3:45 PM - 4:00 PM
[MTT38-02] Estimation of initial atmospheric Lamb wave amplitude in tsunami source region by inversion of pressure observation
Keywords:inverse problem, Atmospheric Lamb wave, Tsunami, micro pressure observation
Background
At the time of tsunami generation, atmospheric Lamb waves are excited and observed as pressure fluctuations that propagate to distant locations at nearly the speed of sound (e.g., Arai et al. 2011). Therefore, if Lamb wave pressure fluctuations are obtained at a large enough number of locations, it is possible, in principle, to reconstruct the pressure distribution of Lamb waves at the tsunami source. Furthermore, there is a direct relationship between the amplitude of the Lamb wave and the wave height of the tsunami source (Nakajima2018). In this study, we investigate a method to estimate the pressure distribution in the tsunami source region at the time of tsunami occurrence from the observed data of Lamb wave pressure as an inverse problem.
Inverse problem of estimating initial pressure in the tsunami source region
We divide the tsunami source region into subregions, each of which is a square of 9km×9km, and consider the pressure distribution in the source region at the time of tsunami occurrence as the superposition of unit pressure anomalies in each subregion. For simplicity, we assume that tsunamis and Lamb waves develop simultaneously and instantaneously at a known time. The initial pressure distribution in the tsunami source region is determined by superposition coefficients of the unit pressure anomaly in the subregions. Let v denote the vector of coefficients and w denote the observed data, and the relationship between the two can be expressed as Mv=w, where M is determined from the solution of the wave equation (linear, horizontal, two-dimensional) for Lamb waves with the unit pressure deviation in each subregion as the initial condition.
Method for Solving the Inverse Problems
The problem of finding v from w cannot be solved in the usual sense because the dimensions of w and v are generally different. We use singular value analysis to find a solution that minimizes the residuals. When the spatial and temporal distribution of observations is appropriate, all the singular values of M are nonzero, the inverse problem becomes an overdetermined problem, and a least-squares solution is obtained. Typically, however, zeros appear in the singular values of M even when there are many observations, and the inverse problem becomes a mixed-determined problem. That is, v contains components that cannot be determined from w (null space). Here, the components belonging to the null space are set to zero (natural solution).
Observation Experiment
We set the initial pressure distribution in the source region as the linier combination of the unit pressure anomalies on the subregions, solve the Lamb wave equation to generate simulated observation data at arbitrary observation points, and examine whether the initial pressure distribution can be recovered from these data.
We found that the initial pressure recoverability can be judged based on whether the singular values of M include zero. That is, if all the singular values of M are non-zero, the initial pressure distribution can be recovered completely, but if any singular value of M is zero or close to zero, the recovery of the initial pressure distribution is incomplete. At the conference, we will discuss examples of cases in which the number and location of observation points, observation period, and initial pressure distribution are varied, with or without observation noise. These examinations will contribute to probe the usefulness of the Lamb waves for tsunami source estimation and to design Lamb wave observation networks.
At the time of tsunami generation, atmospheric Lamb waves are excited and observed as pressure fluctuations that propagate to distant locations at nearly the speed of sound (e.g., Arai et al. 2011). Therefore, if Lamb wave pressure fluctuations are obtained at a large enough number of locations, it is possible, in principle, to reconstruct the pressure distribution of Lamb waves at the tsunami source. Furthermore, there is a direct relationship between the amplitude of the Lamb wave and the wave height of the tsunami source (Nakajima2018). In this study, we investigate a method to estimate the pressure distribution in the tsunami source region at the time of tsunami occurrence from the observed data of Lamb wave pressure as an inverse problem.
Inverse problem of estimating initial pressure in the tsunami source region
We divide the tsunami source region into subregions, each of which is a square of 9km×9km, and consider the pressure distribution in the source region at the time of tsunami occurrence as the superposition of unit pressure anomalies in each subregion. For simplicity, we assume that tsunamis and Lamb waves develop simultaneously and instantaneously at a known time. The initial pressure distribution in the tsunami source region is determined by superposition coefficients of the unit pressure anomaly in the subregions. Let v denote the vector of coefficients and w denote the observed data, and the relationship between the two can be expressed as Mv=w, where M is determined from the solution of the wave equation (linear, horizontal, two-dimensional) for Lamb waves with the unit pressure deviation in each subregion as the initial condition.
Method for Solving the Inverse Problems
The problem of finding v from w cannot be solved in the usual sense because the dimensions of w and v are generally different. We use singular value analysis to find a solution that minimizes the residuals. When the spatial and temporal distribution of observations is appropriate, all the singular values of M are nonzero, the inverse problem becomes an overdetermined problem, and a least-squares solution is obtained. Typically, however, zeros appear in the singular values of M even when there are many observations, and the inverse problem becomes a mixed-determined problem. That is, v contains components that cannot be determined from w (null space). Here, the components belonging to the null space are set to zero (natural solution).
Observation Experiment
We set the initial pressure distribution in the source region as the linier combination of the unit pressure anomalies on the subregions, solve the Lamb wave equation to generate simulated observation data at arbitrary observation points, and examine whether the initial pressure distribution can be recovered from these data.
We found that the initial pressure recoverability can be judged based on whether the singular values of M include zero. That is, if all the singular values of M are non-zero, the initial pressure distribution can be recovered completely, but if any singular value of M is zero or close to zero, the recovery of the initial pressure distribution is incomplete. At the conference, we will discuss examples of cases in which the number and location of observation points, observation period, and initial pressure distribution are varied, with or without observation noise. These examinations will contribute to probe the usefulness of the Lamb waves for tsunami source estimation and to design Lamb wave observation networks.