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A new algorithm for the rotation of horizontal components of shear-wave seismic data

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New algorithm for shear-wave rotation CREWES Research Report — Volume 8 (1996) 12-1 A new algorithm for the rotation of horizontal components of shear-wave seismic data Kangan Fang and R. James Brown ABSTRACT Rotation of horizontal components of shear-wave data is one of the key processing procedures in anisotropy analysis. Several shear-wave rotation algorithms are available and suitable for different situations. In this paper, a new algorithm for the rotation o
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  New algorithm for shear-wave rotation CREWES Research Report — Volume 8 (1996)12-1 A new algorithm for the rotation of horizontal componentsof shear-wave seismic data Kangan Fang and R. James Brown ABSTRACT Rotation of horizontal components of shear-wave data is one of the key processingprocedures in anisotropy analysis. Several shear-wave rotation algorithms are availableand suitable for different situations. In this paper, a new algorithm for the rotation of shear-wave data is proposed. This new algorithm can be used to rotate the horizontalcomponents of shear-waves generated by multiple sources that have differentamplitudes and wavelets in azimuthally anisotropic media. Synthetic data test and fielddata example showed that the algorithm are successful and robust. INTRODUCTION It is now commonly accepted that most upper-crustal rocks are anisotropic to someextent (Crampin, 1981). Anisotropy may be caused by fine layering in sedimentaryrocks, by preferred orientation in crystalline solids, or by stress-aligned fractures orcracks (Crampin & Lovell, 1991). Shear-wave splitting is considered to be the mostdiagnostic phenomenon caused by anisotropy.Shear-wave splitting may degrade the quality of the shear-wave data and cause mis-tie (Alford, 1986). We want to obtain the attribute information of anisotropy, such asnatural polarization directions and degree of anisotropy, by analyzing shear-wavesplitting. Through rotation, effect of anisotropy can be compensated for and the fastand slow shear-wave can be separated. In the case of azimuthal anisotropy caused byvertically-aligned fractures or cracks, the strike of fractures or cracks and the time lag of fast and slow waves can be determined by shear-wave rotation, which is of greatinterest to exploration geophysicists.Several algorithms for the shear-wave horizontal component rotation have beeninvented. Alford’s algorithm is devised especially for four-component rotation analysisand it can determine the orientation of the natural coordinate system (Alford, 1986). Itrequires that the two sources have the same wavelet signature. It has been shown that,for data acquired with a single source polarization, such as converted-wave data,Alford’s rotation method does not work, without modifications (Thomsen, 1988).Much work has been done using hodogram analysis methods to study S-wavesplitting (e.g., Schulte and Edelmann, 1988). As discussed by Winterstein (1989),these methods require both very high signal-to-noise ratio and the presence of a singlewavelet within the analysis window in order to be effective.Other two-component birefringence-analysis schemes that do not involve hodogramshave largely been based upon either the autocorrelation or crosscorreation of rotatedcomponents (Narville, 1986; Peron, 1990). Harrison (1992) presented an algorithmusing the autocorrelation and crosscorrelation of rotated radial and transversecomponents, which is particularly suitable for converted waves and robust in thepresence of noise.The rotation algorithm presented here is similar to Alford’s rotation algorithm, but itinvolves two parameters, the natural polarization direction angle and the time lag  Fang and Brown 12-2CREWES Research Report — Volume 8 (1996) between the fast and slow shear-waves, which can be determined by scanning, ratherthan only one parameter, the former, in Alford’s rotation.It can deal with the situation, where sources have different amplitudes and differentwavelets. Synthetic data test and field data example showed that the algorithm issuccessful and robust. PRINCIPLES Shown in Figure 1, is a plan view of a multi-source, multi-receiver surface line forthe situation where vertical S-wave splitting is assumed to occur. Each shear-wavesource that does not coincide with the natural coordinate axes will split into a fast and aslow waves (Crampin, 1981; Thomsen, 1988). The S 1  direction indicates thepolarization direction along which shear-waves travel at the fastest velocity β 1 , while S 2 taken to be perpendicular