3.1. Phase Velocity Measurement With Two Plane Wave Approximation
[6] To estimate phase velocity, we used a technique developed by Forsyth and Li [2005] which accounts for simple multipathing effects, allowing shorter period Rayleigh waves to be reliably measured. This improves resolution, and has been used successfully in many studies [Forsyth et al., 1998b; Li et al., 2002, 2003; Weeraratne et al., 2003; Fischer et al., 2005]. For each bandpass period, the real and imaginary Fourier components were inverted for phase velocity, using the following steps:
[8] 2. Using the plane wave parameters derived in step 1 and a starting phase velocity model (discussed below), the data are iteratively inverted for updates to the starting model using
where G is the matrix of partial derivatives, C_{mm} and C_{dd} are the a priori model and data covariance matrices, Δd is the difference between the observed and predicted Fourier coefficients, m_{o} is the original starting model, and m is the current model [Tarantola and Valette, 1982]. With each iteration, high misfit events are downweighted to minimize the effects of Rayleigh waves that are too multipathed to be fit by the twoplane wave approximation [Forsyth and Li, 2005].
[9] Ideally, the data would be sufficient to produce a fully resolved twodimensional model of phase velocity variations for each wave period. However, similar to most teleseismic tomography, there were insufficient data to produce a unique two dimensional phase velocity map [Menke, 1989]. Similar to previous studies using the twoplane wave technique, we take a conservative approach to this problem and seek to find the simplest model parameterization that fits the data [Forsyth et al., 1998a; Li et al., 2002; Li and Detrick, 2003; Li et al., 2003; Weereratne et al., 2003; Fischer et al., 2005; Forsyth and Li, 2005; Li and Detrick, 2006]. To accomplish this, three sets of phase velocity inversions were conducted, starting with a uniform phase velocity inversion, proceeding to a regionalized phase velocity, and ending with a twodimensional phase velocity inversion. The regionalization domains are defined based on teleseismic P and S body wave tomograms [Schutt and Humphreys, 2004; Yuan and Dueker, 2005; Waite et al., 2006]. Our most important choice is the width of the hot spot track (Figure 1). The bodywave tomograms constrain the low velocity body beneath the hot spot track above 100 km depth to be 80–120 km wide. But, in the 100–200 km depth range the width of the low velocity anomaly varies between 150 and 200 km. Given that our resolution kernels have little sensitivity below 120 km depth, we have chosen to fix the width of the regionalized hot spot track width to 110 km.
[10] The uniform model is used as initial starting model (m in equation (1), above) for the next more complicated model. Specifically, the following three inversions are performed.
[11] 1. Invert Rayleigh wave observations for mean phase velocity at each wave period.
[12] 2. Use (1) as a starting model, and invert observations for mean phase velocity in each of the three tectonic regions: Basin and Range (BR), Wyoming Craton (WY), and Yellowstone hot spot track (YHT) (Figure 1).
[13] 3. Use (2) as a starting model and invert data for twodimensional phase velocity maps. In these inversions, we use “fat ray” Gaussian approximations to the sensitivity kernels. Ray width is chosen such that 95% of the crosssectional area falls within the first Fresnel zone [Weeraratne et al., 2003; Gudmundsson, 1996].
[14] Thus, at each of the 17 data periods, three sets of inversions are performed with each inversion requiring two steps. Statistics for the inversion of 30 s waves are listed in Table 2. Note that because poorly fit events are downweighted, the variance reduction between models cannot be strictly compared [Forsyth and Li, 2005].
Table 2. Phase Velocity Results at 30 sModel  Velocity Variations  Starting Model  Nobs  RMS Phase Misfit (s)  Total Variance Reduction (%)  Phase Variance Reduction (%) 

1  constant  none  3412  2.14  86.75  99.18% 
2  regionalized  Model 1  3412  2.07  94.16  99.81% 
3  2D  Model 1  3412  1.93  93.56  99.84% 
4  2D  Model 2  3412  2.05  93.72  99.82% 
[15] Figure 3 shows the results of the regionalized phase velocity inversions. The data fit is excellent with 85–95% of the data variance being explained by the three region model (after poorly fit events are downweighted). This suggests that the regionalized velocity structure is a good approximation to the actual structure. To assess whether the mean phase velocity of the tectonic regions is uniquely resolved, the rank of the resolution matrix
is calculated (Figure 3c; variables are the same as in equation (1)). This value diminishes from a value of three for the shortest periods, implying that the mean phase velocity within each region is uniquely resolved, to less than one for the 150 s period. A unique regionalized phase velocity for the longest period waves cannot be resolved because of the relatively narrow width of the YHT and BR regions.
