Annual Report 2000 - WIT
Annual Report 2000 - WIT Annual Report 2000 - WIT
§ ó § 56 sequence. We made tests including two more terms, one for õ§ö©÷ and one for õ , but the stability of the method was reduced. Recall that this criteria should be applied before the smoothing process. So it must work even if there is noise in the parameters values. We have made several tests with this criteria and it really was able to unfold the curve. We deal with the over-determination on the velocity considering a solution in the sense of least squares. By now, we are supposing a model with homogeneous layers, but this approach can be easily extended to incorporate velocity profiles depending for example, linearly on the depth. Currently, we are recovering the interfaces by interpolating the many founded NIPs. Since this could lead to a not so smooth interface, what would be a disaster for the ray tracing process, we apply the same smoothing process described before to the interpolated interface. This really enhances the quality process overall. Following, we present our implementation of the algorithm of Hubral and Krey, with the modifications discussed above. Input data. Recall that after the application of the CRS method, we obtain a simulated zero-offset section in which the most important events (primary reflections) are well identified. This means that the ó traveltimes at each reflection is known. The identified primary reflections will provide the interfaces of the layered model to be inverted. Also recall that at each point of the zero-offset simulated section, we have attached the three parameters òó , õ§ö©÷ and õ extracted from the multi-coverage data. We shall make a consistent use of the CRS parameters along those identified reflections. We finally observe that the medium velocity in the vicinity of each central point is known a priori. § Determination of the first layer. For a given ùÍó central point ó , let be the zero-offset traveltime of the primary reflection from the first interface. òó Also, let be the emergence angle õ and the radius of the N-wavefront curvature of the corresponding zero-offset ray. Draw from a straight line that makes an òó ùÍó angle with the surface ù®ó normal at and has length equals ó to . The extreme of this segment of ray is the NIP. Do this for all central points that illuminate the first interface and then interpolate the inverted s to recover the first interface. To minimize the noise on this first approximation, apply the smoothing process to the curve that describes the interface. We can now say that this first Œª©e« layer is completely determined. Before we go to the determination of the following layers, we will back-propagate all the N-waves associated to each ray. This provides an estimative of the curvature of the reflector at each NIP. Note that the curvature of the reflector will be necessary when we will back-propagate wavefront curvatures down to deeper layers, since to transpose the first interface we need to know its curvature.
¤ ¤ ¤ 57 Subsequent layers. As before, suppose that the model has been already determined . The input data is the CRS parameters for the zero-offset . Trace the rays down to the Y interface up to the layer Y \ rays that reflect at the interface þ þ Y¬©\ and back-propagate the NIP-wavefront along those rays. Applying the focusing conditions, estimate by least squares. Using Snell's law calculate ò Z and, NZ knowing the remaining traveltimes, trace the last segment of each ray. Smooth the interpolated curve of the NIPs to obtain the interface Y!©]\ . Finally, along each zero-offset ray back-propagate the N-wavefront curvature to estimate the curvature of the interfaces. þ SYNTHETIC EXAMPLE The algorithm was applied to the model depicted on Figure 6, which consists of three interface separating four homogeneous layers. The second reflector has a synclinal region between 6.5km and 7.5km that generates caustics. For the three layers, we modeled the CRS parameters by using the dynamic ray tracing package Seis88, designed by jCervený and Pjsenjck (see jCervený, 1985), and other auxiliar softwares developed by ourselves. 0 v 0 = 1.4 km/s Depth [km] 2 4 v 1 = 2.0 km/s v 2 = 3.4 km/s v 3 = 5.5 km/s 6 0 4 8 12 16 Distance [km] Figure 6: Synthetic model with four homogeneous layers. As a validation test, we ran the method with the exact modeled parameters as the input data. As expected, the velocities and interfaces were recovered with the same
- 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 and 44: 31 Dürbaum, H., 1954, Zur Bestimmu
- Page 45 and 46: Ã Ø Ì 0 0 Ì 4 0 Ã 0 G ' 4
- 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 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
- Page 95 and 96: ˆ 83 -0.05 -0.1 Amplitude -0.15 -0
- Page 97: 85 on a second-order approximation.
- Page 100 and 101: 88 kinematic (related to traveltime
- Page 102 and 103: ¨ º Ý È · § À · Á · Á ¼
- Page 104 and 105: ° ´ ´ ã ° ¼ ´ ´ þ Ò Ò ¥
- Page 106 and 107: 94 1 taper value ksi1 0 ksi2 Figure
- Page 108 and 109: 96 0 1 2 3 4 5 Depth [km] CMP [km]
- Page 110 and 111: 98 CONCLUSION As a generalization o
- Page 112 and 113: 100 weight during the stacking proc
- Page 114 and 115: 102 are not illuminated for every o
- Page 116 and 117: 104 obtained from Zoeppritz' equati
¤<br />
¤<br />
¤<br />
57<br />
Subsequent layers. As before, suppose that the model has been already determined<br />
. The input data is the CRS parameters for the zero-offset<br />
. Trace the rays down to the Y interface<br />
up to the layer Y \<br />
rays that reflect at the interface þ<br />
þ Y¬©\<br />
and back-propagate the NIP-wavefront along those rays. Applying the focusing<br />
conditions, estimate by least squares. Using Snell's law calculate ò Z<br />
and,<br />
NZ<br />
knowing the remaining traveltimes, trace the last segment of each ray. Smooth<br />
the interpolated curve of the NIPs to obtain the interface Y!©]\ . Finally, along<br />
each zero-offset ray back-propagate the N-wavefront curvature to estimate the<br />
curvature of the interfaces.<br />
þ<br />
SYNTHETIC EXAMPLE<br />
The algorithm was applied to the model depicted on Figure 6, which consists of three<br />
interface separating four homogeneous layers. The second reflector has a synclinal region<br />
between 6.5km and 7.5km that generates caustics. For the three layers, we modeled<br />
the CRS parameters by using the dynamic ray tracing package Seis88, designed<br />
by jCervený and Pjsenjck (see jCervený, 1985), and other auxiliar softwares developed<br />
by ourselves.<br />
0<br />
v 0<br />
= 1.4 km/s<br />
Depth [km]<br />
2<br />
4<br />
v 1<br />
= 2.0 km/s<br />
v 2<br />
= 3.4 km/s<br />
v 3<br />
= 5.5 km/s<br />
6<br />
0 4 8 12 16<br />
Distance [km]<br />
Figure 6: Synthetic model with four homogeneous layers.<br />
As a validation test, we ran the method with the exact modeled parameters as the<br />
input data. As expected, the velocities and interfaces were recovered with the same
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