PROCESS FDMIGR Document Date: 23 April 2001 Finite-difference Migration using the 45-degree algorithm The finite-difference migration technique is an effective way to handle many types of migration problems. It was developed and made popular by J. F. Claerbout at Stanford University. For most stack sections, finite- difference migration gives results comparable to other schemes; however there are assumptions and stability limitations which must be considered. For certain conditions, frequency domain (process FKMIGR) migration is more effective in resolving typical imaging and positioning problems. LIMITATIONS Steep Dips It is possible to add more terms to the finite-difference equation to obtain successively more accurate equations to deal with the steep dip problem. However, these schemes quickly become impractical to implement due to their cost. Further limitations on dip angle are imposed since the finite-difference method itself introduces errors. The equation used in FDMIGR is known as the 45-degree equation, and is capable of handling dips up to angles of 45 degrees with sufficient accuracy. A certain confusion exists regarding the meaning the meaning of the dips referred to in the 45 degree equation. This is not simply the dip of continuous reflectors. These are the dips included in all events of interest as seen in the F-K domain. A sharp fault, for example, contains dips up to 90 degrees, and the 45 degree algorithm will only properly migrate certain components, with increasing distortion at higher dips. The parameters in the algorithm are set to suppress those dips which are poorly imaged. Velocity Within the finite-difference equation there is no term to describe differences in velocity. Hence, a major assumption of the scheme is that velocity is constant throughout the section. In practice, it is sufficient for the velocity to vary slowly enough that it looks roughly constant within the effective "aperture" of the algorithm. This aperture can be thought of as a box whose time length equals one Tau-step size and whose spatial length equals the effective width of a point diffraction pattern. Boundary Effects Ideally, we would like to perform migration on all of space. But in the real situation, we can only migrate a finite section of the earth, so we must consider the effects of the imposed boundaries. The main consideration is for the sides of the section, where we normally think of the earth as simply ceasing to exist, and the events stopping. This view induces the mathematical equivalent of a vertical reflection coefficient, and events which are migrated towards it will be partly reflected back into the section. In order to suppress, or at least attenuate these undesirable events, a buffer zone, or pad, consisting of a number of traces, is inserted at both sides of the section. The traces are set to zero before migration, and the velocity is the same as the attached traces in the section. Studying the padded traces after migration can sometimes yield valuable information about events close to the edge of the section, especially if other data in the area is available. Comparison with FKMIGR a) Run Time One of the most practical considerations when deciding which migration scheme to use is the difference in cost. Depending on the values of certain parameters used, FDMIGR can run 3 to 4 times as long as FKMIGR. Clearly, if there is no advantage in data quality to be obtained, FDMIGR should not be used. FDMIGR must migrate from time zero, so it replaces the deep water delay with sufficient zeroes. The inserted zeroes are removed after migration so that the output traces will have the same delay as the input traces. Deep water delays do not affect the FDMIGR run time. b) Steep Dips Use of FDMIGR will produce inaccuracies if events are dipping by more than 45 degrees. FKMIGR, the frequency domain approach, migrates all dips with equal accuracy. c) Velocity In general, FDMIGR will perform better in the presence of velocity variations, although both methods assume that velocity is slowly varying. d) Stability While FKMIGR is very stable in almost all conditions, FDMIGR uses parameters which, if mis-used, can cause the migration equation to become unstable. It is also possible to set values for particular data sets in order to control noise on the output section, but you should have a good understanding of finite-difference migration first. In general, the default values will produce stable results. e) Noise Suppression All migration algorithms tend to suppress random noise and enhance coherent events. The result is that the output section will look more "mixed" than the input. The effect will be more prominent as the accuracy of the algorithm increases. For this reason, FKMIGR FKMIGR will generally look more mixed than FDMIGR, which will, in turn, look more mixed than a 15 degree algorithm. Some Important Parameters The parameter Rho is inserted into the expression for the