz equation (in equation ((3))) for the case in the y=0 plane, i.e., with the omission of the transform of the δ(y|ys) term from equation ((2)); in this case, the Fourier synthesis in equation ((4)) is modified to contain a phase shift applied to inline image, which results in the same synthesized wavefield P:

display math(5)

[16] We note that there is a specific region of interest along the integration axis of equation ((4)): Zhou andGreenhalgh[2006] define kcas the critical value of the propagation medium (this point is also made more obliquely by Song and Williamson [1995]). This marks the threshold between propagating waves for values of ky inside the range of −kc<ky<kcand strongly evanescent waves characterized by wave numbers outside this range. The practical application of the transform in equation ((4)) can therefore be concerned chiefly with constructing the equivalent 3D wavefield from those wave numbers within ±kc of the zero wavenumber. Since the solutions to the wave equation of interest are symmetric about ky=0, the discretized version may be simplified to a discrete cosine transform over the reduced space:

display math(6)

[17] The transform in equation ((6)) represents a discretized version of the inverse Fourier transform; the sampling of the spectrum in the transform over ky was the topic of work by Zhou and Greenhalgh [2006]. In that case, synthetic wavefields were calculated for the governing wave equation with values of kycontrolled by the abscissae of the Gauss-Legendre integration formula. We have incorporated this method, which results in somewhat improved numerical stability in comparison to regular sampling, for a constant number of integration components. In this case, the sampling of ky in equation ((6)) is not regular, and Δkyis controlled by the Gauss-Legendre weights.

3.3 Full-Waveform Inversion

[18] The full-waveform inversion stage of the waveform tomography processing is based on the method in Pratt [1999] and related works, and extends the work of Song and Williamson[1995] to account for 3D geometry (this comes from the forward-modeling method discussed above). The inversion is carried out using the adjoint state method; i.e., the gradient descent vector for an iterative descent scheme is computed by multiple forward-modeling steps [Lailly, 1983; Tarantola, 1984; Pratt et al., 1998]. The objective function is based on the logarithmic l2norm, in which the gradient is preconditioned by taking the natural logarithm of the data residuals [Shin and Min, 2006; Kamei et al., 2012]. The gradient points in the direction of fastest (local) increase of the objective function; for any position in model space that is not a local extremum of the objective function, the sign-reversed gradient represents a local descent operator. There exists a perturbation of some size along this vector that will reduce the logarithmic l2 norm of the data residuals. The preconditioned gradient vector indicates the direction of descent, and a perturbation that is a multiple of this vector is applied. If the data residual is not decreased, the step length is reduced and the process is repeated. The conjugate gradient preconditioning method is used to improve the rate of convergence [Polak, 1969], which modifies the gradient for iterations after the first.

[19] In order for the nonlinear inversion process to converge toward the true Earth model, the initial model information (from tomography or earlier stages of FWI) must predict the travel times of a given arrival mode within one half cycle. Since the duration of one cycle is longer at low frequencies, the nonlinear inversion scheme is more tolerant of model errors when processing the low-frequency components of the data waveforms. The initial stages of inversion operate on a set of data including only the lowest frequency components. Subsequent stages of the inversion include information from higher frequency components (i.e., with reduced band limiting), which gradually provide improved resolution and accuracy. This enables a multi-scale approach to inversion, in which coarse model information is resolved prior to inverting for fine structure.

3.4 2.5D, AVO, Source, and Attenuation Considerations

[20] The discretized version of the Fourier transform (equation ((6))) is synthesized from a finite number of component wavefields. The limited number of components results in a periodicity of the source y coordinate, which manifests as numerical ghost sources. The representation of each source along the y coordinate is a numerical approximation to a spatial delta function, which takes the form of a Dirichlet kernel with a shape controlled by the sampling in the discrete Fourier transform. The spacing of the ghost sources in the out-of-plane direction is inline image [Song and Williamson, 1995]. Incorporating additional components in the data synthesis increases this repeat distance. A similar phenomenon is seen in the synthesis of time-domain data from a series of Fourier-component data [Song and Williamson, 1995]. In both cases, the solution of choice is to include additional frequency or wavenumber components. However, because of the implicit periodicity resulting from band-limited Fourier synthesis, some time or space wrap-around effects will still contaminate the data. The application of a frequency-domain exponential damping [Sirgue and Pratt, 2003; Shin et al., 2010] selectively amplifies early-arriving waveforms, while damping late-arriving waveforms. This substantially attenuates the effects of the ghost sources (and time-periodic effects, equivalently) relative to the main arrivals. The use of this method has also been shown to improve convergence by restricting the inversion to fitting the early-arriving waveforms [Brenders and Pratt, 2007], which avoid the later wave coda that may contain wave arrivals with elastic propagation modes. We do not model a free surface, instead extending the near-surface low-velocity region with a homogeneous (non-reflective) half-space of the same velocity and background attenuation above the sources and receivers. Consequently, the downweighting of free-surface multiples by the exponential damping factor is also beneficial, since we do not expect to reproduce these features in the synthetic data.

[21] While the dominant factors that affect data amplitude variation with offset (AVO) behavior—viz., geometric spreading effects—are reproduced accurately by 2.5D forward modeling, a number of residual effects remain that may result in discrepancies between the field data and the equivalent synthetic seismogram. These include elastic effects that are not modeled by the viscoacoustic code (e.g., mode conversions between P waves and S waves), discrepancies in the modeled source and receiver radiation patterns, near-surface coupling effects, etc. In order to accommodate the bulk of these AVO effects, we carry out heuristic normalization of the data amplitudes to those present in forward-modeled data from an early (smooth) version of the model. This follows the method described by Brenders and Pratt[2007]: (1) the real and synthetic data are scanned to determine the bulk AVO response, (2) a least squares linear regression is applied to the logarithms of the amplitude response versus absolute offset, and (3) the field data are scaled based on this factor such that their bulk AVO response approximates that of the synthetic data. In equation ((7)), the scaled data are computed from the original data by an offset-dependent multiplicative scale factor. The offset is r. A0 is the intercept and A1the slope of the log linear regression line, which describes the bulk amplitude difference between the synthetic and real data.

display math(7)

This approach has the benefit of resolving the large-scale AVO effects of approximation errors while retaining small-scale variations in data amplitude that are produced by attenuation or scattering. However, the background attenuation model used in forward modeling affects the computed amplitudes that go into the AVO correction and consequently may affect the attenuation information extracted during inversion.

[22] The source signature for the seismic experiment may in some cases be known, but the apparent encoded source is also dependent on the near-surface response in the source region, which may not be modeled accurately by a discretized finite-difference framework (especially in absence of an accurate free surface). After the method of Pratt [1999], a source signature is estimated from the data in a linearized fashion by finding the complex coefficients at each frequency (i.e., amplitude and phase) that lead to the optimum fit between real and synthetic data. This avoids the need for a known source signature as input to the inversion, and since a unique series of coefficients is found for each source, unmodeled near-surface effects on amplitude and phase can be accommodated to some extent. Other methods for accommodating source and receiver calibration can be considered, e.g., a determination of a scaling factor for each source and receiver that normalizes the amplitude of the data on a per-gather basis. These affect the weighting of the different data elements in the inversion and have recently been advocated by others in the field. However, additional re-weighting of the data amplitudes on a source and receiver basis was not carried out in our case study.

