Geotisica Colombiana pp.71-73 Bogota, D.C.N° 7 dieiembre de 2003 ISSN - 0121 - 2974

True amplitude migration in constant
gradient velocity media

LUIS ANTONIO CASTilLO LOPEZ

Protesor Asistente Universidad Nacional de Colombia, Curso Posgrado de Geofisica

tacultad de Ciencias. Departamento de Geociencias, Grupo de Investigaci6n en Geofisica

e-mail: lacastillol@unal.edu.co

RESUMEN

EI algoritmo de Kirchhoff con trazado de rayo se basa en una funcion de velocidad con stante. Esta solucion incorpora informa-

cion de trayectoria de rayos dentro del algoritmo, el cual puede extenderse a un medio de velocidad variable. El modelo de ve-

loci dad incluye la localizacion de reflectores e interval os de velocidades entre dos reflectores adyacentes, generados a traves

del analisis de velocidad, pero importante en el procesamiento sismico tradicional. Este proceso puede alcanzarse al aplicar un

operador de funcion peso de apilado por difraccion, cuyo desarrollo teorico se bas a en la integral de migraci6n tipo Kirchhoff.

Al escoger el propio peso para el peso de apilado de datos, el resultado del proceso de migraci6n es una seccion sismica donde

se recupera la amplitud y su valor es proporcional al coeficiente de reflexion, lIamado rnigracion con amplitudes verdaderas

(recuperacion de amplitudes verdaderas).

PALABRAS CLAVE: RECUPERACION DE AMPLITUDES VERDADERES, PSDM, MEDID CON GRADIENTE DE VELOCIDAD.

ABSTRACT

The kirchhoff conventional algorithm using ray tracing is based on a constant velocity function. This solution, where is incor-

porated ray path information into conventional algorithm, can be extended to a variable velocity medium. Velocity models in-

clude the locations of reflectors and interval of velocities between any two adjacents reflectors that are generally generated

through the velocity analyses, process important in traditional seismic processing. This process could be achieved by applying

a weighted diffraction stack operator, which theoretical development is based on a Kirchhoff type migration integral. By choo-

sing the proper weight for stacking the data, the result of the migration process is a seismic section where the amplitude are re-

covered and its value are proportional to the reflection coefficient, the so-called true-amplitude migration (True Ampiltude

Recovering). Migration process was applied into different models. the objetive is to reproduce an accurate picture of the sub-

surface model from the data registered in areas where the velocity changes. The migration process was obtained after trying se-

veral testes made with different weigths.

KEY WORDS: TRUE AMPLITUDE RECOVERY, PSDM. GRADIENT VELOCITY MEDIA

INTRODUCTION
By incorporating ray path information into conventional migration
algoritm, the solution can be extended to variable velocity
medium. The rays are traced from each depth point in the survey of
each source-geophone location in order to determine the difraction
traveltime surface difraction (Huygens surface). Then, the
weighted diffraction stack operator could be used to migrate the
seismic data. In this paper we present a fast algorithm that is able to
recover true amplitude in depth migration process, by considering

Manuscrrto recibido para evaluacion el15 de junio de 2003.
Articulo aceptado para publicae ion por el comne Editorial el 36 de octubre de 2003.

GfOFislCA CDLOMBIANA. 7, DICIEMBRE DE 2003

a more particular situation when the seismic velocity model is re-
presented by a linear distribution variable with the depth.

This kind of model is important for simulating many situations
in the seismic exploration. For instance, variations of the velocity
with depth, inversion velocity (negative gradient) and increasing
of velocity in a short layer (positive gradient).

MODELLING
Asymptotic ray theory provide useful extrapolating techniques for
modelling, migration and inversion, using ray theory of coordi-
nates tracing equations which determine the second derivatives of
the traveltime curve. This is desired in order to determine geome-
trical spreading, and then true amplitude recovery. In order to know

71



LUIS ANTONIO CASTILLO L6PEZ

Ray Irajelory with positive gradient velocity

0.5

E
~1.5

!

2.51-------

3L- -'::-- ----'- --,J- -'- ---'
o 0.5 1.5 2.5

Offset (km)

0.5

E
c
-E 1.5

!

2.51------- ...........

3l- -----''-- -' ---'-- -'- ----l

o 0.5 1.5 2
Offset (kml

Figure I. Models for differents gradients in a configuration common shot. Gradient
positive (left) and gradient negative (rigth),

the stability of the diffraction stack migration algorithm, we have
applied it to a set of seismic data in a common-shot configuration.

DIFFRACTION STACK OPERATOR
After data input, two-dimensional diffraction stack operator can be
written following the expression given by Urban (1999), based on
Martins et al. (1998)

72

Seismogram

Distance
o 0 5 (Km) I 1.5 2

I -i------=+=~~~_rT'-i-r-,....,...,~.,....,..-,--L----___+

E 0.5 -I------+-++-I-++-;.+:~~F.l=m"!-'l-+-----____+
f.==

O-l------J-LLLLLLLLLj...LJLULU--'--'--'--,------t

Ray trajectories in constant gradient velocity media with commom shot configuration
o r-~-----''''-'-"T'T'rr-rT'CrroCT>--rT..--.rrr~~~---'

vO=2kml
0.2

I
;S 0.4
c.
OJo

0.6

0.8L--~-~~~~-~-~-~-~-~----J
o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Offset (km)