[16] Another way to assess the regionalized inversions robustness is to compare the results of the regionalized inversion with the two dimensional inversions (Figure 4). For wave periods >50 s, no significant perturbations to the regionalized phase velocities are required to fit the data. This observation implies that it is sufficient to characterize the velocity structure in terms of our three tectonic regions. One can also see the effect of the regionalized starting model on the 2D inversion results by comparing inversions performed using the regionalized starting model (model 4 in Table 2) to those that did not use the regionalized starting model (model 3) (Figure 5). The effect of the regionalized starting model is evidenced by the continuity of the low velocities along the hot spot track. Correspondingly, the nonregionalized model does not show this hot spot track continuity. Our preference for the regionalized model is based upon two reasons. First, the body wave tomograms show that this is an accurate regionalization: in particular the 110 km width of the hot spot track domain is well constrained by the body wave tomograms. Second, comparisons of the variance reductions between the three differently parameterized models (three velocity profile fit and the 2D nonregionalized and regionalized starting model images) shows that all of them provide comparable fits to the data (Table 2).
[17] It might be expected that the strong velocity variations beneath our study area would cause some events to be poorly modeled by the two plane wave approximation to the incoming wavefield. As previously stated, strongly multipathed events are downweighted during the inversion. We note that the two plane wave technique has been used successfully at locations with strong velocity variations, such as Tanzania and the East Pacific Rise [Forsyth et al., 1998b; Weeraratne et al., 2003]. Examination of the wellfit events with a phase RMS misfit <3 s shows that no systematic misfit exists (Figure 6; Table 3). As expected, the wavefield parameters show increased scattering at shorter periods. For instance, the mean azimuthal anomaly (with respect to the great circle path) of the largest of the two plane waves decreases with increasing period. In addition, the amplitude ratio of the smaller plane wave to the larger plane wave also decreases with increasing period. Furthermore, the scatter of the primary wave azimuthal anomalies also decreases with increasing period. If the large mantle velocity anomaly is causing unusual behavior in the two plane wave fits, this would be expected to occur at the periods most sensitive to the depth of the anomaly, namely the 30–66 s period waves. Given that no distinctive increase in misfits occurs for this period range, we expect the main conclusions of the paper are not sensitive to strongly multipathed waves.
Table 3. Statistics From Two Plane Wave Fits^{a}Wave Period (s)  N  Median Amplitude Ratio  Median Azimuthal Anomaly (deg)  Azimuthal Spread (deg) 


150  5  0.096  5.617  98.774 
125  9  0.103  5.874  24.56 
110  15  0.143  3.282  11.468 
90  20  0.222  6.113  12.311 
75  30  0.192  5.026  19.222 
66  33  0.299  3.84  9.316 
50  39  0.296  4.17  10.917 
45  44  0.279  5.563  12.052 
40  48  0.28  4.839  10.786 
35  56  0.36  6.915  8.74 
30  59  0.382  3.678  9.725 
27.5  60  0.362  4.146  8.128 
25  59  0.408  7.398  16.052 
22.5  55  0.454  6.434  11.164 
20  42  0.518  7.332  12.925 
18  42  0.501  7.596  17.109 
15  22  0.638  14.544  22.402 
3.3. Inversions of Phase Velocity for Regionalized Shear Wave Velocity Structure
[19] The regionalized phase velocity curves were used as data to invert for regionalized shear wave velocity profiles. This inversion was done iteratively by solving for perturbations to an initially assumed V_{S} model (equation (1)). Several starting velocity models based on previous observations were tested [Greensfelder and Kovach, 1982; Priestley and Orcutt, 1982; Gorman et al., 2002]. In the end, a simple starting model was chosen that was the mean of the previous observations from the different regions. This starting model has a 42kmthick threelayer crust, and a subMoho V_{S} of 4.16 km/s (equivalent to a V_{P} of 7.5 km/s for a V_{P}/V_{S} value of 1.8) based on receiver function and previous refraction constraints for the YHT [Sparlin et al., 1982; Peng and Humphreys, 1998, Yuan et al., 2006, Stachnik and Dueker, 2006]. Below the Moho, V_{S} was linearly increased from 4.16 km/s to match the V_{SV} of PREM at 200 km depth [Dziewonski and Anderson, 1981]. The V_{P} profile was generated using PREM V_{P}/V_{SV} values [Dziewonski and Anderson, 1981].