[23] Other researchers have discussed the limitations of inverting for velocity and attenuation in some detail [e.g., Mulder and Hak, 2009; Hak and Mulder, 2011]. For linearized inversion, especially in situations when the aperture of illumination of subsurface features is incomplete, an ambiguity exists between the resolution of numerical intrinsic Q and velocity. This ambiguity is reduced when the target is illuminated from many angles (e.g., with near-offset and wide-angle refraction and reflection wave paths) and when strong multiple scattering is modeled correctly, which produces an equivalent effect. Nonlinear inversion enables recovery of accurate high-resolution velocity and attenuation structures and therefore is less susceptible to this ambiguity at late iterations. In order to mitigate remaining non-uniqueness in this case, we begin with a homogeneous (fixed) model of numerical Q and only update this model after the velocity model has already been improved using phase-only inversion. The model of attenuation that is inverted from data waveforms during full-waveform inversion is also not able to distinguish between effects of intrinsic (P wave) Q and coda Q from scattering. This coda Q component is also influenced by mode conversions from P to S waves, which are not modeled by the acoustic forward-modeling method. Consequently, unmodeled scattering effects and mode conversions may result in loss of amplitude in the forward-scattered P wave, which are imaged as high-attenuation anomalies in the numerical model of inverse Q.

4 Case Study: Nechako Basin

[24] We initially developed this 2.5D waveform tomography methodology for use in processing a series of seismic reflection data sets acquired along crooked roads in the Nechako Basin, south-central British Columbia, Canada. In a previous work, we applied waveform tomography processing using the same starting model as this work; however the full-waveform inversion step was carried out on static-corrected data waveforms, using a 2D processing method [Smithyman and Clowes, 2012]. The methodology described in the previous work improves the tractability of the full-waveform inversion step when compared to a methodology that does not account for deviations from 2D geometry. However, the static-correction approach introduces additional errors into the data waveforms and is only applicable for certain wave arrivals (i.e., those that are well modeled by the initial travel time tomography processing). Here we present an improved version of the result from Smithyman and Clowes [2012]; the new 2D methodology uses the logarithmic l2 norm rather than the l2 norm. It results in fundamentally different model perturbations that are more geologically consistent than those seen when the l2norm was used. This revised 2D result allows us to make direct comparison with the 2.5D result that uses the logarithmic l2norm (the topic of this paper), and we detail the advantages offered by the 2.5D full-waveform inversion (FWI) step.

[25] The Nechako Basin is an area of interest for both hydrocarbon and mineral resource exploration. Geoscience BC, in conjunction with the British Columbia Department of Energy, Mines and Petroleum Resources, funded and organized a series of exploratory projects involving both industry and academia from about 2006 to 2012. As a result, a variety of geophysical and geological mapping efforts have been applied, on both the regional and site-specific scales. These include detailed geological mapping, potential field surveys, and a series of seismic reflection profiles along existing roadways [Ferri et al., 2007; Calvert et al., 2009, 2011].

4.1 Geology

[26] The Nechako Basin is located in the Intermontane Belt of the western Canadian Cordillera. The region comprises a series of sedimentary sub-basins. A regional map (Figure 1) depicts the location of our site of interest, and a map of the local surficial geology (Figure 2) provides context for this discussion. The underlying basement is composed of Stikine Terrane sedimentary and volcaniclastic rocks deposited up to the Middle Jurassic. Middle Cretaceous sedimentary rocks of the Skeena group overlie the Stikine Terrane rocks, followed by Late Cretaceous sedimentary rocks of the Sustut group (among others). Both are terrigenous; the former are classified as prospective for hydrocarbon development [Hannigan and Lee, 1994], while the latter are not—both units contain structural traps from compressional tectonics in the mid-Jurassic to mid-Eocene and extensional tectonics in the middle to late Eocene. Eocene volcanic rocks of the Ootsa Lake and Endako groups dominate the near-surface; these were erupted contemporaneously with mid- to late-Eocene extension. Neogene Chilcotin basaltic rocks overlie these in some areas (seldom thicker than 100 m and often considerably thinner). Quaternary deposits of varying thicknesses and types overlie other units.

Figure 1.

Overview of geological terranes in central British Columbia. The study area is indicated. A series of geological mapping efforts and geophysical surveys have been carried out throughout the basin, especially in the central and southeastern portions. Map data from Massey and MacIntyre [2005].

Figure 2.

Surface lithology and the presence of Quaternary cover are shown for the area indicated in Figure 1. Much of the study area is thinly covered by the Neogene Chilcotin basalt, which is seldom thicker than 10 m [Bordet and Hart, 2011]. The shallow seismic response is dominated by the Eocene volcanic rocks of the Ootsa Lake and Endako groups. Stikine Terrane rocks of the post-accretionary Hazelton group outcrop north of the seismic line, and outcrops of Cretaceous sediments are also visible east of the line. The geometry of seismic line 10 is shown, with several other lines from the seismic survey indicated for reference purposes. The endpoints of the 2D seismic profile are indicated along with source and receiver array extents. Shots 200, 350, and 600 are the locations of the vibration points for which waveform data are shown in Figures 79, respectively. Universal Transverse Mercator coordinates are used, with NAD27 map datum.

[27] Smithyman and Clowes [2012] present a combined interpretation of the geology along line 10, using information gleaned from 2D waveform tomography processing of the same data examined in this study. The results of that work are additionally compared with an image from pre-stack depth migration (PreSDM) along the same line [WesternGeco MDIC, 2010], and with a result that includes interpretations of both a post-stack time-migrated seismic section and magnetotelluric data [Calvert and Hayward, 2011].

4.2 Survey Parameters

[28] An industry-style multichannel seismic reflection survey was carried out in the summer of 2008 by CGG Veritas, on behalf of Geoscience BC. This example case deals with 2.5D waveform tomography processing of a subset of data from seismic line 10. The data comprise records from 699 source points, each with 960 live receiver channels arranged in a split-spread configuration (720 receivers along-line to the west of the source and 240 receivers to the east). The vibroseis source points were spaced 40 m apart, and the receiver-group interval was 20 m. Hence, the nominal along-line receiver offsets extended to approximately 14.4 km. However, due to line curvature, the straight-line maximum offset was 12 km on average. The vibroseis sweep was 28 s long (including 0.9 s on- and off-tapers), with frequencies ranging from 8 to 64 Hz. The source consisted of four diversity-stacked sweeps from four vibrators each, distributed over 60 m along-line [Calvert and Hayward, 2009]. The geometry of the seismic line is shown overlaid on surficial geology in Figure 2.

4.3 Model-Building Procedure

[29] The initial model-building process for this study is identical to that discussed in Smithyman and Clowes[2012], with the exception that none of the steps carried out for the static correction were necessary for the 2.5D processing (due to the accommodations for 3D geometry at the FWI stage).

