further reducing model uncertainties at the expense of reduced spatial resolution. For the kinematic problem, we have a posterior ensemble of models, from which m˜ and C˜m can be computed, and we can either use Eq. 2.9 and 2.10, or apply spatial averaging directly on the sample set of the model ensemble.
For the Gaussian posterior in our case, the uncertainty Ei of model parameter mi is obtained from the posterior variance-covariance matrix:
Ei = q
(C˜m)i,i. (2.11)
The spatial averaging operator (e.g., S1R) imposes a minimum length scale Dsi, which we choose as an effective circular diameter for the area of spatial averaging:
Dsi = 2 s
(X
j
Aj+Ai)/π, (2.12)
where the summation is over all the neighboring nodes of parameter mi (based on 1R, 2R or other spatial averaging criteria), andAj is the effective tent area for node j. The impact of averaging m˜ and C˜m wil be illustrated first using synthetic scenarios.
features to be well-resolved while large-scale features may not be when the latter solution is in the null space. Therefore, we use some potentially realistic scenarios in these synthetic tests, and simply aim to obtain a qualitative assessment of the data resolution and an intuitive understanding of how different assumptions of source kinematics and error models might influence the results.
We consider two synthetic source models both of which are kinematic. The two scenar- ios differ in the proximity of maximum uplift to the trench (Figs. 2.4A and 2.5A). In both scenarios, we generate synthetic data with dispersive GF, and consider alternatively quasi- static and kinematic inversions with different combinations of Cd and Cp, and dispersive (DSP) and non-dispersive (NDSP) GF, i.e., a total of eight synthetics. Using non-dispersive GF for a dispersive propagation scenario is motivated by the fact that in real cases, we an- ticipate inaccuracy and limitations of our GF, which makes it necessary to use Cχ instead of just Cd in large problems. Here we use a Cχ that includes CRPp based on the pertur- bation of waveforms using the known reference true model, while forCd we assume 10 cm uncorrelated Gaussian error. We use the entire mesh for the quasi-static problem, while adopting a near-source subset of the seafloor mesh for the kinematic problem, to reduce the number of free parameters and computational demand. We choose a Gaussian prior on the uplift P(m) = N(0,(10 m)2I), since we believe that the real seafloor uplift is unlikely to much exceed 10 m based on experience with marine terrace and sea surface uplift dur- ing large earthquakes (e.g., Plafker and Rubin, 1978; Meltzner et al., 2006), and a uniform priorP(vr) =U(0.5 km/s,2.5 km/s) for the additional parameter displacement propagation velocity vr in the kinematic problem. Although previous studies (Satake et al., 2013) sug- gest that tsunami data are not sensitive to different vr in this range, we allow vr to vary.
All kinematic inversions are done with a fixed initiation point, which we assume to be the
hypocenter location of the Tohoku-oki earthquake (Chu et al., 2011).
In Fig. 2.6, we first demonstrate the effect of spatial averaging on the posterior solutions, including the mean value and uncertainty, using the posterior of a synthetic scenario in which a compact source of uplift occurs near the trench. The posterior mean model becomes smoother with the increase in the range of spatial averaging, accompanied by the reduction of error ellipses associated with the parameters highlighted. In principle, we can apply spatially nonuniform adaptive averaging based on desired resolution or criterion on the absolute or relative uncertainty, in order to eliminate null solutions where the model is less constrained and produce representative models useful for geophysical interpretations. In most of our models, we find that 1R spatial averaging is sufficient to reduce uncertainty to acceptable values and produce appropriate resolution for the source region of our interest, so we adopt the uniform 1R spatial averaging in this study. In Figs. 2.4 and 2.5, all posterior solutions are shown after 1R averaging.