[20] On the basis of the initial V_{P} and V_{S} structure, data kernels ∂c/∂V_{S} (T, z) and ∂c/∂V_{P} (T, z) were calculated where c is phase velocity, T is wave period, and z is depth (Figure 8) [Weeraratne et al., 2003]. Density effects on phase velocity have an insignificant effect and were fixed to nominal values [Weeraratne et al., 2003]. The SP wave velocity perturbation scaling (dV_{S/}dV_{P}) was assumed to be 1.2, appropriate for the near solidus conditions of the YHT [Cammarano et al., 2003; Schutt and Lesher, 2006].
[21] The observed phase velocity maps were inverted iteratively for V_{S} structure with the recalculation of the data kernels between iterations. The iteration was repeated until velocity updates changed <0.001 km/s. The least squares inversion was regularized using diagonal damping. The approximate “elbow” value of the damping parameter was 25 s^{2}, and the best tradeoff between resolution and variance (Figure 9). For consistency, the same damping value was used for the regionalized BR and WY inversions, although this makes these inversions slightly underdamped (Figure 9). The rank of the resolution matrix for the regionalized V_{S} inversions varies from 3.7 for the Wyoming craton region to 2.1 for the Basin and Range region. As expected, the resolution diminishes with depth (Figure 10). Noteworthy is that our Rayleigh wave data set cannot resolve the midcrustal basalt sill beneath the eastern Snake River Plain [Peng and Humphreys, 1998], the bottom of the low velocity plume layer, or the plume conduit.
3.4. Inversions of Phase Velocity for 3D Shear Wave Velocity Structure
[22] Because inversions of phase velocity maps for V_{S} are quite sensitive to crustal velocity and thickness variations (Figure 8), it is important to incorporate a priori information on crustal thickness. We have used measurements of the time difference between the direct P arrival and the Moho converted P_{m}s arrival (hereafter called the P_{m}s time) from receiver functions as crustal thickness constraints [Yuan et al., 2006]. These P_{m}s times were spatially averaged using a Gaussian halfwidth of 40 km to create a smoothed 2D map (Figure 11). The coherent pattern of P_{m}s time variations varies by 2.4 s due to crustal thickness variations between 38 and 54 km [Yuan et al., 2006]. In addition, receiver function reverberation analysis finds a mean crustal V_{P}/V_{S} value of 1.78 [Yuan et al., 2006].
[23] The P_{m}s times are incorporated into our velocity inversions as constraints. For each surface grid point, the V_{S} and V_{P} starting profiles (V_{S} and V_{P} as a function of depth for that grid point) used in the regionalized V_{S} inversion were modified by adjusting the thickness of the lower crustal layer to match the observed P_{m}s times. Using these modified V_{S} and V_{P} profiles, data kernels were generated to invert the observed 2D phase velocity maps (Figure 4) for an updated V_{S} structure. The updated V_{S} structure is then used with the P_{m}s time constraints to estimate a new crustal thickness. This process was iterated three times until the V_{S} and V_{P} profiles matched the observed P_{mS} times within their error bars. Synthetic tests show that the crustal thickness and mean crustal velocity can be resolved within a few percent of their true values. The bootstrap estimated P_{m}s time errors are ±0.2 s and the V_{P}/V_{S} errors are ±0.04. Given these uncertainties, crustal thicknesses are accurate to ±3 km, and the mean crustal V_{S} is accurate to ±0.1 km/s.
[24] Vertical resolution of the velocity inversions can be estimated from the rank of the resolution matrices (Figure 12). Resolution at the depth of the low velocity channel (∼70 km) in the mantle is ±20–30 km and diminishes with depth [Weereratne et al., 2003]. Horizontal resolution varies with depth: within the crust horizontal resolution is ∼40 km and at 150 km depth is 150 km. The 2D phase velocity models produced with regionalized and nonregionalized starting model show how the regionalization affects the lateral resolution of the surface waves (Figure 5). In general, the horizontal and vertical 2D resolution is sufficient to determine crustal velocity variations and the horizontal extent and velocity of the plume layer.