[30] First-arrival travel times were picked for each trace in the 699 source gathers we examined. Picking was carried out on the direct and refracted waves, which possess a data bandwidth from 8 to 16 Hz. The data are generally good quality, but some traces were excluded from further processing because of the presence of noise. This includes random background noise (e.g., due to wind and other ambient conditions on the line swamping signal at long offsets) and systematic noise due to logging trucks and equipment that were active during the data acquisition. Since picking was carried out on the full offset range of the data (nominally 14.4 km), we observed some shift in the phase of the waveform associated with the highest-amplitude arrivals. Unlike with the picking of explosive-source data, the acausal encoded wavelet from vibroseis correlation makes determining the first-arrival phase difficult at widely varying offset ranges (due to differences in the encoded wavelet between near and far offsets, deconvolving with a uniform wavelet is of limited effectiveness). The picking error was between 20 and 30 ms and was controlled by the ability to distinguish a particular arrival phase. Additionally, the seismic records did not contain information before time zero, and as such some of the short-offset traces did not possess the full record of the encoded (acausal) source.

[31] FAST [Zelt and Barton, 1998] was applied to generate a 3D model of seismic velocities based on nonlinear inversion of the travel time data. The model size was 70 km  × 4 km  × 6 km, with a 100 m cell size. The x and y coordinates were oriented such that the y coordinate was transverse to the line of best fit through the plan-view projection of the seismic array. This preserved the geometry, but ensured that the largest components of the sourcevreceiver offsets were represented by the xcoordinate offset. The model was iteratively updated until an RMS travel time misfit of 27 ms was reached. An asymmetric smoothing operator was applied to the model at each iteration in order to encourage the smoothest model that fit the travel time data; the smoothing in the ydirection was sufficiently strong to ensure that the model was homogeneous in that coordinate. This also represents a simplification, with an assumption similar to the 2.5D approximation used in the waveform code: the smoothing along the y direction is strictly valid only if the acquisition line crosses geological strike. The travel time misfit was compatible with the expected picking errors in the data; this represented an average misfit of less than one quarter cycle at 8 Hz, which indicated that the full-waveform inversion stage should be tractable. A slice in x,z coordinates of dimensions 701×61 was extracted, which represented the average model velocities along the y axis. This slice was sub-sampled to a cell size of 50 m  × 50 m and cropped for purposes of FWI to a model size of 901×81 (i.e., dimensions of 45 km  × 4 km).

[32] The waveform data were windowed around the first arrivals, in order to limit the inversion to fitting mainly the refractions. This had the added benefit of eliminating most contributions from elastic propagation modes (S waves and mode-converted arrivals), which generally arrive later in the seismic record. However, loss of amplitude in the P wave arrivals due to elastic mode conversion is still present in the refractions. To avoid contamination of the result by ground roll and the air wave, the data for receivers with offsets smaller than 1 km were not included in the inversion.

[33] In order to compare the results of 2D and 2.5D full-waveform inversion, we undertook a reprocessing of the results from Smithyman and Clowes [2012] using modified preconditioning parameters (viz., logarithmic l2 norm and adjusted smoothing parameters). These parameters were chosen to generate a 2D result with a work flow as close as possible to the 2.5D work flow that is the topic of this study. Full-waveform inversion was carried out to improve the resolution and fidelity of the model of seismic velocity, in each of the 2D and 2.5D cases. The data were preconditioned using an exponential damping factor (τ=0.8 s) tO emphasize early arrivals. In early iterations, we used a small portion of the seismic data, band limited within a few hertz of the 8 Hz minimum frequency. We mainly attempted to fit the waveform phase and did not attempt to solve for complex velocity (and therefore attenuation, parameterized as 1/Q) in the initial stages of inversion. This is important, because both velocity and attenuation models affect the amplitude of arrivals, but the phase of the forward-scattered waveforms is mainly controlled by the velocity model [Brenders and Pratt, 2007; Smithyman et al., 2009]. A fixed model of Q=200 was set for the initial stages of inversion.

[34] In the final stages of 2.5D FWI, we fit both waveform phase and amplitude and additionally allowed the gradient updates to affect the full complex c, thereby to update the attenuation model. For the later stages of inversion including amplitude information, the data were scaled using a log linear correction to fit the bulk amplitude variation with offset as determined by an earlier stage of inversion (Figure 3; see also section 3.4). This normalized the data amplitudes to the unit-amplitude delta-function source employed at early stages of inversion. We applied FWI in several stages. In all cases, frequencies were accumulated in the inversion for maximum stability, and the gradient was formed from the frequency-domain data at 0.25 Hz intervals. The integration over y wave numbers (only present in the 2.5D case) contained 40 components, from 0 to kc (see equation ((6))), selected independently for each temporal frequency component using the Gauss-Legendre method of Zhou and Greenhalgh [2006].

Figure 3.

The natural logarithm of data amplitude is shown for the original correlated seismic data (Field in the upper panel) and an early synthetic result (Synth) using a unit-amplitude delta function source. Colors represent the density of points (magenta indicates the highest density and red the lowest); the variation in data amplitude for a given offset range is higher far away from the source. The mean data amplitude was calculated for each data set at a number of bins over the offset range (indicated by x). From this, a best fit log linear trend (black line) for each data set was determined. The difference between these trends represents a scaling relation that adjusts the bulk characteristics of the field data amplitudes to correspond to the unit-amplitude synthetic data (lower panel; A0, intercept; A1, slope).

[35] The following steps were carried out:

  1. [36] Ten iterations using a constant model for Q=200, including five frequencies from 8 to 9 Hz

  2. [37] Five iterations with Q=200, including 13 frequencies from 8 to 11 Hz

  3. [38] Five iterations with Q=200, including 21 frequencies from 8 to 13 Hz

  4. [39] Eight iterations with inversion for c and Q, including 33 frequencies from 8 to 16 Hz. Updates muted outside 10 km<x<35 km to preserve stability.

[40] We chose to accumulate frequencies (i.e., invert additional frequencies in combination with the frequencies of the previous steps) in the inversion process, for purposes of stability and to reduce the number of potential factors contributing to modeling errors while assessing the capabilities of the 2.5D method; however, in principle, significantly fewer simultaneous frequencies should be required as long as appropriate damping and regularization are applied (e.g., using the method of efficient waveform inversion described by Sirgue and Pratt [2004]).