discretization of the time derivative. This serves to counteract any potential growing waves from the expression for migration, as an explicit damping with time. It can be thought of as a "numerical viscosity". A value of Rho less than 1 reinforces stability. However, any deviation of Rho away from 1, by at most 1 percent, results in some loss of signal as well as noise. In the discretization of depth, the parameter Theta is introduced, with the most natural value being .500. If Theta = 0 is used, there is a tendency to overshoot on variations, whereas Theta = 1 will produce an overdamping of change. To discretize the horizontal distance component, an approximation to the second derivative is found by an iterative method. When the iteration is truncated, the parameter Gamma is introduced, which is allowed to vary between .08 and .17, based primarily on the look of migrated sections. If Gamma is allowed to increase too much more, spurious noise results. In the ideal case, Tau would equal the sample rate of the data, meaning that the entire section would be migrated exactly one sample rate step at each pass through the section. While this scheme reduces the errors, it is impractical due to the huge run-time needed. In practice, Tau should be chosen in the range of 20 to 200 ms. (.02 to .2 secs), with the smaller Tau values producing greater accuracy. It is possible to vary Tau vertically, and this should be done in order to save run-time. Generally, the value of Tau should decrease from shallow to deep data times. This is because greater accuracy is needed in the migration of the deeper events where the greatest movement is taking place. More detailed explanation of the origin of these parameters, and some results of allowing them to vary, may be found in the paper published by H. Brysk (Geophysics: May 1983). PARAMETER DICTIONARY DX - Trace separation distance. This is the distance between reflection points. DX is a constant for the entire seismic line. REQUIRED. range 1.0 to 500.0 e.g. dx 25 FNO - The first shot/rp number the parameter list applies to. Preset = the first shot/rp received. e.g. FNO 101 LNO - The last shot/rp number the parameter list applies to. Preset = the last shot/rp received. e.g. LNO 101 VTP - The rms velocity to use in migration. The rms velocity function is the same as the velocity function used to moveout the data. Given as velocity-time pairs. Velocities not specified are calculated through interpolation and "straight- lining" from the ends. Times must be given in seconds. Preset = none velocity range 350 to 32000 BPAD - The number of zero amplitude traces to insert prior to the first trace. Preset = 1 range 1 to 500 e.g. bpad 10 EPAD - The number of zero amplitude traces to append after the last trace. Preset = 1 range 1 to 500 e.g. epad 10 OPAD - A switch indicating that the pad traces (both bpad and epad) should be output in addition to the migrated input. Preset = no range yes/no e.g. opad yes NRHO - A parameter used to control the Tau step interpolation. Preset = 2.0 range 0. to 10000 FCRHO - A parameter used to control the Tau step interpolation. Preset = .99 range .0001 to 1. RHO - A "hidden" migration parameter discussed above. Preset = .9990 range 0 to .9999 THETA - A "hidden" migration parameter discussed above. Preset = .501 range 0 to 1.0 GAMMA - A "hidden" migration parameter discussed above. Preset = .125 range .08 to .17 TSTEPS - A set of time-delta-tau pairs governing the tau step size (delta-tau) in the time interval terminating with the time given. Up to seven (time, delta-tau) pairs may be given. The delta-tau values will be interpolated between the specified times and will be "straight-lined" at the trace ends. The units of time and delta-tau are seconds. Preset = REQUIRED e.g. tsteps .1 .1 1.0 .2 NX - The total number of traces, including pads, to migrate. The entire seismic line must be transformed from TX (time-space) to XT (space-time). FDMIGR requires much extra disk I/O if the entire seismic line (nx*maxsam) is larger than the computer memory allocated for the transformation (the Cray does not have a virtual memory). NX does not need to be a power of 2. Preset = 4096 e.g. nx 500 MAXSAM - The maximum number of samples per trace, including the deep water delay, to migrate. A trace exceeding MAXSAM will be truncated. Preset = the number of samples plus delay of the first trace. PATH - The pathname (filename) of a scratch file FDMIGR should use for the intermediate transposed data. The purpose of this parameter is to allow the user to specify the exact disk partition to use in case the "current" partition does not have enough space. Preset = a scratch file in the current directory e.g. path /user/scratch/moreroom Copyright (C) by The Regents of The University of California, 1988 Written by Paul Henkart, Scripps Institution of Oceanography, La Jolla, Ca. and by Veritas Seismic Processors, Ltd., Calgary, Alberta. ALL RIGHTS RESERVED.