Figure 2. Ray trajectory model(lower) and synthetic seismogram(upper).

v(M ,t)= rb- fd~w(~,M)~ U (~,t+tD)
v21t A

(1)

with the source and receiver pairs (S,G) described by the vector
parameter and the traveltime curve defined by all the points of on
the earth surface, and a point M in depth. ar~U (~, t +,D ) is the an-
ti-causal half-time derivative (operator that corresponds in the fre-
quency domain to the filter.J -iw) of the seismic trace U (~,t + 'D)
registered on the geophone position G( ~), and it represents the
principal component of the seismic primary reflected wavefield.
The result of the integral (I) is put at the point M into the model.
Providing that we call depth difraction stack migration, with all
lines summed along the stacking line and multiplied by the true
amplitude (weight function) w(~,t-'D) and summed, then the
result of migration process is proportional to the reflection coeffi-
cient recovered.

2.5

DIFFRACTION TRAVELTIME STACK CURVE
Being known the macrovelocity model with a distribution of ve-
locity varying linearly with depth the diffraction traveltime stack
curve can be built, for the diffraction stack processing

v(z)=vo + gz (2)

Where vo is the velocity near the earth surface, g is the gradient
of the velocity function in the direction of the vertical axis z, in
depth. This kind of velocity function is the first approximation to
be considered when investigating the regional variations of ve-
locity for most sedimentary rocks (lapsen, 1993).

GEOFiSIGA GOLOMBIANA, 7, OIGIEMBRE DE 2003



TRUE AMPLITUDE MIGRATION IN CONSTANT GRADIENT VELOCITY MEDIA

1.9,----,------,-----,,---,---r----,--,.--....,

1.85r········· ,........... T···········, L ;..............,............."... ~
~ 1.8 r············ .j ,........... , , ,...........w, ,..,?<. , ~
~ 1.75 r············,··········· "',....... ; ; ; ····:<'A::···············L ~
;:: 1.7 r···········+···············f."-:~~==~4:···w! .]. ~

1.65 r···········t··········· T··········· ; ,........... .I. ; ;.... ~

0.5 1.5 2.5
Distance (Km)

3.5 4.5

Figure 3. Traveltime difraction and traveltime reflection curve for a depth point.

Inside such model with linear variation of velocity, the ray tra-
jectory is circular and the traveltime is given as solution of the in-
tegral (Bleistein, 1986)

I tan ( a~) I
t=-In -In(B,BG)

g tan(a~) g
(3)

being B, = 1 + (g2p + 2g(Ji) / 2cco. All the values of this result
are known; Ii and e, are the circular trajectory and parameter ray,
respectively.

COMPUTATIONAL EXAMPLES
The seismic model consists of by a curved reflector bellow an inho-
mogeneous domo with constant gradient velocity medium, where
the near surface velocity is 2.0 kmls (varying with the depth), and its
gradient is 1.25/s. The shot position is 0.4 km on the left, with 0.5 km
and 1.45 km for the near and far offset, respectively. The source do-
minant frequency is about 70 Hz, and the sample interval is of 2.0
ms. In the Figure 2 are represented the tracing rays(lower) and the
synthetic seismogram(upper) for a asymmetric common shot confi-
guration. The seismic data is noise free. In the figures 3 and 5 we

GEOFislCA COLOMB lANA. 7, DICIEMBRE DE 2003

have the image and depth migrated seismic data, after appliying of
the diffraction stack operator to the respective set of input seismic
data. It is important to note that the proposed algorithm provides a
good image of the target reflector, and it is able to be used for mi-
grating seismic data in constant gradient velocity media.

CONCLUSIONS
In this paper, was develop and tested the diffraction stack mi-
gration algorithm, applying it to a set of common-shot seismic data,
syntheticaly generated by the ray theory for a constant gradient ve-
locity medium. The algorithm was tested providing us a good
image of the target reflector. This result is very important in sense
that the proposed algorithm is fast, stable and suitable to be used for
true amplitude depth migration.

REFERENCES
BLEISTEIN, N., 1986, Two-and -one-half dimensional in-plane wave pro-

pagation. Geophysical Prospecting, 34, 686-703.

JAPSEN, P., 1993, Influence of Lithology and Neogene Uplift on Seismic

Velocities in Denmark: Implications for Depth Conversion of Maps.

The AAPGB, 77, n. 2, 194-211.

SCHLEICHER, 1.; TYGEL, M.; and HUBRAL, P., 1993, 3-D True-Amplitude
Finite-Offset Migration. Geophysics, 58(8): 1820-1830.

SCHNEIDER, JR. W. A.; RANZINGER, K. A.; Balch, H.; and Kuse, C, 1992,
A dynamic programming approach to first arrival traveltime computa-

tion in media with arbitrarily distributed velocities. Geophysics, 57, n.
1,39-50.

URBAN, 1.,1999, Two-Dimensional True-Amplitude Migration and Intro-

duction to 2.5-D Case. Master Thesis. Federal University of Para, Bra-
zil. (In Portuguese).

ACKNOWLEDGMENTS
We would like to thank the seismic group of the Geophysical
Institute, Charles University, Prague, Czechoslovakia, for making
available the ray tracing software SEIS88, and the UNIVER-
SIDAD NAC10NAL DE COLOMBIA for supporting the author.

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