4.4 Results

[41] The initial model from FAST processing of the first-arrival data is shown in Figure 4a. This model image is darkened in the portions not sampled by the ray-tracing algorithm used in travel time inversion; this represents the null space of the travel time inversion. The velocity model resulting from 2D (l2 norm) waveform tomography processing by Smithyman and Clowes [2012] is reproduced in Figure 4b. In order to facilitate comparison under the same norm as the new 2.5D FWI result, a result from 2D full-waveform inversion of static-corrected waveform data by reprocessing of the results of Smithyman and Clowes [2012] using the logarithmic l2norm is presented in Figure 4c. The new result uses processing parameters equivalent to the 2.5D work flow and results in FWI model perturbations that are quite different from those discussed by Smithyman and Clowes [2012]. The earlier 2D waveform tomography study was based on similar processing methods to this study, but used a preconditioned conjugate gradient algorithm to minimize the l2 norm of the waveform data residuals. In both 2D results, large data errors existed due to unresolved geometry errors in the western part of the seismic line. In addition, noise was present in the survey data from nearby industry and environmental effects and was similarly strongest in the western part of the survey. The l2 norm is sensitive to data outliers, and these errors dominated the updates to the model in the previous result due to the l2norm used. The pattern of model updates from the previous study is visible in Figure 5a, and the areas with the strongest updates also correspond to the parts of the line with the largest residual geometry errors due to deviations from a straight line (cf. line geometry in Figure 2). Additional work using the logarithmic l2 norm has yielded FWI updates that are more stable in the new 2D result (Figures 4c and 5b), and this suggests that the FWI updates to the model presented by Smithyman and Clowes [2012] were not imaging geological structure. The data fit in the eastern part of the line in the previous study was comparatively good when compared to the western part. Hence, the assessment of the error under the l2 norm meant that the updates from traces with a nominally acceptable (but not excellent) data fit were not considered (or improved upon) in the presence of larger errors due to limitations in the geometry static correction. The sensitivity of the l2norm to data outliers (e.g., noise bursts or modeling errors) has been documented in research by several workers, who have argued for a variety of treatments to mitigate the problem. Some of these approaches make use of alternate norms, such as the l1 norm [Brossier and Operto, 2009], the logarithmic l2norm [Shin and Min, 2006], and others [Brossier et al., 2010; Aravkin et al., 2011] that reduce sensitivity to data outliers. Another approach involves processing of trace-normalized seismic data under the l2 norm and considers only the phase residuals [e.g., Operto et al., 2004; Bleibinhaus and Hilberg, 2012], which may yield significant benefits for stability because of reduced sensitivity to noise. The use of the logarithmic l2 norm [Shin and Min, 2006] reduces the effect of large data residuals on the gradient update and tends to increase the strength of the perturbations away from the source and receiver arrays (acting as a preconditioning operation on the gradient), which improved the stability of the 2D and 2.5D full-waveform inversions in the present study. However, the difficulties associated with modeling out-of-plane geometry remain for the 2D approach and preclude useful model updates at frequencies above about 13 Hz. Figure 4d shows an intermediate velocity model after 20 iterations of 2.5D FWI, for direct comparison with the result from 2D FWI using the logarithmic l2 norm (Figure 4c). The final models of velocity and attenuation resulting from the new 2.5D waveform tomography processing are presented in Figures 4e and 4f. This new result contains information from inversion of the full bandwidth of the early-arriving (forward-scattered) waveforms (i.e., 8–16 Hz) and does not suffer from the inaccuracies introduced by a static correction using refraction travel times in the previous 2D methodology. Note, however, that much of the structure produced in the velocity model from 2.5D waveform tomography is developed in the first 20 iterations (which are comparable to the 20 iterations carried out in the 2D logarithmic l2 norm processing method); compare Figures 4c and 4d. The contribution to the velocity model from the band of frequencies between 13 and 16 Hz is minor in the 2.5D result (see Figure 11 and surrounding discussion). The main features of the velocity model from 2.5D FWI are substantially established in the first 20 iterations (compare Figures 4d and 4e, which are visually indistinguishable). However, the final eight iterations of the 2.5D FWI process constrain the model of attenuation.

Figure 4.

(a) The starting velocity model from travel time inversion. (b) The velocity model from 2D waveform tomography using the l2 norm in Smithyman and Clowes [2012]. (c) The velocity model from 2D waveform tomography using the logarithmic l2 norm. (d) The velocity model from 2.5D waveform tomography using the logarithmic l2 norm after 20 iterations (for direct comparison with Figure 4c). The detail recovered is higher in the 2.5D result. (e) The velocity model from 2.5D waveform tomography after 28 iterations. Low-velocity features are identified more reliably than in Figure 4a, and depth of investigation is substantially better than with travel time inversion alone. (f) Attenuation model from 2.5D waveform tomography (accompanies Figure 4e). The model updates were zeroed outside of 7.5 km<x<37.5 km; background Q=200. All panels are muted to indicate the portion of the model interrogated by the raypaths in the initial model; recovery of features is most reliable in this region.

Figure 5.

(a) Velocity difference from 2D FWI in Smithyman and Clowes [2012] with l2 norm (i.e., Figure 4b minus 4a). (b) Velocity difference from new 2D FWI in this paper with logarithmic l2 norm (i.e., Figure 4c minus 4a). (c) Velocity difference from 2.5D FWI (i.e., Figure 4e minus 4a). (d) Checkerboard testing result for 1% perturbation on velocity in 2.5D FWI result. (e) Checkerboard testing result for 1% perturbation on Q in 2.5D FWI result.

[42] Figure 5a shows the difference between the final velocity model from 2D full-waveform inversion in the result of Smithyman and Clowes [2012] (reproduced in Figure 4b) and the starting velocity model from travel time inversion (Figure 4a). Equivalently, Figure 5b shows the updates from the re-processing of the 2D work flow using the logarithmic l2 norm (discussed above). The FWI model update from the 2.5D result is shown in Figure 5c, i.e., the difference between Figures 4e and 4a. The increased stability from fitting the 3D geometry and from the improved accuracy in amplitude modeling allows for improved confidence in the imaging throughout the depth range. However, many of the features in Figures 4a–4e are substantially similar between the 2D log l2 norm and 2.5D results (or, for that matter, between the two full-waveform inversion results and the starting model), since each stage of the inversion yields an incremental improvement in resolution and accuracy. The most significant changes in the character of the model updates (compared to the result of Smithyman and Clowes [2012]) come from the change in data norms. In several locations, the 2.5D full-waveform inversion code smoothed or restructured features that were present in the initial model and were left unmodified in the 2D model, e.g., a raypath artifact visible at 1–2 km depth and approximately 23–25 km lateral position (Figure 4). There are also some indications of ringing features in the 2D result (e.g., Figure 4c, 12 km lateral position) that are not present in the 2.5D result or the starting model, although the same is true to a lesser extent in the 2.5D result (e.g., Figure 4e, 9 km lateral position). In both cases, the effect is most noticeable on the flanks of the model where the angles of target illumination are incomplete (see, e.g., Figures 5b and 5c). The areas in which the updates from full-waveform inversion are most complex can be readily determined by examining Figures 5b and 5c. In the case of the 2.5D inversion, several locations seem to show recovery of complex structures with relatively high wave numbers (e.g., 12 km along-line, shallow; 20 km along-line, middle depths; 25 km along-line, shallow).

