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A tripartite array for the study of seismo-volcanic signals. A MATHCAD-11 program

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(1)

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The present worksheet contains a MATHCAD 11 program for the estimate of slowness vector

components of a seismic signal recorded by a short period small aperture tripartite array. I use a

simplified Plane-Wave-Fitting technique (PWF - Del Pezzo and Giudicepietro, Computer and

Geosciences, 28 , 2002, 59-64) based on the estimate of the cross-correlation between signal

pairs in time domain. An example of application to a synthetic signal shows that the estimate of

the slowness vector components is almost unbiased evn in condition of low signal-to-noise ratio.

(2)

7KHPHWKRG

Given a tripartite array of seismic stations located at points with cartesian co-ordinates x i , , I y i make the hypothesis that a wave packet of given slowness vector p ( p x , p y ) crosses the array with the inter-distances between the station pairs sufficiently small to permit a good degree of correlation among the traces. The travel time difference between station i and j is given by the dot product

t i j , := p ⋅ ∆x i j , (1)

where ∆x i j , is the two element vector of the differences between the co-ordinates of station i and j: ∆x i j , := ( x i − x j , y i − y j ) (2)

and t i j , is estimated through the cross-correlation function between the signal pairs a t ( ) i and a t ( ) j :

C ( ) τ i j ,

t a t ( ) i a t ( + τ ) j

⌠ 

⌡ d

:= (3)

The maximum of equation (3) occurs at a time, t max which corresponds to the phase delay between signals at station i and j. t max estimates for any couple i,j t i j , of eq. (1)

I denote with t the one column matrix given by t   , t   , t   ,

 

 

 

 

; consequently p can be estimated solving

the overdetermined equation system (1) by least squares. From the estimate of p, apparent

velocity v and azimuth φ can be estimated by:

(3)

v  ( ) p x  + ( ) p y 



− := 

φ atan p x p y

  

:= 

Errors can be estimated by the covariance matrix of the data (Menke, 1984)

The users who have installed MATHCAD 11 on their PC can change the input parameters to test in

different conditions. Input parameters are enlighted in red color. Set TOL=0.000001 from worksheet

options.

(4)

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3:)SURJUDP 3ODQHZDYHILWWLQJWHFKQLTXH 

predominant frequency of the synthetic signal: f=3 Hz f2 := 

par1 :=  par2 :=  par3 :=  These parameters change the shape of the test signal

signal t ( ) t

 par1

  



exp − ( par2 t ⋅ )

 par3

  

⋅ sin (  π ⋅ f2 ⋅ t⋅ )

:=

quake t ( )  if t ≤  signal t ( ) if t > 

   

:= synthetic signal

x t ( ) := t −  time shift example

(5)

    

 quake t ( ) quake x t ( ( ) )

t

srate :=  Sampling rate

Npti :=  Data points

Signal generation

i :=  Npti ..

t i i srate :=

s2 i := quake t ( ) i

(6)

&RRUGLQDWHVRIWKHDUUD\WHVW

A













 

 

 

 

:= Units in meters

xx A 〈 〉  :=

yy A 〈 〉  :=

 





Array Coordinates

yy

xx

:=

(7)

∆t   , = −  Example of true time shift

sloy =  Co-ordinates of the slowness vector test slox = 

∆t k n , :=  slox xx ( k − xx n ) + sloy yy ( k yy n ) 

sloy 

vapp ⋅ sin az deg ( ⋅ ) :=

slox 

vapp ⋅ cos az deg ( ⋅ )

:= Slowness test in s/m. Azimuth in degrees from East to North

n :=  Nstaz ..

k :=  Nstaz ..

metri / s vapp := 

Test v app

gradi az := 

Test azimuth

Nstaz = 

Nstaz := rows A ( )

(8)

x i k , := t i − ∆t k ,  − 

noise_amplitude :=  Addition of noise. Noiuse is gaussian (function rnorm)

noise 〈 〉 k

rnorm Npti ( , noise_amplitude  , ) :=

s2 i k , := quake x ( ) i k , + noise i k ,

Shift is a function based on the cross-correlation between the signal pairs

, := ← ,

(9)

shift X Y ( , ) XX ← correl X Y ( , ) XXX ← recenter XX ( )

XXXX submatrix XXX round last XXX ( )

 − 

   

, round last XXX ( )

 + 

   

, ,  , 

   

massimo ← max XXXX ( )

index ← v if XXXX v ≥ massimo v ∈  last XXXX .. ( )

for

alfa index round last XXXX ( )

 

  

t_shift alfa srate

← t_shift :=

Data input

i :=  last s2 .. ( ) 〈 〉 

k :=  Nstaz ..

