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[MGI28-03] Automatic detection of atmospheric Lamb waves by inverse problem
Keywords:Atmospheric Lamb Waves, inverse problem, tsunami early warning, generalized inverse
1. Background
Atmospheric Lamb waves are surface trapped and propagate horizontally at nearly the speed of sound which is faster than that of a tsunami (about 200 m/s), so that the development of an extensive micro-pressure measurement network and detection of Lamb waves will be useful for early warning of tsunami. In order to realize early warning, it is highly desirable to be able to automatically detect Lamb waves from observed pressure fluctuations. Here, we discuss a method of Lamb wave detection using an inverse problem and its theoretical background.
2. Methods
Inverse problem is generally a problem of finding the input v given a known system (observation matrix) M and an output w. The solution is obtained using singular value analysis, which decomposes the observation matrix M in terms of singular values and singular vectors. In the Lamb wave detection problem, the output w corresponds to the observed pressure data and the input v corresponds to the spatial distribution of the pressure and wind fields at the desired time (latest time). The governing equations of the Lamb wave (equation of motion and continuity equation allowing for wave motion at c = 310 m/s) are used to construct the observation matrix M. In constructing the observation matrix M, we used the one dimensional d'Alembert solution. The spatial distribution of atmospheric pressure at the latest time estimated by the above system is compared with the actual atmospheric pressure observation data at that time, and if they are in agreement, it can be judged a Lamb wave is arriving. In this study, assuming a relatively compact observation network and assuming that lamb waves are plane waves arriving from one specific direction, we consider an inverse spatial 1D problem.
3. Detection of Lamb Waves with Direction
We investigated the ability of the system to reproduce the latest observations of the pressure distribution, assuming that plane waves traveling at various phase speeds and directions were observed at irregularly distributed virtual observation points. When the direction of the 1D spatial axis assumed by the system coincides with the direction of arrival of the wave, the residual is minimized and the lamb wave can be detected together with the direction of arrival.
4. Correspondence between inverse method and intuitive method
An intuitive method for estimating the pressure distribution that satisfies the governing equation of Lamb waves from observed data is to use the average along the isophase line of the observed data as the estimated value. Therefore, the correspondence to this intuitive method is examined. When the output (observation) space and the input (estimation) space can be directly compared, the observation matrix M can be constructed without interpolation. In this case, the right singular vectors are of localized impulse type, and the left singular vectors clearly show isophase structure in an intuitive way. On the other hand, in general, the observed and estimated spaces cannot be directly compared, and the observation matrix M involves interpolation. In this case, the right singular vector are wave-like and no correspondence with the intuitive method is obscured. However, when the natural general inverse matrix is constructed, elements with large absolute values are localized near the isotropic line for each estimated point, and the correspondence with the intuitive method can be recognized. However, the correspondence is approximate, with negative values at some distance from the isophase line.
5. Remarks
In spite that the presence or absence of interpolation makes a difference in the shape of the singular vectors, in natural general inverse matrices recoveres the correspondence to the intuitive methods. The inverse problem is of great practical value in the sense that it provides guiding principles for constructing solution methods, including a wide range of general cases in which interpolation is indispendable.
Atmospheric Lamb waves are surface trapped and propagate horizontally at nearly the speed of sound which is faster than that of a tsunami (about 200 m/s), so that the development of an extensive micro-pressure measurement network and detection of Lamb waves will be useful for early warning of tsunami. In order to realize early warning, it is highly desirable to be able to automatically detect Lamb waves from observed pressure fluctuations. Here, we discuss a method of Lamb wave detection using an inverse problem and its theoretical background.
2. Methods
Inverse problem is generally a problem of finding the input v given a known system (observation matrix) M and an output w. The solution is obtained using singular value analysis, which decomposes the observation matrix M in terms of singular values and singular vectors. In the Lamb wave detection problem, the output w corresponds to the observed pressure data and the input v corresponds to the spatial distribution of the pressure and wind fields at the desired time (latest time). The governing equations of the Lamb wave (equation of motion and continuity equation allowing for wave motion at c = 310 m/s) are used to construct the observation matrix M. In constructing the observation matrix M, we used the one dimensional d'Alembert solution. The spatial distribution of atmospheric pressure at the latest time estimated by the above system is compared with the actual atmospheric pressure observation data at that time, and if they are in agreement, it can be judged a Lamb wave is arriving. In this study, assuming a relatively compact observation network and assuming that lamb waves are plane waves arriving from one specific direction, we consider an inverse spatial 1D problem.
3. Detection of Lamb Waves with Direction
We investigated the ability of the system to reproduce the latest observations of the pressure distribution, assuming that plane waves traveling at various phase speeds and directions were observed at irregularly distributed virtual observation points. When the direction of the 1D spatial axis assumed by the system coincides with the direction of arrival of the wave, the residual is minimized and the lamb wave can be detected together with the direction of arrival.
4. Correspondence between inverse method and intuitive method
An intuitive method for estimating the pressure distribution that satisfies the governing equation of Lamb waves from observed data is to use the average along the isophase line of the observed data as the estimated value. Therefore, the correspondence to this intuitive method is examined. When the output (observation) space and the input (estimation) space can be directly compared, the observation matrix M can be constructed without interpolation. In this case, the right singular vectors are of localized impulse type, and the left singular vectors clearly show isophase structure in an intuitive way. On the other hand, in general, the observed and estimated spaces cannot be directly compared, and the observation matrix M involves interpolation. In this case, the right singular vector are wave-like and no correspondence with the intuitive method is obscured. However, when the natural general inverse matrix is constructed, elements with large absolute values are localized near the isotropic line for each estimated point, and the correspondence with the intuitive method can be recognized. However, the correspondence is approximate, with negative values at some distance from the isophase line.
5. Remarks
In spite that the presence or absence of interpolation makes a difference in the shape of the singular vectors, in natural general inverse matrices recoveres the correspondence to the intuitive methods. The inverse problem is of great practical value in the sense that it provides guiding principles for constructing solution methods, including a wide range of general cases in which interpolation is indispendable.