to S 1 , is the polarization direction along which S-waves travelat the slowest velocity β 2 . θ Radial (R)Transverse (T)S1S2Acquisition lineSRST θ Radial (R)Transverse (T)S1S2S1S2RadialTransverseU11U22 Figure 1. Illustration of multi-source, multi-receiver acquisition and shear-wave splitting. s R (t) and s T (t) are the radial and transverse source along acquisition coordinate axes,respectively, and can be expressed as matrix: a RT  S  t  ss t t  ( ) =  ( )( )        00 (1) both shear wave sources will split into shear-waves polarized in the S 1  and S 2 directions, which can be illustrated by Figure 2.The relationship of splitting illustrated in Figure 2 can be expressed as matrixmultiplication: cos sinsin coscos sinsin cos θ θ θ θ θ θ θ θ  −             =−        RT  R T  R T  sss ss s 00 (2) or  New algorithm for shear-wave rotation CREWES Research Report — Volume 8 (1996)12-3   R S  T  S   R s cos θ  −  R s sin θ  T  s sin θ  T  s cos θ  1 S direction 2 S direction 1 S direction 2 S direction Figure 2. Illustration of the splitting of acquisition sources into fast- and slow-sources  R S  t S t  a θ  ( ) ã  ( )  =  ( ) (3) where  R  θ θ θ θ θ  ( )  =−       cos sinsin cos  is the vector rotation matrix and  R R R T  − ( )  =  ( )  =  ( )  = −       − θ θ θ θ θ θ θ  1 cos sinsin cos (4) a RT  S  t ss ( )  =        00  is the source matrix in the acquisition coordinate system, and S t s ss s  R T  R T  ( )  =−       cos sinsin cos θ θ θ θ   is the source matrix in the natural coordinate system. Eachrow represents the components of the same direction.Similarly, if U t  uu ( )  =       1122 00  is a data matrix in the natural coordinate system,where u 11  and u 22  are the reflected signals along S 1  and S 2 directions, respectively, thenthe data that would be recorded along acquisition coordinate system can be written as: V t  R U t u uu u ( )  =  ( ) ã  ( )  = −       − 1 11 2211 22 θ θ θ θ θ  cos sinsin cos (5) Based on the above assumptions, we can write in frequency domain that: − ( ) ã  ( ) ã  ( ) ã  ( )  =  ( ) 1  R D R S  V  a θ ω θ ω ω  (6) where,  D f  e f  e ii ω  ω  δ ω  δ  ω ω  ( )  =  ( )( )       −− 12 12 00  represents the travel time delay functionof both fast and slow waves and δ 1 and δ 2 are the two way travel time of fast and slowwave respectively, f1 and f2 are the filter function for the fast and slow wave  Fang and Brown 12-4CREWES Research Report — Volume 8 (1996) propagation, respectively, which may account for the geometric spreading, attenuationand reflection coefficient, etc.By rotation, we would like to have a data matrix that is generated by applying bothsources and receivers along the natural coordinate system, i.e., the output data matrixof the rotation should be: W D S  D R D R V  a ω ω ω ω θ ω θ ω  ( )  =  ( ) ã  ( )  =  ( ) ã  ( ) ã  ( ) ã  ( ) ã  ( ) − − 1 1 (7) If 1 2  f f  ω ω  ( )  =  ( ) , equation (7) can be written as: W w ww we ee ev vv v i ii i RR RT TR TT  ω θ θ θ θ θ θ θ θ θ θ θ θ  ω ω ω ω  ( )  =        =+ −− +       ã        + +− − 11 1221 222 22 2 cos sin sin cos sin cossin cos sin cos sin cos ∆ ∆∆ ∆ (8) where, in w ij  and v ij , i represents the receiver direction and j represents the sourcedirection, and ∆ = −      2 1 1 2 1  z β β  . Multiplication of i e ω  ∆  and − i e  ω  ∆  in frequency domainis equivalent to time shift in time domain.According to equation (8), if we can determine the angle between the direction of the polarization of fast shear wave and the acquisition line, θ , and the time lag betweenfast and slow waves, ∆ , we can rotate the acquisition data matrix V  (t) into W  (t) toseparate the fast and slow shear wave. These two parameters can be determined byscanning.We rotate the input data matrix by a range of angles, θ , and time, ∆ , and computethe norm on off-diagonal elements of rotated data matrix.  p p ije t  pijw t k t k  N  θ θ  , , , ∆ ∆ ( )  = + ( ) =∑      1 1 (9) where N is the number of samples in the scanning window. Then we sum the norm onoff-diagonal elements. If the θ  and ∆  are correct, the sum of the norms will be theminimum.In the process described above, we do not need to assume that the wavelets andamplitudes of radial and transverse sources be the same. SYNTHETIC DATA TEST Figure 3 shows a synthetic data rotation example. Figure 3.a is the input data matrixgenerated by two sources with the same wavelet, both in main frequency andamplitudes. Figure 3.b is the result by our rotation algorithm, while Figure 3.c is theresult by Alford’s algorithm. Both methods work well when the two sources are thesame. For synthetic data rotation test, we just create one CDP and we repeat this CDP20 times for the purpose of display.
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