Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
Elements of the surface-to-surface propagator matrix Ì Ö ¬ ž ž £ × ¯ ž ž £ ž ž ž · · ž £ £ ž à ž · ž £ G à £ · £ ž · £ Ð ž ' ¸ · · & Ð Ã ž · £ ž Ü 32 APPENDIX A Following (Hubral et al., 1992a; jCervený, 1999; Schleicher et al., 2001), we can express for the 2-D case the ÍÀÎ submatrices ÏÑÐ Ð of the ÈÕÀÂÈ surfaceto-surface propagator Ì matrix 1993) form by (Bortfeld, 1989; Hubral et al., 1992a; Schleicher et al., ¤Ò ^Ó ¤Ô (A-1) ° Ö ¯ '@Ö ˆ× (A-2) ˆ× £ka ž × £ Ö ¯ × £ Ö ˆ× (A-3) ¾ ž žI' £ia ž × £ Ö (A-4) £ a ¯ ¿ propagator a ¯ a Ö ¯ where Ö and are the scalar elements of the Ø so-called matrix (Popov and Pjsenjc´ffk, 1978; jCervený, 1985; Hubral et al., 1992a). The matrix is defined in a Cartesian coordinate system and the propagator matrix Ù is defined Ì in a ray centered coordinate system ( jCervený, 1987). We assume a 2-D model (Figures 6.1 and 6.2) with a curved measurement surface. We have a source point ¢ coinciding with the receiver point ¤ . In the paraxial vicinity of the normal ray surface location ¢Œ¤ , where the medium velocity changes only gradually, we have two points ¢£ and ¤:£ . This source-receiver pair is linked by a reflected paraxial ray ¢£O¢¤£M¤:£ . According to ( jCervený, 1987; Schleicher et al., 2001), we can express for the 2-D case the transformation from the local 2-D Cartesian coordinate Ú system to the 2-D ray-centered coordiante Ú system at ¢Œ¤ (Figure A-2) by §1 #Û* Ü žIÝ Ü £ Ÿž (A-5) (A-6) Ÿ,/. (A-7) ž ,f. ¸ ,f. ,f. ¸ ¸ × £ @' In paraxial approximation, the curved surface can be expressed as a parabola. In the , this is representable as local 2-D coordinate system ¯ & · Û (A-8) Û´ ¯
Ã Ø Ì 0 0 Ì 4 0 à 0 G ' 4 ¯ à £ 0 4 0 0 ¯ à à 4 à à £ 0 ž à ž B » B » 4 à 33 where · is the surface's curvature at ¢Œ¤ (Schleicher et al., 2001). is expressed in terms of the NIP and N wavefront curvatures à à and £§¦©¨ The Ù matrix (Hubral, 1983) by £ (A-9) Ö ¯ Ö £§¦©¨ £ & · By substituting eqs. (A-6), (A-7) and the components of matrix Ù (eq. A-9) into eqs. (A-1) we obtain the components °(|¬ |¾ and ¿ of the matrix Ì given by eqs. (7). The matrix Ì also can now be defined in terms of the matrix Ù by Þ©ß a ¯ a 4 £§¦©¨ £§¦©¨ £§¦©¨ £ 4áà ¬ ° ¿ ¾ » B (A-10) » ¯ Inserting the elements of the propagator matrix Ù , we obtain » ž …â Ù 0 …â 4 > ¬ ° ¿ ¾ » B (A-11) Ö …â » ž …â Ö ¯ thus proving the validity of equations (A-1). a ¯ a 4 » ¯ 2h Xg Xs SG m G’ curved surface normal ray S’ paraxial ray . R R’ reflector Figure A-1: Ray diagram for a paraxial ray in the vicinity of a normal ray in a 2D laterally inhomogeneous medium.