[43] To estimate the interpretable resolution of the model, we carried out checkerboard perturbation testing on both the velocity and attenuation for the 2.5D FWI result (shown in Figures 5d and 5e). From checkerboard testing results, it is possible to determine the sensitivity of the seismic experiment to perturbations on the final velocity and Q models. The velocity model and Q model were multiplicatively perturbed by a pattern of rectangular checkers, of amplitude 0.01 (i.e., 1%). Adjacent checkers possessed opposite polarity, and each checker was 1000 m in width and 500 m in height. Each parameter (velocity and Q) was perturbed independently, while holding the other parameter constant. New (noise-free) data waveforms from the perturbed model were synthesized using the same forward-modeling code as that used in the FWI. These synthetic waveforms were processed using the same parameters as the final iterations of the real-data FWI study; however, the use of noise-free viscoacoustic synthetic data means that the resolution estimate is optimistic, compared to the result that would be expected from noisy data recorded in the field. Other limitations include the lack of a free surface, potential out-of-plane scattering, and unmodeled scattering from elastic effects; all of these will also impact practical resolution and are not modeled in this resolution test. Checkerboard tests estimate the modeling resolution based on the recovered Earth model, but do not test the validity of that Earth model (and hence cannot be used as ground truth in a real-data case study such as this). Examination of Figure 5 indicates that the main perturbations in the seismic velocity and attenuation models occurred between lateral positions 10 and 35 km. Because of instabilities in the formation of the gradient at 16 Hz along the flanks of the study (due to incomplete illumination from the seismic array), only a subset of the model was updated in the last stages of the 2.5D real-data inversion. The ambiguities between recovery of velocity and attenuation models are also strongest in the flanks, where the angles of target illumination are limited. The checkerboard pattern is recovered in the region of good illumination, but the updates were muted outside this region for the same reasons as in the field-data inversion. Resolution is highest in the shallowest 1–1.5 km of the model (relative to the source/receiver array) and between about lateral positions 23 and 33 km. Outside this region, striped features indicate an imperfect recovery of small perturbations, with the effect of anisotropic smoothing of the images of the checkers. The checkerboard pattern is recovered best in regions of the model with a strong velocity gradient. This is most likely due to improved reflection amplitude and signal-to-noise ratio, combined with shorter propagation distances for the refracted modes that are important for determining the velocity and attenuation.

[44] In discussing resolution, it is also informative to consider the portions of the model to which each waveform component is most sensitive. Figure 6 presents numerical sensitivity kernels computed for single pairs of sources and receivers with an offset approximately half that of the maximum offset of the seismic survey. These kernels are presented for 8 and 16 Hz frequencies, which represent the lowest and highest usable frequencies present in the forward-scattered field data. In most cases, the anomalies resolved by full-waveform inversion will be on the order of the size of the sensitivity kernel, or some fraction thereof. However, the stacking of multiple kernels (for the set of all sources and receivers) can lead to high model resolution, especially with multiple iterations of nonlinear inversion. The lowest data frequencies (close to 8 Hz) provide a resolution similar to travel time tomography, but with improved physics that leads to large wavenumber model updates at the earlier inversion stages. The highest data frequencies interrogate mainly the near-surface region, due to the rapid decay of amplitude with regard to both offset and depth; however, the model updates resulting from the high-frequency portion of the gradient contain significantly greater information about fine structure than the low-frequency wavefields. Additionally, the use of the log l2 norm can be seen as a preconditioner on the gradient (and therefore on the sensitivity kernels), which increases the strength of updates in the deepest parts of the model at all inversion frequencies.

Figure 6.

Sensitivity kernels are displayed for pairs of sources and receivers 7.2 km apart (i.e., at 50% of maximum offset), calculated in the final model from 2.5D waveform tomography (Figures 4e and 4f). Kernels are shown for the (a and b) 8 Hz wavefield and the (c and d) 16 Hz wavefield corresponding to unit-valued point sources and receivers. Blue represents the positive portion of the kernel; the central blue band is analogous to the first Fresnel zone. Red represents the negative (anti-correlative) portion of the kernel. The wavefields displayed are tapered near the model edges using the same parameters as the gradient preconditioning during inversion; however, no additional preconditioning is applied. Hence, high wavenumber anomalies can be seen in these images that are typically removed by low-pass wavenumber filtering.

[45] In order to assess the quality of the inversion result, we compare the original (field) seismic data and synthetic data generated using the initial velocity (Figure 4a), final 2D logarithmic l2norm velocity (Figure 4c), and final 2.5D velocity and attenuation models (Figures 4e and 4f) for sources 200, 350, and 600 (Figures 7-9, respectively). In each figure, the field data and synthetic data from the 2.5D result are presented for each trace (Figures 7a, 7b, 8a, 8b, 9a, and 9b). Figures 7c–7e, 8c–8e, and 9c–9e show overlay (wiggle) plots for the three synthetic data sets compared with the field data; however, these plots display only a subset of traces to enable readability. These results are presented with trace-normalized amplitudes and in reduced time with a reduction velocity of 4500 m/s. Both real and synthetic data were windowed to 1000 ms records from 200 ms before the travel time pick to 800 ms after the pick. As a result of the vibroseis correlation process, the encoded source signature is acausal and part of the seismic record occurs before the nominal first-arrival travel time (see technical discussion in Smithyman and Clowes [2012]). Real data were low-pass filtered to remove information above 16 Hz prior to inversion (to attenuate noise and avoid aliasing; the first-arrival bandwidth was naturally limited to less than 16 Hz outside of very short offsets), and these band-limited data are shown in Figures 7-9 for comparison with synthetic data. The 2D synthetic data (in Figures 7d, 8d, and 9d) are presented with a reversal of the static correction, which was implemented on the field data for the FWI, to allow for direct comparison with the uncorrected data waveforms.

Figure 7.

A collection of source gathers from source 200 of the Nechako Basin seismic survey. Data are presented time reduced, with a reduction velocity of 4500 m/s. (a) Field data. Letters refer to features discussed in the text. (b) Synthetic data generated from the final result from 2.5D waveform tomography. (c–e) Synthetic data (green) overlaid on field data (black); field data are sub-sampled to every fifth channel and synthetic data are sub-sampled to every 10th channel. (c) Synthetic data overlay generated from the initial model, using 2.5D modeling. (d) Synthetic data overlay generated from model resulting from 2D logarithmic l2 norm waveform tomography of static-corrected data waveforms modeled in 2D, with appropriate reversal of static correction for comparison with unshifted field data. (e) Synthetic data overlay generated from the final result from 2.5D waveform tomography.

Figure 8.

A collection of source gathers from source 350 of the Nechako Basin seismic survey. Data are presented time reduced, with a reduction velocity of 4500 m/s. (a) Field data. Letters refer to features discussed in the text. (b) Synthetic data generated from the final result from 2.5D waveform tomography. (c–e) Synthetic data (green) overlaid on field data (black); field data are sub-sampled to every fifth channel and synthetic data are sub-sampled to every 10th channel. (c) Synthetic data overlay generated from the initial model, using 2.5D modeling. (d) Synthetic data overlay generated from model resulting from 2D logarithmic l2 norm waveform tomography of static-corrected data waveforms modeled in 2D, with appropriate reversal of static correction for comparison with unshifted field data. (e) Synthetic data overlay generated from the final result from 2.5D waveform tomography.

Figure 9.

