Tomographic filtering of geodynamic models: Implications for model interpretation and large-scale mantle structure



[1] The resolution operator equation image is a critical accompaniment to tomographic models of the mantle. equation image facilitates the comparison between conceptual three-dimensional velocity models and tomographic models because it can filter these theoretical models to the spatial resolution of the tomographic model. We compute equation image for the tomographic model S20RTS (Ritsema et al., 1999, 2004) and two companion models that are based on the same data but derived with different norm damping values. The three models explain (within measurement uncertainty) S-SKS and S-SKKS travel times equally well. To demonstrate how artifacts distort tomographic images and complicate model interpretation, we apply equation image to (1) a thermochemical and (2) an isochemical model of convection in the mantle that feature different patterns of shear velocity heterogeneity in the deep mantle if we assume that shear velocity heterogeneity is caused by temperature variations only. equation image suppresses short-wavelength structures, removes strong velocity gradients, and introduces artificial stretching and tilting of velocity anomalies. Temperature anomalies in the thermochemical model resemble the spatial extent of low seismic velocity anomalies and the shear velocity spectrum in the D” region better than the isochemical model. However, the thermochemical model overpredicts the amplitude of shear velocity variation and places the African and Pacific anomalies imperfectly. We suspect that inaccurate velocity scaling laws and uncertain initial conditions control these mismatches. Extensive hypothesis testing is required to identify successful models.

1. Introduction

[2] Seismic tomography and mantle geodynamics are complementary research approaches. Since the pioneering work of Hager and O'Connell [1979, 1981], tomographic images have been key observational constraints to theoretical models of mantle convection. Geodynamical global flow models typically produce downwelling sheets of cold material which are associated with the positive wave speed anomalies seen in global tomographic models [e.g., Grand, 1994; Grand et al., 1997; Fukao et al., 2001]. Although the understanding of instantaneous, present-day global flow patterns relies on tomographic images to define a preexisting density structure [e.g., Hager et al., 1985; Hager and Richards, 1989; Becker and O'Connell, 2001], time-dependent geodynamical models have failed to self-consistently generate the broad (and presumable hot) regions in the lower mantle beneath Africa and Pacific comparable to those observed seismically [e.g., Dziewonski et al., 1977; Tanimoto, 1990; Su et al., 1994; Li and Romanowicz, 1996; Masters et al., 1996; Ritsema et al., 1999; Gu et al., 2001]. Whether these seismic structures are (poorly imaged) thermal upwellings in an effectively isochemical mantle [e.g., Bunge and Richards, 1996; Bunge et al., 1998; Lithgow-Bertelloni and Richards, 1998; Becker and Boschi, 2002; Schubert et al., 2004] or whether they are signatures of intrinsically dense piles or superplumes in the deep mantle [e.g., Thompson and Tackley, 1998; Davaille, 1999; Kellogg et al., 1999; Tackley, 2000; Forte and Mitrovica, 2001; Jellinek and Manga, 2002; Tackley, 2002; Davaille et al., 2003; Jellinek and Manga, 2004; Nakagawa and Tackley, 2004; Tan and Gurnis, 2005] remain outstanding questions.

[3] The significant differences in resolution complicate comparisons of geodynamical and seismic models. Geodynamical models are typically calculated on uniform meshes with a dense grid spacing on the order of 20 km. The parameterization of tomographic models is at least an order of magnitude larger in scale and the tomographic model resolution is inherently spatially heterogeneous due to the incomplete and uneven seismic sampling of the mantle (Figure 1). In fact, tomographic model resolution can be counterintuitive and it cannot be fully characterized with few, generic resolution (e.g., “checkerboard” or “spike”) tests [e.g., Lévêque et al., 1993].

Figure 1.

Estimates of vertical resolution of shear velocity heterogeneity in S20RTS at 150 km, 650 km, 1600 km, and 2500 km depth from the vertical extent of Backus-Gilbert resolution kernels K (r, r0). The resolution kernel is defined by m(r0) = ∫VdVK(r, r0)mt(r) and describes how the value obtained in the generalized model m(r0) at r0 is a spatial average over the true structure mt(r). Vertical resolution is defined by the area under the vertical cross section through K that occupies 50% of the resolution kernel [see Ritsema et al., 2004]. While lateral variations in resolution are due to the uneven distribution of seismic stations and earthquakes, the variation in depth can be attributed to the variable resolving power of the various data types.

