THE INVERSE PROBLEM OF RECONSTRUCTING A TSUNAMI SOURCE WITH NUMERICAL SIMULATION T.Voronina Institute of Computational Mathematics and Mathematical Geophysics.

THE INVERSE PROBLEM OF RECONSTRUCTINGA TSUNAMI SOURCE WITH NUMERICAL

SIMULATION

T.Voronina

Institute of Computational Mathematicsand Mathematical Geophysics

SB RAS

In this paper, we make an attempt to answer the following questions: How accurately a tsunami source can be reconstructed based on recordings at a given tide-gauge network?

Is it possible to improve the quality of reconstructing a tsunami source by distinguishing the “most informative” part of the initial observation system?

Mathematically, the inverse problem to infer the initial sea displacement in the source area is considered as a usual ill-posed problem of the hydrodynamic inversion of tsunami tide-gauge records. The direct problem; Inverse problem; Numerical experiments

Satake (1987, 1989,2007) , (Johnson et al., 1996; Johnson, 1999), Pires and Miranda (2001) A. Piatanesi, S. Tinti, and G. Pagnoni (2001) and others.

1. Kaistrenko V. M. : Inverse problem for reconstruction of tsunami source. In: Tsunami waves. Proc. Sakhalin Compl.Inst. 1972, is.29. P.82-92.

The direct problem, i.e. the calculation of synthetic tide-gauge records from the initial water elevation field, is based on a linear shallow-water system of differential equations in the rectangular coordinates:

| 0 00; 0;t t tW W

(1)

(2)

coast

0W

n

(3)

, ,c x y gh x y

W (x, y, t) is a water elevation above the mean sea level

h(x,y) - is the depth of the ocean

c(x,y) – is the velocity of the tsunami wave

f (x, y, t) describes the movement of the bottom in the tsunami area.

f (x, y, t ) = (t) (x, y), where (t) is the Heavyside function (x, y) - the initial bottom elevation

0 , ,GW W x s y s t

{ : , ; 0 };G x s y s s L

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T.A.Voronina ,V.A.Tcheverda: Reconstruction of tsunami initial form via level oscillation. Bull.Nov.Comp.Center,Math.Model.in Geoph., 4(1998), p.127-136

12 2: 0, 0,A W D L L T :

Ladiejenskaya O.A. Boundary-value problems of mathematical physics., M., Nauka, 1973, 407 p.

Воронина Т.А. Определение пространственного распределения источников колебаний по дистанционным измерениям в конечном числе точек // Сиб.Ж.Выч.Мат. 2004,Т.7,№3, С.203-211.

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dim sol K dim( )data L

In the “ model” space

in the “data’ space

1

, 1, ,K

lk k lk

a c f l L

, , , ,KL x y x y K L

Cheverda V. A., Kostin V.I.: r-pseudoinverse for compact operators in Hilbert space: existence and stability. In: J. Inverse and Ill-Posed Problems, 1995, V.3, N.2, pp. 131-148.

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-the right singular vectors of the matrix make a basis in the space of solutions;

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[6] Tsetsokho V.A., Belonosov A.S., Belonosova A.V. On one method to construction of r-smooth approximation for multivariable functions // Proceedings of the Seminar «Computational Methods of Applied Mathematics» (headed by G.I. Marchuk), Novosibirsk, 1974. V. 3, pp. 3-13 (in Russian).

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400; 500; 200; 300;

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Our approach includes the following steps:

First, we obtain the synthetic tide gauge records from a model source, which form we are to reconstruct. These can be records observed at real time instants.

The original tsunami source in the area in question is recovered by the inversion of the above wave records.

We calculate mariograms from the earlier reconstructed source. To define the ”most” informative” part of the initial observation system for a target area, we compare synthetic mariograms, obtained in two cases in the same locations (so, synthetic and real recordings will be compared) at all available sea-level tide gauges.

Next, we consider the observation system, which contains only good matching stations.

Now we can again restore the tsunami source using only the tide-gauge records that were determined as being the ”most” informative” part. This “improved” tsunami source can be proposed for the use in further tsunami calculation.

km

m

km

fmax-1,959; fmin=-0,67 ;

p12

p13p3

r=72; err.=0,37; fmax-1,549; fmin=-0,6591 ;

P6:{3,4,10,11,12,13}

P9:{3,4,5,8,9,10,11,12,13}

Conclusion

Based on the carried out numerical experiments we can conclude:

• The quality of the source restoration strongly depends on the number of records used and their azimuthal coverage;

• To obtain a reasonable quality of source restoration we need at least 5-7 records smoothly distributed over the space domain that is comparative in size to the projection of the source area onto the coast line;

• Complexity of a source function and the presence of the background noise imply serious limitation on the accuracy of the restoration procedure, more complex sources require a larger number of wave records and finer computational grids used for the calculation of synthetic waveforms;

• The application of r-solutions is an effective means of regularization of an ill-posed problem. The number of r basic vectors applied appears to be essentially lower than the minimum dimension of a matrix. This, in fact, enables us to avoid instability of the problem dealing with a sharp decrease of singular values of the matrix.

Thank for your attention