A collection of source gathers from source 600 of the Nechako Basin seismic survey. Data are presented time reduced, with a reduction velocity of 4500 m/s. (a) Field data. Letters refer to features discussed in the text. (b) Synthetic data generated from the final result from 2.5D waveform tomography. (c–e) Synthetic data (green) overlaid on field data (black); field data are sub-sampled to every fifth channel and synthetic data are sub-sampled to every 10th channel. (c) Synthetic data overlay generated from the initial model, using 2.5D modeling. (d) Synthetic data overlay generated from model resulting from 2D logarithmic l2 norm waveform tomography of static-corrected data waveforms modeled in 2D, with appropriate reversal of static correction for comparison with unshifted field data. (e) Synthetic data overlay generated from the final result from 2.5D waveform tomography.

[46] Figure 7 shows a series of source gathers corresponding to source 200, the location of which is indicated in Figure 2. In general, the phase of the real-data arrivals is predicted more accurately by the result from 2.5D FWI (Figure 7e) than in either the starting model (Figure 7c) or the model from 2D logarithmic l2 norm FWI of static-corrected data (Figure 7d). In several locations (labeled A), there are features on the first-arriving waveforms that are not well replicated by the initial or 2D data but are closely recovered in the result from 2.5D processing. However, the result from 2D FWI also shows substantial improvement over the result from the initial model, and the fit is quite similar to that from 2.5D FWI at a number of points on the first arrival (e.g., labeled B). In some locations (labeled C), there are strong diffractions present in the field data that are partially recovered by 2.5D FWI but not in the initial model or result from 2D FWI. These features are probably recovered in part during the final eight iterations of 2.5D FWI, when the bandwidth of the inversion is maximal and attenuation is imaged.

[47] Figure 8 shows data corresponding to source 350 (see Figure 2 for location). The field data (Figure 8a) are quite complex in this region, as the westward traveling waves mainly interrogate the (interpreted) basin structures (viz., the synformal features from about 12 to 26 km lateral position). The data waveforms are not recovered accurately in either the initial model or in the result from 2D logarithmic l2 norm FWI (Figures 8c and 8d, respectively). However, the 2D FWI result improves on the fit from the starting model over most of the offset range toward the west, at least for waveforms closely following the first arrival (those labeled D). The result from 2.5D FWI does somewhat better (Figure 8e) and reproduces several secondary arrivals (E) as well as generally fitting the first arrival with improved accuracy. A complex feature on the first-arrival (F) is only fit in the 2.5D result. In all cases, some phase shift is present on the eastern end of the gather, although less than a cycle in the case of the 2.5D result, which fits the wave interference from multiple arrivals (G) to some extent.

[48] Figure 9 shows data corresponding to source 600 (see Figure 2 for location). Overall, the data fit is quite good for synthetic data generated in both the 2D (Figure 9d) and 2.5D (Figure 9e) models. The eastern traveling waves are predicted accurately from the initial model and are improved upon in both FWI results. For western traveling waves, a partial cycle-skip is visible at about 4.5 km offset (H). This indicates that a region of the model is recovered with a velocity that is slightly too high (corresponding to the deeper portions of the high-velocity region visible at 20 km lateral position in Figures 4a–4e, where some numerical artifacts are visible at about 2 km depth). The first arrival is modeled more accurately from 2.5D FWI than from the initial model or 2D FWI throughout the longer-offset portion of the gather (about 5.5 km onward beginning at I), which is representative of different propagation paths that do not interrogate the region affected by numerical artifacts.

[49] The data waveforms are predicted more accurately in part because of the incorporation of the full 3D geometry information by 2.5D waveform tomography; in several cases, there are noticeable perturbations in the shape of the travel time curve which are not accurately modeled with a 2D projected geometry (see Smithyman and Clowes [2012] for discussion of travel time data and example gathers without reversal of the static correction). Waveform tomography also benefits from fitting secondary arrivals that are not visible to travel time-inversion methods. Low-moveout emergent arrivals from deeper layers, which are modeled more accurately by the waveform tomography process, are observed at several points in Figures 7-9. The emergent waveforms are representative of wave paths that pass through comparatively fast earth materials, and control the first-arrival travel times at offsets greater than the crossover point. The expression of these waveforms at offsets shorter than the crossover point is exemplary of a phenomenon that is not considered by travel time inversion, but is fit by the full-waveform inversion process.

[50] Comparison of time-domain waveforms alone may not be sufficient to assess the success of the nonlinear waveform inversion step, as the synthetic data waveforms may appear visually quite similar to the real data waveforms while still being generated in a flawed model [e.g., Pratt, 2008]. Also, each shot gather shows only a small proportion of the information in the full data set. In order to develop a more complete view of the model quality, it is beneficial to consider both the frequency-domain data fit and the proportion of the model illuminated by each frequency group. Figure 10 shows the amplitude and phase at 12 Hz (i.e., middle bandwidth) for the preconditioned real data and the synthetic data generated in the final 2.5D FWI model (i.e., Figures 4e and 4f). The sensitivity can be inferred from Figure 6, which shows that the deepest parts of the model are not well constrained by the high-frequency parts of the data set, whereas the shallowest parts of the model are interrogated by waveforms with a broader bandwidth.

Figure 10.

Frequency-domain panels showing amplitude and phase for each source/receiver pair at 12 Hz. The amplitude is shown for each datum in the (a) AVO-corrected field data and (b) synthetic data from the final 2.5D result. Amplitudes are displayed on a logarithmic scale. The phase is shown in radians for each datum in the (c) field data and (d) synthetic data.

[51] To assess the relative efficacy of the 2D and 2.5D waveform tomography methods, the reduction in each respective objective function is presented in Table 1. The objective function in each case was based on the log l2norm and values are tabulated for several sets of iterations. Since the 2D method was only stable at lower frequencies, the last stage of inversion including bandwidth from 8 to 16 Hz and allowing updates to the model of attenuation was only carried out in the 2.5D waveform tomography work flow. In all cases, the percent reduction in the objective function was higher in the case of 2D full-waveform inversion processing of static-corrected field data. However, since the data were static corrected before 2D FWI, the absolute values of the data residuals are not comparable between the two methods. While a comparison between the absolute errors is not meaningful in this case (due to the differences in the field data exposed to each work flow), a numerical measure of the relative model quality is desirable. The velocity model result after 20 iterations was recorded for each approach (Figure 4c for 2D and Figure 4d for 2.5D). Synthetic data were calculated after forward modeling in 2.5D for each case (i.e., using identical numerical methods) and compared to the uncorrected field data. The model from 2D processing produced a data residual 12.1% higher than the model from 2.5D processing, under identical conditions. In the final eight iterations that were not included in the 2D work flow, the 2.5D waveform tomography work flow further reduced the misfit by 19.9%.

Table 1. Relative Reduction of Objective Function for the 2D and 2.5D Inversion Stagesa
 Iter 01–10Iter 11–15Iter 16–20Iter 21–28
  1. a

    As the error is measured under the log l2norm, the numerical size of the (dimensionless) misfit reduction is small when compared to errors calculated using the l2norm.