[4] To ensure a meaningful interpretation, it is critical to properly filter the geodynamic model so that it features structural heterogeneity with the same spatial resolution. Mégnin et al. [1997] and Bunge and Davies [2001] mimic the effects of such a filter by inverting synthetic seismic observables computed for a geodynamic input model with the same procedures applied to real data. This approach is somewhat cumbersome because it requires that the synthetic data (e.g., waveforms and travel time picks), the forward modeling approach (e.g., waveform synthesis) and the inversion procedures (e.g., data weighing, earthquake relocation, parameterization, and regularization) are reproduced precisely. It would be valuable to nonexperts if tomographic models are accompanied by their resolution operator. This would enable them to compare any hypothetical model of mantle heterogeneity to seismic images without an understanding of the intricate details of seismic data selection and tomographic inversion procedures.

[5] Leaving a more complete comparison to a forthcoming publication, we illustrate here the effects of the tomographic resolution operator on thermochemical and isochemical end-member models of the deep mantle. These models have recently been discussed by McNamara and Zhong [2005]. While these authors suggest that direct comparisons of S20RTS [Ritsema et al., 1999, 2004] images to temperature fields indicate that the thermochemical mantle model yields the best match, we reevaluate these comparisons after the geodynamical models have been tomographically filtered. We begin with a summary of the determination of the seismic resolution operator following classic least-squares inversion methodology. Subsequently, we discuss the seismic resolution of velocity heterogeneity in the deep mantle and illustrate how the geodynamic models of the deep mantle are seen through tomographic eyes.

2. Seismic Resolution Operator

[6] In tomography, we typically relate seismic data d to a model m of wave speed variation in Earth in a linear fashion

equation image

and estimate m by damped least-squares inversion [e.g., Tarantola, 1987; Menke, 1989; Scales et al., 2001; Aster et al., 2005] to minimize

equation image

The solution to (1) is

equation image

where G is the generalized inverse of G. If UΛUT is the eigenvalue decomposition of GTG, we can define the generalized inverse by

equation image

where Λ−1 = (Λ + εI)−1. Combining (1) and (3) yields

equation image

where we have defined the resolution operator equation image = GG that specifies how the true Earth mt is mapped into the tomographic model m. The spatially heterogeneous resolution, an attribute of any tomographic model, is fully described by equation image.

[7] By design, damping (via ε) suppresses the magnitude of shear velocity variation in m at the expense of data fit. Choosing an appropriate value for ε requires knowledge of data errors. Instead of ε, we can express the amount of damping by N, the trace of the resolution operator equation image. For model S20RTS we chose ε = 0.035 on the basis of the combined travel time and phase velocity fit. This damping is equivalent to choosing N = 2932, meaning that 2932 effective unknowns have been resolved. It is straightforward to derive a tomographic model and its resolution equation image for any ε. Ideally, the inspection of tomographic models and the application of the tomographic filter to geodynamic models is carried out for the full range of values for ε that yield acceptable data fits. Using (4), this is a relatively straightforward task.

[8] Table 1 summarized the results for three models derived using damping values ε = 0.075 (Model 1), ε = 0.035 (Model 2), and ε = 0.015 (Model 3). Model 2 is identical to S20RTS. The number of effective unknowns (N) increases from 2083 to 4000 as ε decreases. The extreme damping factors are chosen based on the fit to 8700 S-SKS and 1900 S-SKKS differential travel times. These travel times are primarily affected by shear velocity heterogeneity in D” [e.g., Kuo et al. 2000]. All models explain the travel time data equally well. The average S-SKS delay of 3.64 s and S-SKKS delay of 5.49 s are reduced after inversion to values that range from 2.65 to 2.42 s and 3.46 to 2.84 s, respectively. These ranges are small compared to the estimated 0.5 s measurement errors.

Table 1. S-SKS and S-SKKS Travel Time Residuals and Model Fits
ModelεNδVS in D”, %Average Residual Delay, sMisFit Red., %
S-SKS Avg. Delay = 3.64 sS-SKKS Avg. Delay = 5.49 s
10.0752083−1.2 to +1.22.653.4649
2 (s20rts)0.0352932−1.6 to +1.52.523.1055
30.0154000−2.1 to +1.92.422.8459

[9] Figure 2 shows the shear velocity heterogeneity in D” for Model 1, Model 2, and Model 3. Since Model 1 is damped the most and Model 3 is damped the least, the shear velocity variations in D” are smallest (−1.2 to +1.2%) in Model 1 and highest (−2.1 to +1.9%) in Model 3. In addition, shear velocity variations are smoothest in Model 1 due to relatively strong damping. The uncertain amplitude of velocity heterogeneity is a consequence of measurement uncertainty and scatter. It is inherent to any tomographic models given the integral constraints by travel times [see also Montelli et al., 2004].

Figure 2.

(a) Surface projected segments of S and diffracted S wave ray paths through the bottom 200 km of the mantle. The segments are colored red (blue) if the corresponding S-SKS or S-SKKS differential travel time is higher (lower) than the PREM value. The coherent geographic distribution of the travel time anomalies is well reflected by the shear velocity variation in D”, shown in Figure 2b. (b) Shear velocity variations at 2750 km depth in Model 1, Model 2, and Model 3 derived using different norm damping parameters ε (see Table 1).

