Plain Waves HowTo

Geophysics Department TU Clausthal

Applied Geophysics : Seismics

Plain Waves in Layered Media

Applet ( in separate window, ca. 535 x 340 Pixel )

The applet simulates transmission and reflexion of plain waves in a layered medium.

The sequence of transmission and reflexion coefficients of all interfaces

is calculated in the time domain,

including thin "lamellas" of constant travel time δT ≤ Δt/2

( Δt = sampling interval DEL-T = 1 / F-SMP ),

representing layers of constant velocity gradient or density gradient.

Next the complex response matrix of the model

is calculated recursivly in the frequency domain for every source-receiver-combination,

and

is multiplied by the Fourier transform of the source signal.

The inverse transformation of these products back to the time domain results in the seismic traces recorded at the receivers.

- Table of Content -

Comments :

Model + Recording Configuration
Layers with Gradients
Response Matrix

HOWTO :

Dialogue Window :
Recording Configuration ( SHT / GEO CONFIG )
Model Parameters ( U.H.S / LAY_1 PARAM )
Time Scale + Fourier Transformation ( TIME SCALE + LEN FFT )
Source Signal ( SOURCE SIGNAL )
Graphic Windows :
Model ( DISP MODEL )
Seismic Traces ( DISP TRACES )

Download

- Comments -

Model + Recording Configuration :

The Model is composed of

1 ... 30 parallel layers¹) ( LAY_1 ... LAY_30 ) `

between

an upper halfspace ( UHS, Z < 0 )

and

a lower halfspace ( LHS ).

Both halfspaces are homogenious and of infinite extension, i.e.

reflexions occur

at the surface of the model ( TOP, Z = 0 ),

at the bottom of the model ( BOT, = boundary to the lower halfspace )

and

at interfaces of layers / "lamellas" inbetween.

¹) The number of reflexion coefficients ( layer and "lamella" boundaries ) is limited to 4096.

The applet simulates three different recording configurations :

1 shotpoint and 2 receiver positions ( 1 SHT + 2 GEO ),

with possible shot positions

in the upper halfspace ( UHS ),

at the surface of the model ( TOP ),

at the bottom of the model ( BOT )

or

in the lower halfspace ( LHS ),

and receivers

at the surfac ( TOP )

and / or

at the bottom of the model ( BOT ).

( For SHT = UHS / LHS the shotpoint lies inside the respective halfspace, but "close" to the boundary :

the amplitude of the signal reflects the transmission loss of the boundary

without traveltime difference to SHT = TOP / BOT. )

Two additional configurations simulate VSP ( Vertical Seismic Profiling ) recordings :

1 shotpoint and 6 receiver positions ( 1 SHT + 6 GEO )

includes additional traces of up to four borehole receivers ( GEO = G_1 ... G_4 ),

6 shotpoints and 1 receiver ( 6 SHT + 1 GEO )

calculates the traces, recorded at the surface ( TOP ) or at the bottom ( BOT ) of the model,

resulting from up to four additional shotpoints in a borehole ( SHT = S_1 ... S_4 ).

The equaly spaced geophone / shot array can be posioned arbitrary between surface and bottom of the model, the length of the array ( ≤ Z(BOT) - Z(TOP) ) can be adjusted.

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Gradients :

Layers with velocity and / or density gradients ( grad_V, grad_ρ ) are replaced by an equivalent sequence of "thin", homogenious "lamellas" of constant traveltime δT ≤Δt/2,
where Δt is the selected sampling intervall for the source signal, the Fourier&nbs;transform ( FFT ) and the seismic traces.

The one way traveltime of a layer with given thickness, starting and ending velocities is calculated by integrating

V(z) = V(Z_0) + grad_V × ( z - Z_0 )

=> T(z) = log{1 + grad_V × ( z - Z_0 ) / V(Z_0) } / grad_V

and subdevided to N steps δT ≤ Δt/2,
where δT is chosen to preserve the traveltime ( = N × δT ) and the thickness ( = sum of "lamella" thicknesses δZ(i), i = 0 ... N-1 ) of the layer.

