Resolution test

A number of resolution tests with various checkerboard sizes implanted in the model space are performed to examine the robustness and minimum resolvable wavelength of the model solution as well as to assess the differences of the model recovery ability between the kernel- and ray-based tomography. The implanted synthetic structures are the cube-like checkerboards assigned with the alternating anomaly pattern of constant positive and negative velocity perturbations. Each side of the checkerboards occupies eight to four grid units with the dimension varying from ~250 km to ~120 km. Based on the available path coverage from the observed travel-time data, the synthetic data and the data kernel for the inversion are computed using either finite-frequency theory or ray theory.

Figs. 4-3 to 4-8 display the recovered velocity images for three sets of the resolution tests with the checkerboard sizes of 248 km, 186 km and 124 km, respectively. The relative fast and slow velocity perturbations of these synthetic models are ±2%. The

recovered images are obtained with the same parameterization and regularization chosen for the optimal realistic P and S velocity structures. The 2-D cross section views of the recovered synthetic structures are displayed in the same manner as those for the velocity models shown in Figs. 4-1 and 4-2. The degree of the model recovery viewed on the vertical cross sections can be somewhat misleading because the slice directions are not necessarily perpendicular to the edges of the checkerboards.

As seen from the comparison of the recovered images using two different forward theories and the multi-scale parameterization, the ray-based model tends to produce elongated streaks along the directions with no or spares ray crossings. This is particularly evident in the BB’ cross section along the La Ristra array, where most of the incoming rays from the circum-Pacific subduction zones are parallel to the strike of the cross section and leave the uncrossed rays in the peripheral areas of the models at greater depths. The finite frequency model, on the contrary, yields better recovery of the locality and amplitude of implanted velocity anomalies in both the vertical and lateral directions (see the comparison of the recovered images of the S-wave models in Figs. 4-4 and 4-6).

For the case with the largest size of the checkerboards about 250 km wide in each side, the implanted P-wave velocity structures using both ray theory and finite-frequency theory to compute the synthetic data and data kernel are well recovered at least down to the depth of 500 km. The ray-based model yields the weaker amplitudes of recovered velocity anomalies and the imaged checkerboard patterns becomes blurred at the depth of 562 km. For the kernel-based S-wave model, the recovered checkerboard pattern remains resolute in well-defined rectangular shapes in the upper 500 km depth and becomes invisible at the depth greater than 600 km. In contrast, the ray-based S-wave model produces the vague checkerboard image with very weak recovered amplitudes.

The region with decent resolution is essentially located in the vicinity of the La Ristra

array along the BB’ cross section. For the case of intermediate-sized checkerboards having six grids, 186 km wide on each side, the resolution in the western half region for the kernel-based P-wave model remains fairly good in the upper 400 km, as a result of the dense station coverage from the USArray. The same checkerboard test fails for S-wave models where only small portions of the implanted structures are recovered. For the smallest, four grid sizes of the checkerboards, the resolution of both the P- and S-wave recovered structures is very poor at the depths greater than 100 km, regardless of the forward theories employed in the inversion.

Summarizing the results of the checkerboard resolution tests, it can be concluded that the finite-frequency theory provides the superior recovery ability of the implanted velocity structures than ray theory and yields the models with improved resolution to reconstruct the geometry and amplitude of 3-D velocity heterogeneity. In addition, the results of the synthetic tests suggest that the 3-D P-wave velocity structure under the study region is robustly-resolved in the upper mantle at least down to the depth pf ~600 km. The laterally, minimum resolvable scale length for the P model is on the order of 150 km. The maximum depth extent of the resolvable S-wave velocity model is similar to the P-wave model. Though, because of the naturally broader sensitivity for the S travel-time data, the laterally resolvable wavelength for the shear velocity variation is larger on the order of ~250 km.

(a) Ray theory

(b) Finite frequency theory

Fig. 4-1. Tomographic images of P-wave velocity perturbation for the (a) ray theoretical and (b) finite-frequency models. The 3-D velocity variations are viewed on three vertical cross sections (middle column) and six constant-depth maps (two right columns) from 94 km to 531 km depth. The areas with the relative sampling density lower than 1% of the maximum density are masked out. Different symbols represent the used stations from various networks. The Rio Grande Rift, the Colorado Plateau and late Cenozoic volcanic fields (red-colored shades) are outlined in the station map

(a) Ray theory

(b) Finite frequency theory

Fig. 4-2. Tomographic images of S-wave velocity perturbation for the (a) ray-theoretical and (b) finite frequency models. See Fig. 4-2 for the detailed description of the symbols and figure setting.

(a) Ray theory

(b) Finite-frequency theory

(c)

Fig. 4-3. Resolution tests for the P-wave models using available path coverage of the observed travel-time data. The input model shown in (c) has an alternating checkerboard pattern of positive and negative velocity perturbations, ±2% with eight grid sizes, ~248 km wide in each side. The recovered images using (a) ray theory and (b) finite frequency theory to compute the synthetic data and data kernel are plotted in the same way as those in Figs. 4-1 and 4-2.

(a) Ray theory

(b) Finite-frequency theory

(c)

Fig. 4-4. Resolution tests for the S-wave models using available path coverage of the observed travel-time data. The input checkerboard model is the same as in Fig. 4-3.

The recovered S-wave models are displayed in the same manner as shown in Fig. 4-3.

(a) Ray theory

(b) Finite frequency theory

(c)

Fig. 4-5. Resolution tests for the P-wave models using available path coverage of the observed travel-time data. Each checkerboard occupies six grid sizes, ~186 km wide on each side. See Fig. 4-3 for the detailed description of the figure display.

(a) Ray theory

(b) Finite frequency theory

(c)

Fig. 4-6. Resolution tests for the S-wave models using available path coverage of the observed travel-time data. Each checkerboard occupies six grid sizes, ~186 km wide on each side. See Fig. 4-4 for the detailed description of the figure display.

(a) Ray theory

(b) Finite-frequency theory

(c)

Fig. 4-7. Resolution tests for the P-wave models using available path coverage of the observed travel-time data. Each checkerboard occupies four grid sizes, ~124 km wide on each side. See Fig. 4-3 for the detailed description of the figure display.

(c) Ray theory

(d) Finite-frequency theory

(c)

Fig. 4-8. Resolution tests for the S-wave models using available path coverage of the observed travel-time data. Each

checkerboard occupies four grid sizes, ~124 km wide on each side. See Fig. 4-4 for the detailed description of the figure display.