3. Application to End-Member Convection Models

[10] We illustrate the effects of the tomographic filter for a thermochemical, TCIN, and an isochemical, ICIN, model of the deep mantle [McNamara and Zhong, 2005]. The filtered versions are TCOUT and ICOUT, respectively. The convection models are developed using the thermochemical extension [McNamara and Zhong, 2004] of CitcomS [Zhong et al., 2000] and employ Earth-like convection vigor, temperature, and depth-dependent rheology and surface plate motions for the past 120 million years [Lithgow-Bertelloni and Richards, 1998]. Model TCIN includes a relatively dense layer in the lowest 225 km of the mantle that has been perturbed by convective motions. Its intrinsic density contrast is denoted by the buoyancy ratio [McNamara and Zhong, 2004]. Apart from the addition of this basal layer TCIN employs model parameters identical to ICIN.

[11] We assume that shear velocity variations are only caused by temperature variations in the mantle. We scale the temperature from the geodynamical models to wave speed by first determining the average temperature at each depth. We define the wave speed anomaly to be 0 at that temperature. Lateral shear velocity variations are determined by scaling departures from the average temperature to shear velocity perturbations using dVS/dT = −7.0 × 10−5 km s−1 K−1 following Forte and Mitrovica [2001].

[12] The geodynamical models predict fundamentally different shear velocity patterns. The intrinsically dense basal layer in the thermochemical model is focused into a large ridge-like structure beneath Africa and a superposition of ridges beneath the Pacific [McNamara and Zhong, 2005]. These dense piles are significantly hotter than the ambient mantle and produce broad thermal anomalies with a similar shape and location as the low shear velocity anomalies seen tomographically. The isochemical model comprises a network of thin, linear upwellings that are clustered beneath Africa and the Pacific.

3.1. D” Heterogeneity Maps

[13] D” maps (Figure 3) illustrate how equation image filters the geodynamic input models (TCIN and ICIN) into models (TCOUT and ICOUT) with the resolution of S20RTS. Low-pass filtering is a first-order effect of equation image. The strong variations (20%) of shear velocity in TCIN, strong gradients along the edges low velocity anomalies, and the narrow, linear anomalies of ICIN in D” are smoothed and suppressed especially in regions (e.g., Africa) with sparse data coverage. However, the models retain their contrasts after tomographic filtering. For example, the linear low velocity anomalies in ICIN are smoother in ICOUT but they are not projected into a single low velocity anomaly.

Figure 3.

Shear velocity variations at 2750 km depth in (upper left) TCIN and (lower left) ICIN and, to their right, the tomographically filtered models TCOUT and ICOUT determined for equation image that correspond to values for the damping parameter ε of 0.015, 0.035, and 0.075. The color scale is the same as in Figure 2. The shear velocity varies from −X% to +X% in order to display the shear velocity variations across the full color spectrum. X = 8% for TCIN and ICIN, X = 6% for TCOUT, and X = 2% for ICOUT. Triangles are hotspots from the compilation of Ritsema and Allen [2003].

[14] equation image accentuates the degree-2 D” heterogeneity in TCOUT and ICOUT that characterizes the seismic models. Figure 4 shows the spectrum of shear velocity in the D” region as a function of model damping. Model TCOUT matches the strong reduction of spectral amplitude seen in the spectrum of Model 2 better than ICOUT. However, the shear speed variations in TCOUT are at least a factor of two larger than in Model 2 and the ring structure of the “African anomaly” in TCOUT is not seen in Model 2.

Figure 4.

Spectral amplitude as a function of spherical harmonic degree of shear velocity heterogeneity in D” for Model 2 (S20RTS), (a) TCIN and TCOUT, and (b) ICIN and ICOUT. TCOUT and ICOUT are determined for ε = 0.035.

3.2. Whole-Mantle Cross Sections

[15] Vertical cross sections through the mantle (Figure 5) demonstrate several artifacts that equation image introduces in the images. For example, the central Pacific low-velocity anomalies in ICOUT and TCOUT extend higher than in ICIN and TCIN, respectively, and they are artificially tilted toward the northeast due to the predominant northeasterly ray coverage in the Pacific mantle. However, the easterly tilt of the African anomaly in S20RTS [Ritsema et al., 1999] cannot be dismissed as an artifact since tilting is not seen in the images of TCOUT. Smaller-scale instabilities that erupt from the edges of the thermochemical piles [e.g., Davaille, 1999; Jellinek and Manga, 2002; Davaille et al., 2005] in TCIN cannot be resolved tomographically (i.e., TCOUT) at the resolution of S20RTS. High-velocity anomalies in D” that feature prominently in TCIN to the east and west of the African anomaly are missing or have significantly reduced amplitude in TCOUT due to the lack of seismic sampling in this region [e.g., Wysession, 1996].