From the results of synthetic tests, we find that inversions of quasi-static models with only Cd tend to bias the solutions toward more localized and larger uplift with peak value offset in space from the input model (Figs. 2.4B,E and 2.5B,E). Spurious features of subsi- dence to the south are also observed in these models, and are particularly bad in Fig. 2.5B,E, because source kinematics and the dispersive nature of tsunami both introduce waveform complexity, and ignoring finite displacement propagation velocity would force these addi- tional features into the model in order to fit the waveform, as is also reported by Hossen et al. (2015). Such a bias is amplified for the case of a more dispersive tsunami wave ex- cited at the trench (see Fig. 2.20). However, we observe that with the incorporation of more realistic Cχ (Figs. 2.4C,F and 2.5C,F), the bias due to the quasi-static assumption is reduced, and the model uncertainties are reasonably increased, so that the resulting model
is more compatible with the true model within uncertainty. In the quasi-static problem, we notice that by using NDSP GFs (Fig. 2.4E,F and 2.5E,F), not only are spurious features worse, but uncertainties are also underestimated more than their counterparts with DSP GF (Fig. 2.4B,C and 2.5B,C), suggesting that an inaccurate GF without an appropriateCp
can lead to a biased mean solution and underestimated uncertainties.
Inversions of kinematic models recover the first synthetic scenario well (Fig. 2.4D,G,H,I), due to the spatially non-uniformvr. These extra degrees of freedom lead to improved fits to the waveforms, even in cases where non-dispersive GF are used (Fig. 2.4H,I). Uncertainties of model parameters in the kinematic models is smaller than those in quasi-static counterparts, partly because the causality constraint imposed by the deformation front requires that distant regions do not experience deformation and thus reduces the plausible parameter space for the problem. The use ofCp yields more realistic estimate of model uncertainties when incorrect GF is used. For the second synthetic scenario, the combinations of Cd and NDSP GF (Fig. 2.5H) produce a solution with a similar uplift pattern as the one with correct DSP GF (Fig. 2.5D) and yet significant deviations in the inferred deformation fronts. In the cases with Cp (Fig. 2.5G,I), short-wavelength uplift (about 20 km) at the trench can not be fully resolved, but the overall long-wavelength pattern could still be retrieved with well- recovered deformation fronts and more realsitic error estimates. Generally, these results show that the tsunami data can indeed resolve features of offshore sources over the length scale of tens of kilometers.
50 km
141˚E 142˚E 143˚E 144˚E
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(I) Kin. + Cχ + NDSP
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Figure 2.4: The effect of source kinematics, error structure and inaccuracy in GF on the inversion of a synthetic scenario with maximum uplift away from the trench. The synthetic data is produced from (A) a kinematic scenario with maximum uplift landward of the trench and dispersive GF. Eight inversions (B-I) are conducted for quasi-static and kinematic sce- narios with the combinations ofCd orCχ, and dispersive (DSP) or non-dispersive (NDSP) GFs. The contours represent inferred kinematic deformation fronts with intervals of 30 sec.
50 km
141˚E 142˚E 143˚E 144˚E
37˚N 38˚N 39˚N
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(D) Kin. + Cd + DSP
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38˚N 39˚N
(F) Static + Cχ + NDSP
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(G) Kin. + Cχ + DSP
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(H) Kin. + Cd + NDSP
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37˚N 38˚N 39˚N
(I) Kin. + Cχ + NDSP
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Uplift (m)
Figure 2.5: The effect of source kinematics, error structure, and inaccuracy in GF on the inversion of a synthetic scenario with maximum uplift at the trench. The synthetic data is produced from (A) a kinematic scenario with maximum uplift at the trench and dispersive GF. Eight inversions (B-I) are conducted for quasi-static and kinematic scenarios with the combinations of Cd or Cχ, and dispersive (DSP) or non-dispersive (NDSP) GFs. The contours represent kinematic deformation fronts with intervals of 30 sec.
50 km 141˚E 141˚E
142˚E 142˚E
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144˚E 144˚E
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37˚N 38˚N 39˚N
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42 02 46 108 12
Uplift (m) 10 m
50 km 141˚E 141˚E
142˚E 142˚E
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Uplift (m) 10 m
50 km 141˚E 141˚E
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(C)
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Uplift (m) 10 m
Figure 2.6: Spatial averaging of posterior solutions and model uncertainties. Posterior mean models are shown with (A) no spatial averaging and (B) “one-ring” (1R) and (C) “two-ring”
(2R) spatial averaging for a synthetic example. The mean values and uncertainties of uplift are plotted as vertical arrows and circles, respectively, for several near-source nodes. The red star indicates the assumed hypocenter of the scenario event.