NAME
swdrt - generalized Semblance Weighted Discrete Radon
Transform from (t,x) to $( tau ,p)$ and (with -R option)
from $( tau ,p)$ to (t,x) space.
SYNOPSIS
swdrt [ -Nfile_in ] [ -Ofile_out ] [ -pminpmin ] [ -pmaxpmax
] [ -xapxap ] [ -t1t1 ] [ -t2t2 ] [ -t3t3 ] [ -t4t4 ] [
-f1f1 ] [ -f2f2 ] [ -f3f3 ] [ -f4f4 ] [ -lf3lf3 ] [ -lf4lf4
] [ -zrefzref ] [ -sigma1sigma1 ] [ -sigma2sigma2 ] [ -mxmx
] [ -wsembwt ] [ -minliveminlive ] [ -nxtapernxtaper ] [
-gammagamma ] [ -cccc ] [ -scsc ] [ -plpl ] [ -maxitermax-
iter ] [ -L ] [ -P ] [ -H ] [ -nlsw ] [ -R ] [ -C ] [ -I ] [
-V ] [ -? ]
DESCRIPTION
swdrt takes irregularly sampled seismic (t,x) gathers and
forward transforms them into the generalized $( tau ,p)$
domain. The user models the seismic data with a series of
linear, parabolic or hyperbolic curves which are fit to the
data in a least squares sense by using an iterative time
domain constrained conjugate gradient technique. While con-
siderably more expensive than unweighted transforms (such as
program radonf) by careful use of semblance weighting, one
can deal effectively with spatially aliased data.
swdrt gets both its data and its parameters from command
line arguments. These arguments specify the input, output,
the temporal design window, the type of transformation to be
used, the minimum and maximum ray parameters to be modeled,
the number of parameters (curves) to fit, tapers, and mode
of regularization and/or semblance weighting if desired.
Command line arguments
-N file_in
Enter the input data set name or file immediately after
typing -N unless the input is from a pipe in which case
the -N entry must be omitted. This input file should
include the complete path name if the file resides in a
different directory. Example
-N/export/data2/china/cdp.gather tells the program to
look for file 'cdp.gather' in directory
'/export/data2/china'.
-O file_out
Enter the output generalized $( tau ,p)$ domain data
set name or file immediately after typing -O. This
output file is not required when piping the output to
another process. The output data set also requires the
full path name (see above).
-R If present, calculate the reverse transfrom from $( tau
,p)$ to (t,x) space.
-C If present, place the x origin of the signed distances
to be at the center of the gather.
-I If present, interpolate dead traces during the reverse
transform (trace distances must be correct).
-t1 t1
Begin roll-in time of analysis window in ms (Default =
0 ms).
-t2 t2
End roll-in time of analysis window in ms (Default
t2=t1).
-t3 t3
Begin roll-out time of analysis window in ms (Default
t3=t4)
-t4 t4
End roll-out time of analysis window in ms (Default:
Last sample of trace).
-pmin pmin
Minimum ray parameter to be modeled in units of ms/m
(ms/m**2 for the -P option). (Default, pmin = 0.)
-pmax pmax
Maximum ray parameter to be modeled in units of ms/m
(ms/m**2 for the -P option). (Default, pmax = 0.)
-xap xap
Nominal seismic aperture in m. For a common shot or
midpoint gather, this would be the cable length. (No
Default).
-f1 f1
Frequency (Hz) at which we begin roll in of a Hamming
zero phase band pass filte r (default = 5Hz)
-f2 f2
Frequency (Hz) at which we end roll in of a Hamming
zero phase band pass filter (default = f1).
-f3 f3
Frequency (Hz) at which we begin roll out of a Hamming
zero phase band pass fil ter (default = f4).
-f4 f4
Frequency (Hz) at which we end roll out of a Hamming
zero phase band pass filter. The number of ray parame-
ters, p, will be calculated such that the Nyquist cri-
teron is met for the values of (pmax-pmin), xap, and
f4. The cost increases with the value of f4. (default =
Nyquist)
-lf3 lf3
Begin roll out frequency (Hz) used in generating a low
frequency constraint in the $( tau ,p)$ transform. This
value will also low pass filter the data prior to any
semblance weight calculation. (Default: lf3=f3).
-lf4 lf4
End roll out frequency (Hz) used in generating a low
frequency constraint in the $( tau ,p)$ transform. This
value will also low pass filter the data prior to any
semblance weight calculation. (Default: lf4=f4).