- Page 1 and 2: Wave Inversion Technology WIT Annua
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- Page 11 and 12: i TABLE OF CONTENTS Reviews: WIT Re
- Page 13 and 14: Wave Inversion Technology, Report N
- Page 15 and 16: 3 theory. It relates properties of
- Page 17: Imaging 5
- Page 20 and 21: 8 two hypothetical eigenwave experi
- Page 22 and 23: 7 ¢¥£§¦©¨ ; < calculate sear
- Page 24 and 25: 7 7 searches ¡ HNJOL for ¢IHNJML
- Page 26 and 27: 14 Trace no. 1000 1500 2000 0.5 1.0
- Page 28 and 29: 16 CDP number 3100 3000 2900 2800 2
- Page 30 and 31: 18 REFERENCES Höcht, G., de Bazela
- Page 32 and 33: 20 INVERSION BY MEANS OF CRS ATTRIB
- Page 34 and 35: 22 For synthetic data, it is easy t
- Page 36 and 37: — J J J J J J G J - J J G
- Page 39 and 40: Wave Inversion Technology, Report N
- Page 41 and 42: · · v 0 B 0 £ & Ä Ã & v
- Page 43: 31 Dürbaum, H., 1954, Zur Bestimmu
- Page 47 and 48: Wave Inversion Technology, Report N
- Page 49 and 50: 0 0 ˜ ” Z ˜ ” Z 4 4
- Page 51 and 52: 39 strategy by combining global and
- Page 53 and 54: 41 elled data is presented in Figur
- Page 55 and 56: 43 0.2 Distance [m] 1000 1500 2000
- Page 57 and 58: 45 In Figure 10 we have the optimiz
- Page 59: 47 Gelchinsky, B., 1989, Homeomorph
- Page 62 and 63: § ¢ ø ø for a paraxial ray, ref
- Page 64 and 65: Œ § … Z ‹ Z § õ ø \ ø §
- Page 66 and 67: 54 As stated in Hubral and Krey, 19
- Page 68 and 69: § ó § 56 sequence. We made tes
- Page 70 and 71: 58 precision of the modeled input d
- Page 72 and 73: 60 Cohen, J., Hagin, F., and Bleist
- Page 75 and 76: Wave Inversion Technology, Report N
- Page 77 and 78: £ 65 The details of the theory inv
- Page 79 and 80: 67 of the figure. On both sides, a
- Page 81 and 82: 69 (a) (b) Depth (m) 600 800 Depth
- Page 83 and 84: 71 Langenberg, K., 1986, Applied in
- Page 85 and 86: È Û Ñ¿ = ¨ Í Ñ>* = ¿ + ð
- Page 87 and 88: Wave Inversion Technology, Report N
- Page 89 and 90: g ý g [ ^ â g [ g g g g g â h [
- Page 91 and 92: g § » ¹ [ƒŽ g 79 We now subst
- Page 93 and 94: ê 81 ing was realized by an implem
Elements of the surface-to-surface propagator matrix Ì<br />
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32<br />
APPENDIX A<br />
Following (Hubral et al., 1992a; jCervený, 1999; Schleicher et al., 2001), we can express<br />
for the 2-D case the ÍÀÎ submatrices ÏÑÐ<br />
Ð of the ÈÕÀÂÈ surfaceto-surface<br />
propagator Ì matrix<br />
1993) form by<br />
(Bortfeld, 1989; Hubral et al., 1992a; Schleicher et al.,<br />
¤Ò<br />
^Ó<br />
¤Ô<br />
(A-1)<br />
° Ö ¯<br />
'@Ö ˆ×<br />
(A-2)<br />
ˆ× £ka<br />
ž<br />
× £ Ö ¯<br />
× £ Ö ˆ×<br />
(A-3)<br />
¾ ž<br />
žI'<br />
£ia<br />
ž<br />
× £ Ö <br />
(A-4)<br />
£ a ¯<br />
¿ <br />
propagator<br />
a ¯ a Ö ¯ where Ö and are the scalar elements of the Ø so-called<br />
matrix (Popov and Pjsenjc´ffk, 1978; jCervený, 1985; Hubral et al., 1992a). The matrix<br />
is defined in a Cartesian coordinate system and the propagator matrix Ù is defined<br />
Ì<br />
in a ray centered coordinate system ( jCervený, 1987).<br />
We assume a 2-D model (Figures 6.1 and 6.2) with a curved measurement surface.<br />
We have a source point ¢ coinciding with the receiver point ¤ . In the paraxial vicinity<br />
of the normal ray surface location ¢Œ¤ , where the medium velocity changes only<br />
gradually, we have two points ¢£ and ¤:£ . This source-receiver pair is linked by a<br />
reflected paraxial ray ¢£O¢¤£M¤:£ . According to ( jCervený, 1987; Schleicher et al., 2001),<br />
we can express for the 2-D case the transformation from the local 2-D Cartesian<br />
coordinate Ú system to the 2-D ray-centered coordiante Ú system<br />
at ¢Œ¤<br />
(Figure A-2) by<br />
§1<br />
#Û*<br />
Ü žIÝ Ü £<br />
Ÿž<br />
(A-5)<br />
(A-6)<br />
Ÿ,/.<br />
(A-7)<br />
ž ,f.<br />
¸<br />
,f.<br />
,f.<br />
¸<br />
¸<br />
× £<br />
@'<br />
In paraxial approximation, the curved surface can be expressed as a parabola. In the<br />
, this is representable as<br />
local 2-D coordinate system ¯<br />
& ·<br />
Û<br />
(A-8)<br />
Û´<br />
<br />
¯
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