Thickness δZ and velocity V of a "lamella" are determined by

δZ(i) = exp(i × δT×grad_V) × { exp(δT×grad_V) - 1 } × V(Z_0) / grad_V and V(i) = δZ(i) / δT

The density ρ corresponds to the arithmetic mean

ρ(i) = ( ρ(Z(i)) + ρ(Z(i+1)) ) / 2 with &rho(z) = ρ(Z_0) + grad_ρ × ( z - Z_0 ) or ρ(i) = ρ(Z_0) = const., if grad_ρ = 0

Layers with constant velocity V_0 and a density gradient grad_ρ ≠ 0 are replaced by "lamellas" of constant velocity and thickness

V(i) = V_0 = const and δZ(i) = V_0 × δT = const

( the density ρ(i) corresponds to the arithmetic mean as above )

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Response Matrix :

A special solution of the elastic wave equation for the displacement w in a perfectly elastic, isotropic and homogenious medium is a plain harmonic compressional wave travelling in Z direction :

w(z,t) = D × exp{ jω × ( t - z / V ) } + U × exp{ jω × ( t + z / V ) }

with angular frequency: jω = j × 2 × π × f, ( f = frequency, j^2 = -1 )
and amplitudes: D of a wave travelling downward ( = pos. Z direction ), and; U of a wave travelling upward ( = neg. Z direction ) .

The amplitudes D and U are frequency dependent and can be derived from boudary conditions.

At an interface z = Z_0, separating two different media 1 and 2, two conditions have to be satisfied for all times t :

the continuity of displacement w(z,t)

w_1(Z_0,t) = w_2(Z_0,t)

and the continuity of normal stress σ(z,t)

σ_1(Z_0,t) = σ_2(Z_0,t).

Defining the accoustic impedance

I = ρ × V,

this leads to

a reflexion coeffizient

R = ( I_1 - I_2 ) / ( I_1 + I_2 ) |R| ≤ 1

and a transmission coefizient

T = 2 × I_1 / ( I_1 + I_2 ) = 1 + R.

The above boundary conditions, applied to the lower boundary Z(k) of layer ( or "lamella" ) k of

thickness H(k) = Z(k) - Z(k-1),

constant velocity V(k),

constant density ρ(k), hence

acustic impedance I(k) = ρ(k)×V(k)

relates the complex amplitudes

D(k+1) and U(k+1) in layer k+1, acustic impedance I(k+1), below the interface Z(k)

to

D(k) and U(k) in layer k above the interface.

This relationship is described by a 2×2 matrix, the so called layer matrix m(k) :

The repeated application of this operation for k = 0 ... K-2,

i.e. to all of the K-1 interfaces between the K layers / "lamellas" ( 0 ... K-1 )

relates

the lower halfspace ( D(K-1), U(K-1) )

the upper halfspace ( D(0), U(0) )

via the 2×2 matrix M ( = response matrix ),

corresponding to the product of layer matrices m(0)×m(1) ... m(K-2) :

Example :

Assuming a wave of amplitude 1, travelling downward in the upper halfspace

D(0) = 1

and

U(0) = Rpp

( response of the model, refexions including all multiples travelling upward ),

and in the lower halfspace

D(K-1) = Tpp

( all waves travelling downward, leaving the model )

and

U(K-1) = 0

( no waves travelling upward, i.e. coming back )

leads to

i.e.

and

with

Det(M) = I(0) / I(K-1)

( = det(m(0))×det(m(1))× ... ×det(m(K-2)), with det(m(k)) = I(k) / I(k+1) )

and

a factor 1 / M_22, common to both functions and describing the multiples.