-L Enter the command line argument '-L' to use linear
curves in the data model (No default. Other options are
-P and -H).
-P Enter the command line argument '-P' to use parabolic
curves in the data model. (No default. Other options
are -L and -H).
-H Enter the command line argument '-H' to use time
invariant hyperbolic curves in the data model (No
default. Other options are -L and -P).
-zref zref
Reference depth for time invariant hyperbolic curves if
-H option selected. Hyperbolae defined as in Foster
and Mosher, Geophysics, 1992 . Default: zref=xmax).
-sigma1 sigma1
If present, reject data whose local semblance $sigma
bar <= sigma sub 1 = sigma1$ in the frequency band
(f1,f2,lf3,lf4). See algorithm description below. This
weighting will accentuate coherent/continuous events
and suppress incoherent events. (Default: sigma1 =
0.05)
-sigma2 sigma2
If present, pass data whose local semblance $sigma bar
>= sigma sub 2 = sigma1$ in the frequency band
(f1,f2,lf3,lf4).. See algorithm description below.
This weighting will accentuate coherent/continuous
events and suppress incoherent events. (Default: sigma2
= 0.10)
-nlsw
If present, use a nonlinear semblance weighting tech-
nique modified after the work by Yilmaz and Taner
(1994).
-mx mx
Half width, $ m sub x $, in traces, of the running win-
dow semblance calculation. See algorithm description
below. (Default: mx = 5.
-w w Half length, $w sub t =K DELTA t$, in ms, of the run-
ning window semblance calculation. (Default: w = $5 *
DELTA t$)
-cc cc
Convergence critereon. Terminate iteration procedure if
the error between successive reconstructions ${d sup
(k+1) - d sup (k)} over d sup (0) $ < cc, where $ d sup
j $ is the reconstructed data at the j-th iteration.
(Default: cc=.01).
-pl pl
Percent of 'idealized' Lagrange multiplier used in the
linearly constrained methods described by Nemeth and
Marfurt (1995). (Default: pl=100.)
-sc sc
Constraint scaling factor, s, desribed in Nemeth and
Marfurt (1995). Small values (eg sc=0.1 produce a
broad constraint mask while large values (eg sc=.90)
produce a highly focussed (and possibly overcon-
strained) constraint mask. (Default: sc=.25).
-gamma gamma
Parameter used to relax the constraints at each itera-
tion. For linearly constrained method $epsilon sup (k)
= gamma sup k epsilon sup (0) $ where $epsilon sup (0)$
= pl described above. For nonlinear semblance weight-
ing method, $sigma sub 1 sup (k) = gamma sup k sigma
sub 1, and sigma sub 2 sup (k) = gamma sup k sigma sub
2 $. Described in Nemeth and Marfurt (1995). (Default:
gamma=0.95)
-nxtaper nxtaper
Spatial taper length (traces). This taper is applied to
both the start and the end of the analysis window.
(Default: nxtaper =0, or no taper)
-minlive minlive
Enter the minimum number of live traces accepted in a
gather. Gathers having less than minlive live traces
will be zeroed out and flagged dead. (Default: minlive
= 3).
-maxiter maxiter
Enter the maximum number of iterations allowed in an
attempt to reach the convergence critereon given by the
-cc option described above. (Default: maxiter = 5).
-V Enter the command line argument '-V' to get additional
printout.
-? Enter the command line argument '-?' to get online
help. The program terminates after the help screen is
printed.
ALGORITHM DESCRIPTION
The Generalized $( tau ,p)$ Transform:
We define the time domain, non-orthogonal, generalized for-
ward and inverse $( tau ,p )$ transform pairs by:
$m ( tau ,p sub n ) = sum from j=1 to J { d( tau -p sub n theta sub j , x sub j )} $
and
$d (t,x sub j ) = sum from n=1 to N { m( t +p sub n theta sub j , p sub n )} $,
where:
$p sub n = $ the nth apparent moveout in the x direction,
$x sub j = $ the position of the jth trace ,
$theta (x sub j ) = {x sub j } $ for a linear transform,
$theta (x sub j ) = {x sub j } sup 2 $ for a parabolic transform,
$theta (x sub j ) = [{x sub j } sup 2 + {z sub ref } sup 2 ] sup 1/2 $ for a hyperbolic transform, and
$ z sub ref $ = an appropriate reference depth.