2D Result29.6%12.2%8.0% 
2.5D Result24.3%7.7%6.3%19.9%

[52] Figure 11 shows the data residual at various frequencies and iterations in more detail. Figure 11a summarizes the character of the reduction in data fit in each group of iterations. For the first 20 iterations of 2.5D FWI, only the phase residual was considered (and hence, the misfit in the amplitude portion of the logarithmic l2 norm is not significant in the model updates). In the final eight iterations, the norm is the linear combination of the (unweighted) phase and amplitude residuals. Figure 11a shows that while the phase residual increases modestly in the final eight iterations, the fit to the data amplitude is significantly improved and the corresponding amplitude and total residuals are reduced. This trade-off is not unexpected, and a comparison between Figures 4d and 4e shows that the changes to the velocity model are not significant. Note that the misfit reduction shown in Figure 11a is representative of the reduction within each band of frequencies, but that the first iteration in each band produced a large reduction that is not shown in the jump between bands (Table 1). Figure 11b shows the phase residual calculated at each frequency from 8 to 16 Hz in 0.25 Hz increments, for each iteration. However, only the regions labeled A–D are significant, and region E is presented only to provide further insight into the behavior of the wide-band misfit function. The misfit is mainly decreased in the band of frequencies considered at each iteration, although some improvement is seen within perhaps 1 Hz above the top inversion frequency (Figure 11b, regions A–C). The amplitude residual (Figure 11c) is only considered in the final eight iterations of inversion, and the decrease is significant within region D. The initial 20 iterations produce no significant update in the amplitude residual. The total residual (Figure 11d) is actively considered only in the final eight iterations of inversion (it comprises the contributions of both phase and amplitude, weighted equally). Hence, the behavior in region E is due only to the reduction of the phase residual in the first 20 iterations of inversion. The front edge of region D at iteration 28 in Figure 11d gives a rough guide to the data fit with respect to frequency in the final result. The contribution from the highest frequency group (13–16 Hz) is minor and almost nonexistent in the phase residual (Figure 11b) although the increased bandwidth helps in constraining the amplitude information needed in the reconstruction of the attenuation model. A slightly elevated residual in the lowest ∼1 Hz range (8–9 Hz) is expected and is consistent with the lower signal to noise ratio in that part of the data set.

Figure 11.

The data residual is shown for all iterations (0–28) and all frequencies (8–16 Hz). The (a) combined residual comprises the sum of all the residuals for the frequencies processed at a given iteration. Each individual band of iterations (in Figure 11a) comprises a different number of discrete frequencies (see regions in Figure 11b for a representation of the number of summed frequencies), and the iterations with wider bandwidth therefore minimize a larger total objective function. (b) Phase, (c) amplitude, and (d) total residuals are illustrated. Five separate regions are indicated, labeled A to E. Region A corresponds to iterations 1–10 and comprises five frequencies from 8 to 9 Hz. Region B corresponds to iterations 11–15 and comprises 13 frequencies from 8 to 11 Hz. Region C corresponds to iterations 16–20 and comprises 21 frequencies from 8 to 13 Hz. Regions A–C (iterations 1–20) consider only the phase residual. Region D corresponds to iterations 21–28 and comprises 33 frequencies from 8 to 16 Hz. Iterations 21–28 update the model based on the magnitude of the combined amplitude and phase residuals (i.e., the total residual). Region E is presented to illustrate the effects of low-frequency inversion stages on the data fit at higher frequencies and the effects of phase-only inversion on the amplitude fit; however, no information from Region E is considered by FWI.

4.5 Interpretation

[53] Figure 12 shows the velocity and attenuation models resulting from 2.5D waveform tomography (reproduced from Figures 4e and 4f with a 3× vertical exaggeration), with our interpretation overlaid. The labels indicate the basins and faulted blocks that may exist in the region as a result of Eocene extension and earlier compression. The attenuation model from 2.5D full-waveform inversion (Figure 12b) provides information that is not available from previous geophysical research in this study area. A number of other geophysical methods have been applied along the same profile as this work. Comparison with the pre-stack depth migration image of WesternGeco MDIC [2010] and the magnetotelluric inversion model of Spratt and Craven [2010] is informative [see also Smithyman and Clowes, 2012], but our interpretation is based predominantly on the geology and the models resulting from this study. Recent petrophysical analyses summarized by Kushnir and Andrews [2012] provide information about the expected physical properties of the rocks in the region, although we are cognizant that the in situ samples may vary widely from laboratory results.

Figure 12.

Velocity and attenuation results, reproduced from Figures 4e and 4f, here with 31 vertical exaggeration. Interpreted faults and unit contacts are indicated, along with suggested rock types. A pull-apart basin is interpreted, filled by approximately 1.5 km of Cretaceous sedimentary rocks, and bounded to the east and west by Hazelton group rocks of the Stikine Terrane. The shallowest 0.5–1.5 km appears to be dominated by Eocene volcanic and sedimentary rocks, which show low velocity and attenuation in comparison to the older units.

[54] The shallowest 1 km of the model shows seismic velocities ranging between 3000 and 4000 m/s and relatively low apparent attenuation (Figure 12). Based on these comparatively low seismic velocities, this region most likely represents a mixture of Eocene volcanics and probably some sedimentary rocks (e.g., the Ootsa Lake group), consistent with the results of Kushnir and Andrews [2012]. Although Chilcotin group basaltic rocks overlie much of the study area (Figure 2), other workers [Bordet and Hart, 2011] have indicated that these are comparatively thin, on the order of tens of meters. Accordingly, we consider it unlikely that the Chilcotin group rocks contribute measurably to the seismic response.

[55] A sharp contrast in seismic velocity and attenuation is observed between the Eocene volcanics and the underlying blocks that show rugged paleotopography (Figure 12). Below the volcanics, the seismic velocity increases to greater than 5000 m/s, which is diagnostic of either intact sedimentary rocks (e.g., the Cretaceous Skeena group or Sustut group rocks) or underlying basement rocks. The velocity gradient is particularly high from about 25 to 32 km along the line (Figure 12), an area which also corresponds closely with the presence of volcanic and volcaniclastic Hazelton group rocks of the Stikine Terrane that outcrop north of the seismic line (Figure 2). We infer that Hazelton group rocks most likely extend below the seismic line at a depth of 1 km; however, it is also possible that the seismic response is directly due to these rocks north of the line, since some model non-uniqueness exists in the off-line direction.

[56] Based on the velocity and attenuation models, several features can be interpreted as faults. For example, the model characteristics at 12 km are interpreted as an extensionally reactivated thrust fault (arrows at 12 km on Figure 12). In most cases, these features show up as low-attenuation anomalies (i.e., high Q), with very high wavenumber content; the features in the model of attenuation sometimes correspond with areas of low velocity, but the anomalies in the velocity model appear less distinct. The coincident change of these properties may indicate unmodeled variability in the model of rock density, which is determined from velocity using the heuristic Gardner's relation. From about 13 to 24 km along the line, the velocities at depths from 1 to 2.5 km are moderate (4800–5500 m/s), values which are likely diagnostic of intact basin sediments. Thus we interpret this area on the cross-section as a pull-apart sedimentary basin (Figure 12). It appears to be composed of several blocks separated by westward dipping faults and is substantially deeper in the western part of this region. To the east, the interpreted basin is bounded by the strong features interpreted to represent rocks of the Hazelton group, as noted previously. To the west, the basin is bounded by shallower high-velocity rocks for which the velocities are not as well constrained as those to the east (due to the limitations of the seismic array sensitivity in the data subset we used). However, examination of the surficial geology map (Figure 2) shows that outcrops of Stikinia are present northwest of the end of our line. Thus, we speculate that this high-velocity material may also represent Hazelton group rocks.