The inverse Fourier transform of the complex refectivity Rpp(jω) and transmissivity Tpp(jω),

calculated for all frequencies n × D_FRQ = n / (L-FFT×Δt), n = 0 ... L-FFT-1

leads to the impulse responses of the model

to a source in the upper halfspace ( UHS ),

recorded

at the surface ( TOP, Rpp ) and

at the bottom ( BOT, Tpp )

of the model.

The seismic traces for the source signals implemented are calculated by multiplying the corresponding Fourier transforms by the response functions in the frequency domain before transforming back to the time domain.

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- HOWTO -

Dialogue window :
( Screenshot )

The applet dialogue allows to chose

a recording configuration ( SHT / GEO CONFIG ),

the parameters of the upper halfspace and the first layer ( U.H.S / LAY_1 PARAM ),

the time scale and length of transformation for the calculation ( TIME SCALE + LEN FFT ),

and

type and parameters of the source signal ( SOURCE SIGNAL ).

At the lower edge of the window ( backgrond = orange )

the graphic displays of the model ( DISP MODEL ) and of the seismic traces ( DISP TRACES )

and

the listing of model parameters ( LIST MOD ) to the screen / the JAVA-Console

is started.

At the upper edge

INFO

displays informations to the actual cursor postition,

HELP

displays hints to actually possible mouse interactions

in both graphical windows, and

RESET MOD

resets the model to its starting configuration ( 1 layer between UHS and LHS ).

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Recording Configuration :
( 3 Screenshots )

The screenshots show the recording configurations implemented.

1 SHT + 2 GEO :
one shotpoint
in the upper halfspace ( UHS ),
at the surface of the model ( TOP ),
at the bottom of the model ( BOT )
or
in the lower halfspace ( LHS )
and max. 2 geophones,
at the surface
and/or
at der bottom of he model.

1 SHT + 6 GEO :
as above
+ additionally an array of 4 equally spaced borehole geophones ( G_1 ... G_4 ).

6 SHT + 1 GEO :
max. 6 shotpoints
in the upper halfspace or at the surface,
and / or
at the bottom of the model or in the lower halfspace,
+ additionally an array of 4 equally spaced shotpoints in a borhole ( S_1 ... S_4 ).
and 1 geophone
at the surface or at the bottom of the model.

Position and extend of the shot / geophone array are adjusted in the graphic display window DISP MODEL using the mouse.

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Model Parameters :
( 3 Screenshots )

Two sets of typical elastic parameters can be preselected for the upper halfspace and the first layer.

U.H.S :	FREE SURF	=> V =	333 [m/s]	ρ =	0.0013 [g/cm^3]
	WATER	=> V =	1500 [m/s]	ρ =	1.0 [g/cm^3]


LAY_1 :	WATER	=> V =	1500 [m/s]	ρ =	1.0 [g/cm^3]
	SEDIMENT	=> V =	2000 [m/s]	ρ =	2.0 [g/cm^3]

For both layers USER SEL allows to adjust the parameters ( as for all other layers of the model ) in the graphic display DISP MODEL using the mouse.

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Time Scale + Length of FFT :
( Screenshot )

The time scale of calculations, source signal and seismic traces can be fitted to the model dimensions by selecting

a sampling interval Δt from 0.125 [ms] to 50 [ms], or

a corresponding sampling frequency from 8 [kHz] to 20 [Hz].

The parameter L-FFT ( = 2^n, => trace length ) is selectable from 256 [Smp] to 32768 [Smp] to fit the calculations

to the total number of layers ( incl. "lamellas" of gradient layers ),

to the length of the source signal ( i.e. type SWEEP ),

and

to the in general very low decay rate of amplitudes of multiple refexions

( no material damping, no geometrical spreading ).

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Source Signal :
( Screenshot )

In the menue area SOURCE SIGNAL the type of source signal is selected :: a single spike
or: one of 9 time functions,; with a charackteristic frequency F_SIG ( EXP*SIN, RICKER ),; or characteristic frequencies FRQ_1 and FRQ_2 ( KUEPPERS, SWEEP )
determined by the sampling interval Δt.
These frequencies are adjusted: raw by F_SCAL x 5 ... x 0.2, and; fine by F_TUNE INC / DEC
( i.e. tuning to fit the signal to the thickness of a thin lamella ).