Figure 5.

Shear velocity heterogeneity in whole-mantle cross-sections through (a) the Pacific and North America, and (b) the Atlantic and southern Africa according to (left) Model 2 (S20RTS), (middle) TCIN and TCOUT, and (right) ICIN and ICOUT (filtered using ε = 0.035). The color scale is the same as in Figure 2. The shear velocity varies from −X% to +X% in order to display the shear velocity variations across the full color spectrum. X is given below each cross section.

[16] The model comparison indicates several characteristics of TCIN and ICIN that are inconsistent with Model 2 even after equation image has been applied. The convection models predicts a relatively strong high-velocity anomaly in the lower mantle beneath North America (the signature of Farallon subduction [e.g., Grand, 1994; Tan et al., 2002]), that dips in a direction opposite from what is observed tomographically.

4. Discussion and Conclusions

[17] It is critical that the interpretation of tomographic models incorporates comprehensive studies of model resolution. The resolution of tomographic models is spatially heterogeneous. The resolution operator equation image can be computed in a straightforward manner when GTG is determined by, for example, singular value decomposition (equation (4)). Application of equation image to conceptual models of shear velocity in the mantle allows for a more meaningful comparison to tomographic models because it filters them to the same resolution and introduces modeling artifacts that distort the tomographic images.

[18] Applications to a thermochemical (TCIN) and isochemical (ICIN) model of convection illustrates that low-pass filtering is a first-order effect of equation image. Amplitudes of especially short-wavelength shear velocity variations are reduced and sharp velocity gradients are suppressed. However, the key differences between TCIN and ICIN are retained in TCOUT and ICOUT. The D” spectrum of TCOUT is dominated by the lower harmonic degrees while ICOUT features elongated (albeit smoother) linear low-velocity structure in D” that reflect narrow upwellings. These characteristics are robust over a range of damping values for which comparable misfit reductions of S-SKS and S-SKKS travel times are obtained. Tomographic model resolution is therefore sufficiently high to assess the applicability of models TCIN and ICIN.

[19] TCOUT is the preferred model as it provides a better match to the shape of the seismic spectrum of D” heterogeneity than ICOUT. In addition, the shear velocity structure in Model 2 correlates best with the structure in TCOUT throughout the lower mantle (Figure 6). These results corroborate other seismological observations that point to thermochemical heterogeneity in the deep mantle [e.g., Su and Dziewonski, 1997; Kennett et al., 1998; Ritsema et al., 1998; Ishii and Tromp, 1999, 2004; Masters et al., 2000; Forte and Mitrovica, 2001; Wen et al., 2001; Ni et al., 2002; Deschamps and Trampert, 2003; Trampert et al., 2004; Wang and Wen, 2004].

Figure 6.

Cross-correlation coefficient between Model 2 (S20RTS), (sold line) TCOUT, and (dashed line) ICOUT for a damping value ε = 0.035.

[20] However, we readily acknowledge that there is considerable mismatch between the tomographic models and TCOUT. This is obvious in the shear velocity images of Figures 2 and 5 and it is reflected by low correlation values for spherical harmonic degrees larger than 6. This indicates either tomographic model artifacts that are not represented by equation image (i.e., errors in the forward problem) or inaccuracies in the geodynamic models. For example, the purely temperature dependent wave speed conversion may not be justified and may underly the large mismatch in the amplitude of shear velocity variations in TCOUT and the tomographic models. Moreover, prescribed initial conditions, especially uncertainties in the plate velocities in the geologic past, may have a major effects on the location of anomalies in the mantle. In ongoing research, we are investigating the role of composition and depth-dependent temperature to wave speed conversions [e.g., Stixrude and Lithgow-Bertelloni, 2005] and we examining the effects of uncertain initial conditions.

[21] It is key not to draw general conclusions concerning the applicability of thermochemical and isochemical model aspects on the basis of two resolution tests. We believe that we have demonstrated that seismic tomography can be used to discriminate between isochemical and thermochemical structures but the generality has to be explored with a large number of cases. To this end, we make equation image available to interested research groups for further quantitative comparison of dynamic models and S20RTS.


[22] All figures were generated with the GMT software of Wessel and Smith [1995]. The IRIS and GEOSCOPE Data Centers provided seismological data. We thank Matt Fouch for valuable discussions and Jeannot Trampert, Jun Korenaga, and Michael Ritzwoller for constructive reviews. This research has been funded by NSF grants EAR-0609763 (JR) and EAR-0510383 (AKM).