Large values of Q and difficulties of estimating attenuation

uation

In the previous examples we have shown that we can resolve an attenuation model with our current dataset and we have shown the importance of noise estimation. However, all our synthetic examples include a relatively small variation between low values of Q, namely 200 and 300. Clearly, our inversion works well for these cases and it shows that we should be able to resolve these small variations. This is not surprising: Figure 5.5.9 shows how the logarithm of Q changes with t∗ across the range of epicentral distances used in this study, assuming ak135 model and a source depth of 500 km. The figure shows base 10 logarithm of Q. We can see that large attenuation (large values of t∗, low values of log(Q)) can be relatively easily distinguished between epicentral distances. For example, if we have an extremely large attenuation of Q= 50, this is equivalent to≈1.7 on the log scale, and Figure

0 1 2 3 4 5 6 7 1 1.5 2 2.5 3 3.5

t* (s)

log(Q)

Figure 5.5.9: Quality factor Q as a function of the t∗ parameter across the range of

epicentral distances used in this study, assuming ak135 model and a source depth of 500 km.

5.5.9 shows that t∗ for that case varies between ≈1.0 for the epicentral distance of 149◦ and ≈ 2.0 for the epicentral distance of 155◦. This makes sense when one is reminded that thet∗ parameter is a function of travel time which is obviously larger with longer travel paths over larger distances.

In the case of Q = 200 and Q = 300 on the log scale this equates to ≈ 2.3 and ≈ 2.5, respectively. We can see that in that case the difference in t∗ parameter is much smaller across all the epicentral distances – for these values of Q t∗ is lower than 1.0 and generally clustered around the value of 0.5. However it is still possible to make some distinction between the two values, and we have also shown this in the previous section.

With really low attenuation (Q equal to or larger than 1000) the value of the t∗

parameter is virtually indistinguishable between the epicentral distances, in com- parison to larger values of attenuation (low Q). In the case of Q = 2000 and

Q = 3000, which on the log scale equates to ≈ 3.3 and ≈ 3.5, Figure 5.5.9 shows that thet∗ across epicentral distances is so similar it is almost impossible to tell the difference between these values of Q.

What does this mean in terms of tomography? It should be expected that the in- version should easily discriminate between the values of Q, although this may prove more difficult when the values of Q are large. A more concrete expectation is that large recovered values of Q (low attenuation) will come with a larger error in com- parison to smaller values of Q(large attenuation), when both extremes are present in the model. We will now show two synthetic examples which will demonstrate this.

In the first example we assume interchanging values of Qbetween 2000 and 2500 in a checkerboard model as above. As Figure 5.5.9 shows this is not too big a variation in terms of thet∗parameter, but both values are high and implying low attenuation. Figure 5.5.10 shows the results of this test. The noise added to the synthetic dataset was 2% of the range of the data, and we imposed damping regularisation. At the top of this Figure we again observe poor recovery and large errors for the first 10-12 parameters, as expected and as observed previously. The rest of the parameters show good recovery with similar errors as in the case of Q = 200 and Q = 300 (compare with Figure 5.5.4). Note how the bottom of Figure 5.5.10 shows much smaller t∗ values overall - they are lower than 0.1. This result is in line with one of our expectations: the inversion discriminates between values of Q, and errors are similar to the ones observed when the values of Q are low.

§5.5 Synthetic examples 107

board model with interchanging values of Qbetween 2000 and 300. This is a much larger variation between somewhat extreme values of large and low attenuation. We again added 2% noise and imposed damping regularisation. The top of the Figure shows us again that the inversion easily distinguishes between these values of Q, however error estimates are different: we observe significantly larger error for large values of Q. The value of 2000 is always within error, with the exception of poorly covered regions (first 10-12 parameters). It is interesting to note that in those re- gions attenuation is more than likely overestimated (lower Q).

These examples confirm two really important facts: 1) we can resolve Q variations and 2) large values of Q come with large errors, with possible overestimation of attenuation in poorly covered regions.

Figure 5.5.10: Top: The recovered model (black filled circles) and its 3σ confidence intervals (error bars) in a synthetic test assuming a checkerboard model with interchanging

values of Q = 2000 and Q = 2500. Bottom: The relationship between observed and

predicted data shows good recovery of the input model.

Figure 5.5.11: Top: The recovered model (black filled circles) and its 3σ confidence

intervals (error bars) in a synthetic test assuming a checkerboard model with interchanging

values of Q = 2000 and Q = 300. Bottom: The relationship between observed and