The resolution operator is a critical accompaniment to tomographic models of the mantle. 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 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 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. 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.
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 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].
 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.  and Bunge and Davies  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.
 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 . 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
 In tomography, we typically relate seismic data d to a model m of wave speed variation in Earth in a linear fashion
where G† is the generalized inverse of G. If UΛUT is the eigenvalue decomposition of GTG, we can define the generalized inverse by
where Λ−1 = (Λ + εI)−1. Combining (1) and (3) yields
where we have defined the resolution operator = G†G 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 .
 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 . 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 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.
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
δVS in D”, %
Average Residual Delay, s
MisFit Red., %
S-SKS Avg. Delay = 3.64 s
S-SKKS Avg. Delay = 5.49 s
−1.2 to +1.2
−1.6 to +1.5
−2.1 to +1.9
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].
3. Application to End-Member Convection Models
 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.
 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 .
 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
 D” maps (Figure 3) illustrate how 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 . 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.
 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.
3.2. Whole-Mantle Cross Sections
 Vertical cross sections through the mantle (Figure 5) demonstrate several artifacts that 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].
 The model comparison indicates several characteristics of TCIN and ICIN that are inconsistent with Model 2 even after 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
 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 can be computed in a straightforward manner when GTG is determined by, for example, singular value decomposition (equation (4)). Application of 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.
 Applications to a thermochemical (TCIN) and isochemical (ICIN) model of convection illustrates that low-pass filtering is a first-order effect of . 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.
 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 (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.
 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 available to interested research groups for further quantitative comparison of dynamic models and S20RTS.
 All figures were generated with the GMT software of Wessel and Smith . 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).