The parameter values F_SIG and evtl. ALPHA or FREQ_1 and FREQ_2,
as well as the length of the source signal L-SIG, measured as [Smp] and [ms] / [sec],
are displayed.

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Graphic Display :
( Screenshot )

DISP MODEL starts the graphic display of model and recording configuration,

a plot of velocity V ( VEL ) and

density ρ ( RHO )

versus depth Z. ( see window MODEL )

Additionally

COF includes the reflexion coefficients,

IMP includes the acustic impedance, and

SIG includes a depth dependent display of the source signal.

LIST MOD

lists the parameters of the model and,

enabled by +COF, the reflexion coefficients

to the screen / to the JAVA-Console.

DISP TRACES starts the graphic display of

seismic traces,

reflexion coefficients ( COF ) and

source signal ( SIG ),

plotted versus time. ( see window TRACES )

The menue item RAW TRACES AS RECORDED ... modifies the display of seismic traces :

RAW TRACES ...

shows the traces as recorded at the geophone positions,

SUPPRESS DIRECT WAVE

suppresses the direct waves, to improve the amplitude resolution for primary reflexions and multiples,

ADD DIR WAV TRAVEL TIME

shifts each trace according to the travel time of the corresponding direct wave, to align reflexions travelling in opposite direction,
(i.e. for a shot at the surface or in UHS, reflexions from interfaces below are aligned ),

SUP DIR + ADD TRV TIME

combines effects of SUPPRESS DIRECT WAVE and ADD DIR WAV TRAVEL TIME to improve the amplitude resolution.

If SIG = SWEEP is selected :

CORR

crosscorrelates the seismic traces and the source signal.

INFO and HELP ( upper edge of the applet dialogue ) affect both graphic windows :

INFO displays informations concerning the actual postion of the cursor,

HELP shows hints to actually possible mouse interactios.

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Window MODEL :
( Screenshot,
  U.H.S,  Z < 0 [m]; :  V=333 [m/s]i   ρ=0.0013 [g/cm^3]
  LAY_1,  Z = 0 ... 150 [m] :  V=1500 [m/s]   ρ=1.0 [g/cm^3]
  L.H.S,  Z > 150 [m] :  V=2500 [m/s]   ρ=2.5 [g/cm^3]
  recording config. = 1 SHT + 6 GEO,   borehole receivers :  Z = 30, 60, 90 and 120 [m]
  source signal :  KUEPPERS_2,  L-SIG = 60 [Smp] = 120 [ms] )

The window shows a plot versus depth Z of

velocity ( VEL ),

density ( RHO ),

and additionally

acustic impedance ( IMP = ρ×V ),

reflexion coefficients ( COEFF, here 2 : TOP and BOT )

and

a snapshot of the source signal travelling in pos. Z direction

( SIGNAL,

travelling downward, travel time T = 0 [ms] => start of signal at Z = 0 [m],

snapshot : T approx. 61 [ms] => start of signal at Z approx. 92 [m] ).

The surface of the model ( TOP ) is marked red, additional interfaces ( here : BOT only ) are marked blue.

The recording configuration ( herer : 1 SHT + 6 GEO ) is displayed left to the model :

shotpoints as circles,

geophones as triangles

( active shot points / geophones as filled symbols, colors corresponding to trace display ).

Configuration of the Model :

The model displayed above shows the initial state of the applet ( or after RESET MOD ),

the number of layers ( ≥ 3, max. 32 ),

the dimensions

and

the elastic parameters

are modified and adjusted using the mouse.

!!! Hints concerning the mouse activities by activating HELP !!!,

!!! Informations to the actual position of the cursor ( HAND_CURSOR ) by activating INFO !!!