The Semblance weighted $ ( tau
For spatially aliased data, it is useful to consider weight-
ing the input data u(t,x) by weights $w ( tau ,p sub n , x
sub j ) $ proportional to the time averaged semblance in a
running window ranging from $-m sub x$ to $+m sub x$ and
from $-K DELTA t$ to $+K DELTA t$ along the same summation
curves used to calculate $m ( tau ,p sub n ) $:
$ m sub sigma ( tau , p sub n ) = sum from j=1 to j=J w[ sigma bar ( tau -p sub n x sub j , x sub j , p sub n )] d( tau -p sub n x sub j , x sub j )
$,
where
$w ( sigma bar )$ is defined as:
$w ( sigma bar ) = 0.$ if $ sigma bar <= sigma bar sub 1 $,
$w ( sigma bar ) = 1 over 2 [ 1 + cos({ sigma bar - sigma bar sub 1 } over {sigma bar sub 2 - sigma bar sub 1} pi )]$, if $ sigma bar sub 1 < sigma bar < sigma bar sub 2 $,
$w ( sigma bar ) = 1.$ if $ sigma bar >= sigma bar sub 2 $, and
$ sigma bar ( tau - p sub n x sub j )={ sum from {k=-w/ DELTA t} to {k=+w/ DELTA t} [ sum from {m=-m sub x } to {m= +m sub x } d ( tau -p sub n x sub j+m ,x sub j+m )] sup 2 } over {(2m sub x +1) sum from {k=-w/ DELTA t} to {k=+w/ DELTA t} sum from {m=-m sub x } to {m=+m sub x } [d ( tau -p sub n x sub j+m , x sub j+m )] sup 2 }$
Orthogonalization via Least Squares:
The two transforms above are in general non-orthogonal. We
can define the forward and inverse transforms symbolically
by:
$ m = L sup T W sub sigma sup T d $, and
$ d = W sub sigma L m $,
where
L = the interpolation matrix, and
$ W sub sigma $ = the semblance weighting matrix.
We wish to define a $( tau , p) $ transform $m ( tau ,p)$
that will reconstruct d(t,x) in a least squares sense. Mul-
tiplying the equation for d above by $ L sup T W sub sigma
sup T $ we obtain:
$ L sup T W sub sigma sup T d = (L sup T W sub sigma sup T W sub sigma L + epsilon sup 2 C sup T C) m $,
where C is a constraint matrix and $epsilon sup 2$ is a Lagrange multiplier.
Numerical experimentation shows that $ W sub sigma sup T W sub sigma = I $ where I is the identity matrix.
Solving for m we obtain:
$ m = ( L sup T L + epsilon sup 2 C sup T C ) sup {-1} L sup T W sub sigma sup T d $.
We solve this system of equations in a reasonably efficient manner by exploiting a conjugant gradient technique.
REFERENCES
Nemeth, T., and Marfurt, K. J., 1995, Antialias Discrete
radon transforms: Geoscience Research Bulletin F95-G-30.
Marfurt, K. J., Schneider, R. V. and Mueller, M. C., 1994,
Challenges in seismic processing of irregularly sampled,
finte aperture aliased data using discrete Radon $( tau ,p)
$ transforms: APR Geos. Res. Bul. F94-G-14.
Marfurt, K. J. and Cottle, D. A. 1994, A comparison of
coherency weighted $( tau ,p) $ filtering: Application to
poststack and common offset data: APR Geos. Res. Bul. F94-
G-16.
Yilmaz, O. and Tanner, T. 1994, Discrete plane wave decompo-
sition by least-squares method, Geophysics: 59, 973-982.
Stoffa, P. L., Buhl, P. Diebold, J. and Wenzel, F., 1981,
Direct mapping of seismic data to the domain of intercept
time and ray parameter - A plane-wave decomposition: Geophy-
sics, 46, 255-267.
BUGS
No bugs known at present.
SEE ALSO
radonf,radonr,rmmult,swfilt3d
AUTHORS
Kurt J. Marfurt (EPTG, Tulsa) and Tamas Nemeth (Univ. of
Utah). (1995)
DEVELOPMENT AGREEMENT
Seismic Signal Analysis (D-95-2518). 3rd Quarter Deliver-
able. Thank you for your support!
COPYRIGHT
copyright 2001, Amoco Production Company
All Rights Reserved
an affiliate of BP America Inc.
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