THE INVERSE PROBLEM OF RECONSTRUCTING
A TSUNAMI SOURCE WITH NUMERICAL
SIMULATION
T.Voronina
Institute of Computational Mathematics
and 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:
(1)
W|t 0  0;
W
n
Wt t 0  0;
0
(2)
(3)
coast
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
c  x, y   gh  x, y 
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
Let us assume : the support of the function ( x, y) is included in the rectangle
and the function h (x, y) is continuously differentiable
 ( x, y) W ( D)
1
2
W
G
 W0  x  s  , y  s  , t 
{G :  x  s  , y  s   ;
0  s  L};
D
  ( x, y) : 0  x  600;0  y  400
   x, y  : 400  x  500, 200  y  300
Y­


® X
Z Y
A  x, y   W0 (s, t )
(4)
 ( x, y) W21 ( D)
Ladiejenskaya O.A. Boundary-value problems of mathematical physics., M., Nauka, 1973, 407 p.
A: W21  D   L2   0, L    0, T  
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
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.
k  x, y 
W21  D 
In the “ model” space
 ( x, y ) 
space L2   0, L    0, T  
l  x, yCheverda
 in theV. A.,“data’
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.


k 1
ck   , k  , W0l = W0 , l  ,  Ak , l   alk

 c ( A , )  W
k
k 1
k
dim  sol   K
K
a

k 1
KL
l
0l
, l  1,..., 
dim(data)  L
c  fl , l  1,
lk k
 x, y     x, y  ,
,L
K, L  
ck k
Ac  f
(5)
f  (w11 , w12 , ..., w1Nt , ..., wi1, ..., wiNt , wP1, ..., wPNt ) )T
wij  W0 (si , t j ); i  1,...P; j  1,...Nt ; c  (c1,..., cK ) .
T
sj
vi
A U  V
T
dim(U )  L  L; L  P  Nt
dim( A)  L  K ;dim(V )  K  K ;
-the right singular vectors of the matrix make a basis in the space of solutions;
uj
- the left singular vectors make a basis in the space of the right-hand
K
c 
j 1
 f , u v
r
j
sj
j
 ( x , y )[ r ]  
j 1
K
Vj ( x, y )   v jll ( x, y );
l 1
 f , u  V ( x, y )
j
sj
j

sk
r  max k :
s1


 d 

K
c 
j 1
K
 
k 1
  u   v
j
sj
j
K
k
s
 f   , u v
k
  k uk
k 1
[6] Tsetsokho V.A., Belonosov A.S., Belonosova A.V. On one method to construction of rsmooth 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).
2 k
2 l
akl sin
 x  X 0  sin  y  Y0 

X
Y
m 1 n 1
M
N
X  X max  X min , Y  Ymax  Ymin ,
X min  400; X max  500; Ymin  200; Ymax  300;
dim  sol   K  M  N
dim(data)  L  P  Nt
observe points
coast
200 km
Ymax
*
O  x0 , y0 
100 *
0
*
ФФ
Ymin
X min


X max
150
300 km
observe points
coast
200 km
Ymax
*
O  x0 , y0 
100 *
0
*
ФФ
Ymin
X min


X max
150
300 km
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.
m
fmax-1,959; fmin=-0,67 ;
km
km
p12
p13
p3
P6:{3,4,10,11,12,13}
r=72; err.=0,37; fmax-1,549; fmin=-0,6591 ;
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 illposed 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