Elastic Parameters :

Using the left mouse button ( cursor symbol = LEFT_RIGHT_MOVE )

inside a layer

both velocities ( V_TOP + V_BOT ) or

both densities ( R_TOP + R_BOT )

of this layer are modified synchroniously,

close to an interface

V_TOP or R_TOP is adjusted at the upper boundary of a layer,

V_BOT or R_BOT at the lower boundary,

to implement

a velocity gradient ( V_TOP ≠ V_BOT ), and / or

a density gradient ( R_TOP ≠ R_BOT )

( except in UHS and LHS ).

Close to an interface ( cursor = LEFT_RIGHT_MOVE )

the right mouse button sets

the velocity / density of the neighbouring layer to the corresponding value of the actual layer.

Model Geometry :

Inside a layer ( except UHS, cursor = CROSSHAIR )

the right mouse button inserts a new interface.

Both new layers, above and below the new boundary, have constant velocities and densities, derived from the corresponding values of the original layer :

above V_LAY = V_TOP and R_LAY = R_TOP,

below V_LAY = V_BOT and R_LAY = R_BOT

( gray lines in the graphic display ).

If the mouse button is kept pressed ( => cursor = UP_DOWN_MOVE ),

the depth of the new interface can be adjusted,

or

the new boundary can be removed without modification of the original model by dragging it to one of the boundaries above / below.

Any interface marked blue ( i.e. except TOP ) can be moved vertikally ( cursor = UP_DOWN_MOVE ) :

using the left mouse button,

a layer is removed by dragging its lower boundary to its upper boundary ( or vice versa ), using the left mouse button,

and

using the right button

all layers below an interface can be moved simultanously preserving thicknesses and elastic parameters

( removal of the layer above the interface as explained above ).

!!! Removal of layers only, if the resulting number of layers ( incl. UHS and LHS ) ≥ 3 !!!

Zoom :

In the depth scale right to the model display

the left mouse button sets a vertical zoom range,

the right button resets the window.

Recording Configuration :

At both ends the array of geophones / shotpoints

is lengthened / shortened using the left mouse button,

is positioned using the right button,

or

it is positioned using either button in the middle part of the the array.

( cursor = UP_DOWN_MOVE )

Signal :

The startig depth of the signal ( marked orange, in panel SIGNAL )

can be moved ( cursor = UP_DOWN_MOVE ) using the left mouse button, and

can be set ( cursor = CROSSHAIR ) to the respective depth using the right mouse button.

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Window TRACES :
( 3 Screenshots,
  model and recording config. as in window  MODEL,
  source signal Kueppers_2,  pos. ampl. = compression,
  seismic traces :  pos. ampl. = ground displacement upward )

RAW TRACES AS RECORDED :

SUPPRESS DIRECT WAVE :

SUP DIR + ADD TRV TIME :

At the time axis left to the seimic section

the surface of the model ( TOP ) is marked red,

the bottom( BOT ) blue, and

the positions of the borehole geophones ( G_1 ... G_4 ) are marked cyan.

The time scale at the left edge of the display window

covers the time range - L-TRC/4 ... + L-TRC,

( here : -2048 [ms] ... +8192 [ms], L-FFT = 4096, Δt = 2 [ms] ),

the time window displayed in the seismic section is marked orange.

The zoom range ( orange ) can be

moved and / or lenthened / shortened using the mouse,

and

left to the time scale of the seismic section a zoom range can be

set using the left mouse button, and

reset using the right button.

In one of the headlines SIG, GEO=TOP, GEO=G_1 ...

( cursor = HANDCURSOR )

the display of amplitude and phase versus frequency for the respective trace can be started.

( In config. 6 SHT + 1 GEO : for SHT=TOP, SHT=S_1 ... )
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- Download -

Class and html files for a local installation of the applet are available as a zip file and as a tar.gz file.

More applets at : the author's Homepage

Rev. 10-sep-2006

Comments to Fritz Keller
( ned gschempfd isch globd gnueg )

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