(10)

t i i srate :=

g2 := s2 cols s2 ( ) =  o :=  cols s2 .. ( )

g2 〈 〉 o

g2 〈 〉 o

o ⋅ 

+

:= Data plot. 16 traces maximum are allowed to be plotted

(11)

   













 Synthetic traces

time (s)

a.u.

g2 〈 〉 g2



〈 〉

g2



〈 〉

t Coordinate := augment xx yy ( , )

∆XX

xx  − xx  xx  − xx  xx  − xx 

yy  − yy  yy  − yy  yy  − yy 

 

 

 

 

:= xx







 

 

 

 

= yy







 

 

 

 

=

(12)

Data selection. Set the number of windows in which the signal is segmented n_win := 

t_w t_win max t ( ) − min t ( ) n_win

tin n ← min t ( ) + ( n −  ) t_win ⋅ tfin n ← tin n + t_win

n ∈  n_win ..

for

augment tin tfin ( , ) :=

Define the start and end time for each window

signal

signal n submatrix s2 t_w  , ( 〈 〉  ) n srate , ( t_w 〈 〉  ) n srate , cols s2  , ( )

 

← 

n ∈  n_win ..

for

signal

:= Segmentation of

the signal

Npti last signal  (  ) 〈 〉 

:= 

(13)

The observable is the time value corresponding to the max of cross-correlation function obtained from shift function

Generation of the data Matrix and covariance Matrix

s :=  last signal .. ( )

m :=  last xx .. ( ) 〈 〉  n :=  last xx .. ( ) 〈 〉  vector

Data s ,  shift  ( signal s ) 〈 〉  , ( signal s ) 〈 〉 

← 

Data s ,  shift  ( signal s ) 〈 〉  , ( signal s ) 〈 〉 

← 

Data s ,  shift  ( signal s ) 〈 〉  , ( signal s ) 〈 〉 

← 

s ∈  last signal .. ( ) for

Data

:=

(14)

Datacolumns

A 〈 〉 s

vector T

( ) 〈 〉 s

s ∈  last signal .. ( ) for

A :=

6ROXWLRQRIWKHLQYHUVHSURUEOHP

P

P 〈 〉 s

∆XX T ⋅ ∆XX

( )

 −   ∆XX T Datacolumns 〈 〉 s

s ∈  last signal .. ( ) for

P :=

velapp

slowp s 

P 〈 〉 s

( ) 

  

 + ( ) P 〈 〉 s  

s ∈  last signal .. ( ) for

slowp :=

backp

angolo s ∈  last signal s .. angle ( ( ) P ) 〈 〉 s  , ( ) P 〈 〉 s 

for

angolo

:=

(15)

errdata   srate

 ⋅

  

:=  Estimate of the error on data

sigmaq := errdata identity ⋅ ( ) 

covP  ( ∆XX T ⋅ ∆XX )  ∆XX T

sigmaq  ( ∆XX T ∆XX )  ∆XX T

T

⋅ :=

covP   × 

  × 

  × 

  × 

 

 

=

erx := ( covP   , ) 

ery := ( covP   , ) 

(16)

errvapp

errvapp s −  ( ) P 〈 〉 s  P 〈 〉 s

( ) 

  

 + ( ) P 〈 〉 s  

   





 

 



 

 



 erx 

⋅ −  ( ) P 〈 〉 s  P 〈 〉 s

( ) 

  

 + ( ) P 〈 〉 s  

   





 

 



 

 



 ery 

⋅ +

 

 

 

 

 

 





s ∈  last signal .. ( ) for

errvapp :=

errback

continue ( ) P 〈 〉 s  P 〈 〉 s

( ) 

on error

errback s 

P 〈 〉 s

( )   ( ) P 〈 〉 s   P 〈 〉 s

( ) 

  

+ 

 

 

 

 

 

 



 

 



 erx 

( ) P 〈 〉 s 

P 〈 〉 s

( ) 

  

  ( ) P 〈 〉 s   P 〈 〉 s

( ) 

  

+ 

 

 

 

 

 

 

 

 

 

 

 ery 

 +

 

 

 

 

 

 





← s ∈  last signal .. ( ) for

errback

:=

(17)

cent_time s ( t_w 〈 〉  ) s + ( t_w 〈 〉  ) s

:=  k :=  rows P .. ( )

    





 Apparent velocity

vapp velapp

velapp errvapp + velapp errvapp −

cent_time

(18)

    











 Back - azimuth

az backp

deg

backp errback + deg backp errback −

deg

cent_time

(19)

  .



 .



  .



 .







 .



 .



 .



 .





Input test values and solution

x-slowness component

y-slowness component

P

T

( ) 〈 〉



sloy

P

T

( ) 〈 〉



, slox

Riferimenti

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