5 Discussion

[57] The use of a 2.5D assumption introduces some simplifications in comparison to 3D, which must be taken into account at the processing stage (and ideally at the survey-design stage). The y coordinate of geometry is handled by phase-shifting individual plane-wave components in the Fourier composition that generates the 3D wavefield from component 2D wavefields. However, the assumed support of each (sinusoidal) component is infinite in the y direction; i.e., the 3D point sources constructed at a given location are only valid in the case when the y coordinate is parallel to geological strike. Without this criteria being fulfilled, the plane-wave decomposition of the 3D wave equation is invalid and the waveform forward modeling and inversion will not be accurate. In practical terms, however, an active-source seismic survey only interrogates Earth structures at some maximum distance from the source and receiver. In the Nechako Basin case study, the maximum offset is 14.4 km, the moveout velocity for the main part of the offset range is 4500 m/s and the end mute for the data waveforms is 800 ms after the first arrival. Hence, the maximum path length for a scattered wave arrival to be included in the windowed data is on the order of 18 km, and the maximum off-line distance interrogated is perhaps 5.4 km at the midpoint. While this calculation simplifies the geometry and the physics considerably, it is a reasonable estimate for the order of magnitude of the effects involved. Hence, the requirement that the true Earth be homogeneous in the y direction can be relaxed to a requirement that the Earth be smoothly varying in that direction in the locality of the survey, such that the bulk properties are close to those in the 2D model and there are no reflections generated. It is perhaps worth noting that the same limitations apply in 2D modeling and inversion studies—which use only the zero-wavenumber plane-wave component—but are seldom mentioned. The most straightforward means of ensuring these conditions is to select survey configurations that cross geological strike wherever possible. The coordinate system of the 2.5D approach may also be rotated to accommodate strike at an oblique angle to the cross-line direction, with appropriate choice of numerical parameters to ensure that the synthetic wavefields are not aliased in the projected xz plane. This also illuminates a practical limit to the degree of line curvature that can be accommodated in 2.5D processing: switchbacks in the acquisition array may result in wave paths that are normal to the finite-difference grid and synthetic wavefields with wave numbers too high to model practically. However, elimination of some data may prove sufficient to mitigate these problems. In this study, the line is oriented perpendicular to some geological features (e.g., outcropping Hazelton group rocks), but does not necessarily cross geological strike at depth.

[58] Whereas 2.5D waveform tomography has been demonstrated to yield improvements over 2D processing, the increased computational cost may still prove prohibitive for cases when rapid turnaround of results is necessary. In offshore studies, when the seismic array may be deployed in a straight-line configuration, the array geometry can usually be parameterized in 2D without difficulties; other researchers have shown compelling successes in 2D FWI processing of data collected in these circumstances. A 2.5D approach also provides improvements in modeling accurately the amplitude and phase of a 3D wavefield by simulating point sources; this remains a reason to consider 2.5D processing as an improvement over the 2D equivalent. Corrections may be applied to partially accommodate the problems of 3D geometry and maintain the computational advantages of a 2D processing work flow, e.g., the static correction approach discussed previously, or a reprojection of sources and receivers on a per-trace basis to simulate 3D offset. In this case, proper treatment of any unresolved errors is of paramount importance and may necessitate processing with an alternative to the l2 norm that is typically used. Full 3D waveform tomography processing (with appropriate model constraints to accommodate the data null space) will almost certainly yield results with improved accuracy, especially when the array configuration allows for cross-line information to interrogate the 3D earth. However, as noted previously, the computational expense of 3D processing for a crooked-line 2D survey may not be justified.

6 Conclusions

[59] A method is developed for applying waveform tomography to early-arriving refraction waveforms from seismic reflection data sets that possess deviation from an ideal 2D survey configuration, e.g., a typical crustal seismic profile acquired along existing and crooked roads. Thus, this new method is particularly applicable to data sets from academia or small-scale industry surveys that do not make use of 3D seismic acquisition. The use of 2.5D full-waveform inversion, in combination with 3D travel time inversion, enables the method to account for 3D geometry in the source and receiver arrays. This is of greatest benefit when the seismic array configuration lends itself to a 2D interpretation stage, but the 3D geometry makes conventional 2D full-waveform inversion processing intractable. In addition, the 2.5D implementation of the full-waveform forward modeling and inversion steps results in improved fidelity in modeling the amplitude and phase effects of geometric spreading when compared to a 2D wave-equation method. The primary benefits are greatly improved velocity models relative to travel time inversion and generation of attenuation models within the region interrogated by the seismic waveforms. These improvements result in moderate computational overhead, which in practice is typically on the order of 40–50 times the cost of 2D inversion for each iteration but significantly less than application of a full 3D algorithm.

[60] A case study from a crustal reflection profile in the Nechako Basin in south-central British Columbia, Canada, exemplifies and validates the newly developed approach. Models of velocity and viscoacoustic attenuation provide valuable information that can be useful for interpretation, including identification of rock types and structural features. The high resolution exhibited by these models provides significant improvements with respect to fault delineation and definition of structural boundaries when compared to conventional tomography techniques. The use of low-frequency refracted waveforms limits the resolution when compared to an industry-style pre-stack depth migration work flow; however, the high-resolution model of seismic velocity makes the placement of features comparatively accurate. As well, the inversion of refraction waveforms makes it possible to produce a detailed image including near-surface structures that may be invisible to a conventional stacking or migration work flow.





acoustic velocity;




temporal frequency;


time-domain acoustic wavefield;


3D frequency-domain acoustic wavefield;

inline image

2D Fourier component;


time-domain source-time function;


frequency-domain source term;


Kronecker delta function;


number of components in y wavenumber summation;


original data;


AVO scaled data;


source-receiver offset;


log linear regression intercept;


log linear regression slope.


[61] Thanks to Gerhard Pratt, Rie Kamei, Drew Brenders, and Michael Afanasiev at the University of Western Ontario for endless discussions and support. We also thank Andy Calvert at Simon Fraser University and many others at conferences and meetings for some great ideas and feedback. Thank you also to the Associate Editor, reviewer Florian Bleibinhaus, and an anonymous reviewer at JGR Solid Earth; their detailed and constructive comments greatly improved the quality of this manuscript. Geoscience BC provided funding for much of the early work of this project. This work was also supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Canada Graduate Scholarship—Doctoral (to B. Smithyman) and Discovery grant (to R. Clowes). Computational facilities for this research were provided by WestGrid, part of Compute Canada; in particular, these resources were vital to fast turnaround times when running computations during the revision process..2