The main publication devises a simple and intuitive modification of the regular cell tomographic model, so that ray coverage is improved. Structural synthetic tests demonstrate that more reliable and correct tomograms result.
This electronic supplement focuses on the resolution matrix for the alternative inverse problems. It is well-known that ray density or hit-count is not identical to the degree of tomographic resolution. However, this analysis shows that improvement of the uniformity in ray density in sections is a good guideline towards higher resolution in these sections except for obvious exceptions like the edge of a ray fan where rays are almost parallel.
1- The resolution matrix
2- Synthetic Model 1: stations on a straight line
2.1 Ray adapted modeling
2.2 Regular cells modeling
3- Synthetic Model 2: 2D scattered array
The resolution matrix is defined as
R = (GT *G + BTB)−1) * (GT * G)
Where BTB delivers regularization and G is:
d = G * m
Where m is the model vector (slowness at each cell) and d is the data vector (arrival time residual for each event-stations pair). R is an M by M matrix (M is the length of the model vector). Each row of R shows the dependency of each model parameter on all other model parameters. Diagonal element of R ranges from close to 0 (no resolution) to close to 1 (perfect resolution, well resolved).
Therefore, the resolution matrix is commonly applied as a measure of how well our estimated model reflects the true model. In particular, the diagonal element is commonly regarded as a relative proxy of the distribution of resolution in the model estimate. This is exemplified in Fig S1a.
Figure S1a shows two rows of the resolution matrix in the ray-adapted model. Each row represents a so-called resolution kernel. For the cell in the middle we see moderate vertical smearing, and the kernel value is relatively high at the cell to be resolved, i.e. the diagonal element relating to this row in the resolution matrix. Conversely we see expectable strong smearing for a cell at the edge of the ray coverage, which nicely is reflected also in a low diagonal element for the row associated with this cell.
Figure S1b shows the diagonal elements of the resolution matrix for the ray-adapted model. The maximum diagonal element of R is 0.27. This value decreases with depth and some stratification is seen due to subtle variations in the ray crossing pattern seen in Figure S1c.
Figure S2a shows two rows of the resolution matrix in the regular model; one in the middle and one at the edge. Unlike the ray-adapted model, the cell in the middle shows no smearing towards shallower or deeper levels. This does not mean a higher resolution in this model, this is an artifact due to lack of rays (no rays), see ray density in main text Figure 2c. The smearing goes to cells in the other vertical sections along the oblique dominant ray direction. In fact this cell has lower resolution matrix diagonal element ~ 0.1, than that of the ray-adapted model ~ 0.25 (compare Figure S1a and Figure S2a). Same story applies for the cell at the edge, where we see no smearing in the vertical section because the smearing goes to the adjacent vertical sections.
Figure S2b shows the diagonal elements of the resolution matrix in the regular model at 9 different vertical sections between latitudes 60.5° north, where rays hit the model at the top, and 62.5° north, where rays hit the depth of 450 km. Here the maximum diagonal element of resolution matrix is 0.1392 (note that same color scale is used in all figures).
It is obvious that not only has the regular cell model a very uneven distribution of rays in all vertical sections, also the diagnostic diagonal element of the resolution matrix is unevenly distributed in almost the same way as the ray density. Finally, the values of these diagonal elements are all smaller than the resolution diagonal elements in the ray-adapted model. This is partly due to longer ray segments at each cell.
Figure S3 illustrates this in more detail. The segment of the ray in the regular model which crosses the depths of 250-350, marked by a green ellipse, is shorter than that of the ray-adapted model (compare Figure S3a and Fig 1c). In the ray-adapted model the ray segments in each layer are almost the same (Fig S1c), therefore resolution is mainly controlled by the directional pattern of the rays at each layer. Figure S3c shows the diagonal elements of the resolution matrix at the blue-dash marked vertical section in Figure S3a. Note the lower resolution value in the regular model compare to ray-adapted model.
The straight line synthetic test is discussed in detail in the previous section. It may be questioned if such dense profile array layout is better than a more scattered 2D distribution of the same number of stations. We test this by perturbing the station locations by up to +/-0.75° (~80 km) perpendicular to the profile. We keep the same number of events as in first synthetic test.
Figure S4a shows the station locations and rays down to depth of 600 km, seen from above. Figure S4b and c are showing the model and rays from east in the regular and ray-adapted models respectively.
Figure S5a shows the diagonal elements of resolution matrix of regular cell modeling of the 2D scattered array in Figure S1. Here the maximum of diagonal element is ~ 0.1, much lower than that of the first synthetic test (line profile station array) (Figure S4c). At each section, cells with higher resolution seem to be more scattered which is due to the scattered station pattern.
Figure S5b shows the diagonal elements of resolution matrix in the ray adapted model. Again, we see that resolution is condensed more evenly into sections, in this case sections at 60.25° and 60.75° north, with higher value for resolution matrix, up to 0.1735.
We conclude that the dense profile seems to make the best of the resolution available for the given narrow-aperture event distribution, as was analyzed in section 2. The more scattered distribution analyzed in this section 3 produce a more diffuse resolution distribution. However, we note that also in this situation the ray adapted tilting of the cells delivers sections with improved resolution distribution.
In practice the choice of dense profile or scattered array layout may be controlled by logistic conditions like road access or distribution of land in an archipelago.
Figure S1. a) Visualization of one row in the resolution matrix for two cells at depth 150 – 200 km for the ray-adapted model. One cell at where we have cross firing of rays and smearing is moderate, and one at the edge of the model where rays are almost parallel and smearing is very strong. Colorbar is set to 0.25 (the highest value of Resolution matrix in the ray adapted model is 0.27). b) Diagonal elements of the resolution matrix, Colorbar is set to the maximum of the diagonal elements of the resolution matrix. b) South view (left) and east view (right) of the rays in the Ray-adapted model. Note the length of the ray segments in each cell is almost the same.
Figure S2. a) Visualization of a row in the resolution matrix for two cells at depth 150 – 200 km for the regular model. The color bar is set to the level of the ray-adapted model (the highest value of resolution matrix in the regular model is 0.13). b) Diagonal elements of the resolution matrix in 9 different sections from latitude 60.5 o to 62.5 o. Colorbar is set to the maximum of the diagonal elements of the resolution matrix in ray-adapted model.
Figure S3. a) Ray distribution in the regular model viewed from east. Location of the vertical east west section is shown with dashed blue line. Diagonal elements of the resolution matrix in the regular model at b) longitude 10.5° and c) latitude 61.75°.
Figure S4. a) Stations geometry and rays down to depth of 600 km in a 2D scattered array. View from east to the b) regular and c) ray-adapted model.
Figure S5. Diagonal of resolution matrix in 2D scattered synthetic test in a) regular and b) ray-adapted models.
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