An Application of T-X Domain Filter in Extracting Water Bottom

Multiple elimination is one of the important topics in the field of seismic data processing. Since the water bottom multiples usually in-terfere with the primary reflections and induce overmigration during the migration stage, we propose one simple and easy method to extract the 五 rst order water bottom multiple. This method is an extended application of the time space (T-X) domain dip filter combined with the residual migration. In the area with rugged water bottom, this method is more effective, than conventional multiple suppression methods, for the suppression of multiples. It erases the water bottom multiples from the record, but also any undesired energy. Since this mehod is applied in the post-stack stage and the recognization of the multiple is subjective, so the application of this method should be made very carefully. When parameters are properly chosen, the application of this method can make an noticeable improvement in the 白間 J results.


INTRODUCTION AND PREVIOUS STUDIES 193
The conventional multiple elimination techniques are all based on the facts listed below (Yilmaz, 1987): a.The move�mt difference between primaries and multiples.b.The dip difference between primaries and multiples.c.The frequency content between primaries and multiples.d.The periodicity of multiples.
Multiple suppression techniques based on these criteria include the slant stack multiple suppression technique (Alam andAustin, 1981, Treitel et al., 1982), the velocity discrimination technique and the application of predictive deconvolution in the r-P domain (Taner, 1980), etc.Although these techniques all have sound conceptual bases, their performance is sometimes unsatisfactoy.Possible explanations may be that there are not significant moveout differences between primary reflections and multiples or that there are structural irregular ities in the lateral direction (i.e.rugged water bottom).The geometry divergent correction using velocity functions may also enhance the amplitude of multi ples.All these factors described above may cause the conventional suppression technique to fail in removing multiples from the seismic data.As for water bottom multiples, they can be easily removed by using a pre dictive deconvolution filter if the water bottom is flat.In areas with a deep and rugged water bottom, the prediction lag and operator length should be varied with the relief of the water bottom.These fa ctors make it hard to decide on deconvolution parameters during the data processing procedures.Furthermore, the RMS velocity of the water bottom multiple is approximatly equal to the RMS velocity of water.This means that) if there exist multiples in the stacked section, they may cause overmigration by using the RMS velocity in migration operation and obscure the primary reflections.
Since the performance of the conventional multiple suppression technique is often disappointing for deep water bottom multiples one direct and easy way to extract multiples is provided in this paper.The procedure of migration is divided into two passes -water velocity migration in the first pass and residual migration in the second.Firstly, the stacked section is migrated with the RMS water velocity.In the first pass migration, the water bottom reflections and multiples are migrated to their "correct" positions (Fig. la).Then, the time values of the multiples are picked and digitized.With these time values, the traces are shifted to make all the multiples fl.at (Fig. 1 b ).The flattened multiples can now be removed by a T-X domain dip filter (Fig. le).The filtered traces have to be shifted back to their original positions (Fig. ld).Since the primaries are still not migrated completely, the whole section has to be migrated with the calculated residual velocities of the primaries (the second pass migration).After applying all these procedures, the multiples that can be recognized on the original seismic section could completely be removed.

METHOD
Two operations are involved in this paper.They are space-time (T-X) do main dip filtering and two-pass finite difference migration.
The generation of the water bottom multiple is due to seismic reflections happening more than once within the water layer (Fig. 2a).According to the number of reflections, the multlple can be defined as the first order, the second order, etc.The seismic energy can also be reflected within one layer and generate the other type of multiples such as pig-leg multiples shown in Fig. 2b.The multiple discussed here is concerned with the first order type, i.e. the water bottom multiple, for the occurrence of this type of multiple is easy to recognize in the seismic section.
The arithmetic equations concerned in this paper are described below : The conventional dip filter is operated in a frequency-wave number (F-K) domain.The seismic data are firstly Fourier transformed into an F-K domain and a filter in this domain is designed, then the filter is applied and the data are then transformed back into the T-X domain.Although the F-K dip filter is effective in eliminating dip events, it may introduce troublesome coherent noise.Furthermore, the F-K domain dip filter is usually applied for the whole seismic section.For the dip events confined in a certain time range, it is unnecessary to apply the dip filter to the whole section.Hale (1983) suggested to apply one T-X domain dip filter only to a certain part of the seismic data that need to be filtered, with the remainder of the data being unfiltered.
Hale's Butterworth dip filter can be simplified as the Fourier transforms of the following equations.and ( 2 ) where P, Qare the input and output wave functions respectively, Dis the cutoff dip, Wis the angular frequency and K is the wavenumber.The finite difference method will transform the Fourier transformed equation into difference forms.
Then the'dip filtering can be simplified as the recursive solution of a tridiagonal system.
In this paper, we design one high-dip-pass filter and apply it to the flattened multiple events.For the time variant nature of this filter, the seismic data outside the time gates chosen by the user remains unfiltered.

b. Two pass finite difference migration
The finite difference migration was first innovated by Claerbout (1 97 6, 1985).This technique is derived from the finite difference solution of the scalar wave equation.Since the RMS velocities of the multiples are smaller than the ve locities of the primaries, we have to migrate the multiples and water bottom reflections to their "correct" positions first by using the RMS velocity of the sea water (about 1500 m sec-1 ) in order to prevent the occurrence of overmigration.This is the first pass migration.After this, the water bottom multiples will be recognized by their two-way travel times and apparent dip at about twice the time of the water bottom reflections.After all the multiples have been removed, we have to migrate the primaries into their real position since they are still in undermigrated conditions (the second pass migration ) .The velocities used in the second pass migration can be easily derived from the following equations: ( 3 ) and ( 4) Where P is the wave function, kz, ky are the wave numbers in z and CMP direction, w is the angular frequency and v is the veloeity function in the z direction.
The 15 degrees parabolic approximation is the basis of finite difference mi gration.From the Taylor expansion of equation ( 4), we get Practically, the migration operation used is still performed in the time do main ( time migration ), and where, Wr is the output angular frequency.By substituting equation ( 6) into (5), we have v2k2

w
The first pass output angular frequency will be and the second pass output angular frequency will be Let w be equal to wT.From equations (7), ( 8) and (9), we get The equations listed above are only parts of the mathematical foundation for the residual migration.The water bottom multiples occurring in the deep and rugged water bottom have long been a trouqlesome problem in order to process the marine seismic data.Since the conventional methods sometimes failed to solve this problem, we have developed the technique described above to remove these multiples existing on the seismic records.After applying this technique to the seismic data, the quality of the resulting seismic sections becomes noticeably improved.

EXAMPLES
We are going to show two examples here which were obtained in a deep water survey of the southern Taiwan Straits.Due to the deep and rugged water bottom in this area, the long period multiples appear on most of the seismic lines.
For the first example, Fig. 3 represents the final section of a seismic line.Figs. 4 to 7 represent the results of every main step included in this technique.
Fig. 4 is the migrated section of Fig. 3 without any multiple elimination being applied.Obviously, the multiple M has been overmigrated and interfered with the primary P in this figure .Although mutiple M can be identified, the     does not induce any overmigration effect, the trend has intersected with the primary reflections and degraded the qu ality of processing.The filter used here is of dip angle 2 m s/trace with a time gate of 100 ms .It should be noted that the seismic data outside the multiple M still preserved their original superficial appearances even after the dip filter had been applied.

DISCUSSIONS
This paper provides an alternative method for extracting multiples from seismic records, and preventing overmigration during the migration procedure.Compared with other before-stack multiple suppression technques, this method is simple and time saving, especially, when the water bottom of a survey area is rugged.a.When multiples embedded in or parallel to primaries or the sea bottom are not va ried too much, it is difficult to distinguish any differences between them.The predictive deconvolution is prefered in this condition, especially, if the sea bottom of survey area is flat.b.The parameters of the T-X domain dip filter should be selected carefully.
With too wide of a time gate or too steep a high pass dip, the filter will harm the data which are not to be filtered.c.The recognization of the multiple is very subjective, so the multiple must be prudently defined.Events which are not related to multiples must not be removed.d.If the overmigration of the multiple is not serious after the migration pro cedure with RMS velocity functions of the primaries, the T-X domain dip filter can even be applied into the migration section directly and there is no need to migrate seis _ mic data with the two pass mode.
By carefully using this technique, all multiples can be removed and the quality of.the seismic section can be obviously improved.It is suggested that this technique be applied in parallel with other multiple suppression techniques.
Even if all other methods failed, this technique still can provide satisfactory results.

CONCLUSIONS
The technique introduced here can be considered an extended application of the T-X domain dip filter.The time variant characteristics of the dip filter and two-pass finite difference migration are combined to extract water bottom multiples which are hard to remove with the other suppression techniques.This technique can also be applied in other respects, for example, to remove any

Fig. 1 .
Fig. 1.Procedures of multiple extraction.Events A, Band C represent water bottom, water bottom multiple and primary reflection respectively.(a) Sketches of A, B and C after the first pass of migration.(b) B is flattened by time shift.(c) B is extracted by dip filter.(d) Correct position of A and C after the second pass of migration.

Fig. 2 .
Fig. 2. Generation of multiples.(a) The primary water bottom reflection A and the first order water bottom multiple reflection B. (b) A multiple is generated within a layers.

Fig. 3 .
Fig. 3.The final stack section of Example 1.The multiple M located at 2900 m sec can not be recognized clearly in this figure.

Fig. 4 .
Fig. 4. The migrated section of Fig. 3 without any multiple extraction proce dure being applied.M is water bottom multiple, P represents primary reflection.Overmigration of M has already intersected with P at 2550 m sec.

Fig. 5 .
Fig. 5.After the first pass migration operation, the multiple M can be identified clearly in this figure.

Fig. 5
Fig. 5 is the first-pass migration section of Fig. 3 with the use of the RMS velocity of the sea water.The multiple M occurring at about twice the two-way travel time for the water bottom reflection can easily be recognized in comparison with Fig. 4. The reason is that the migration operation has moved the multiple M and water bottom reflection to their "true" dip positions and their appearance has some kind of similarities.

Fig. 6
Fig.6is the section where the multiple M has been removed by the dip filtering and shifting procedures.The dip filter used is a high pass type with dip 2 m sec/trace having a time gate of 100 m s.The seismic events greater than this dip angle or outside the time gate are unaffected.

Fig. 7 Fig. 6 .
Fig.7is the result of the migration of Fig.3.All events have been moved to their correct positions after the second pass migration operation.Comparing Fig.7to Fig.4, the quality of this section has obviously been improved, especially the events between 2 seconds to 3 seconds and the overrnigration of

Fig. 7 .
Fig. 7.The final resulting section of multiple extraction, after the second migration, shows that all the events have been migrated to their true spacial positions.

Fig. 8 .
Fig. 8.The migrated section of another line without applying any multiple extraction procedures.The multiple M extending downward is obviously seen in this figure.