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Synthetic land data 1

The first set of examples are of synthetic land data. The data are a single shot gather, forward modeled using an elastic isotropic migration code which utilizes the stress-displacement formulation, and a staggered-grid finite-difference scheme (Virieux, 1986). The medium parameters are shown in Figure 2. The source was a pressure source, located on the surface at the center of the model. A free boundary condition was used for the upper boundary, and absorbing boundaries were used for the bottom and the sides. The receivers were split-spread around the source, up to a maximum offset of $ 1800m$ . The direct arrival has been muted from the synthetic data.

The recorded vertical and horizontal displacements are shown in Figures 3(a)-3(b). These represent the geophone data. The P-wave and S-wave shown in Figures 3(c)-3(d) are calculated from the displacement field with the Helmholtz separation operator in equations 3-4.

vp-land-a
Figure 2.
Medium parameters of synthetic land data.
vp-land-a
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lu1-0L lu1-1L lp1-0L lp1-1L
lu1-0L,lu1-1L,lp1-0L,lp1-1L
Figure 3.
Synthetic land data. (a) vertical displacement, (b) horizontal displacement, (c) P-wave, (d) S-wave.
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Figure 4 shows the arrangement of the virtual sources and the receivers during the inversion. The virtual-source line is longer than the receiver line to better facilitate the modeling of high-wavenumber waves, which may be present at the edges of the recorded data. The medium itself is homogeneous, and surrounded by absorbing boundaries. The elastic propagation finite-difference scheme used in the inversion is different from the one used for generating the synthetic data. It utilizes a displacement-only formulation (as in equation 11), on a regular grid. The finite-difference approximation is $ 2_{nd}$ order in time and in space.

The vertical and horizontal virtual-source models resulting from the inversion are shown in Figures 5(a) and 5(b). When these source functions are injected as displacement sources into the homogeneous medium which was used in the inversion, the results are the reconstructed displacement data. If the inversion converged, then these reconstructed data will be equal to the recorded geophone data. The purpose of Figures 6(a)-6(h) is to show that the inversion managed to converge, even when the medium parameters used were different from the true ones.

src-rec-locations
Figure 4.
Qualitative sketch depicting the relative locations of the receivers and the virtual sources used in the inversion. The virtual source line has a greater horizontal extent than the receivers in order to enable convergence to a solution when high wavenumbers exist at the sides of the recorded data.
src-rec-locations
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su1-0 su1-1
su1-0,su1-1
Figure 5.
The virtual source functions that generate the observed land data. These ``models'' are the inversion results. (a) vertical displacement source functions, (b) horizontal displacement source functions.
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The P-wave velocity at the receiver level in the medium used for forward modeling was $ 1700 \frac{m}{s}$ , S-wave velocity was $ 850 \frac{m}{s}$ and density was $ 1.7 \frac{gr}{cm^3}$ . Figures 6(a) and 6(b) are the forward modeled displacements. Figures 6(c) and 6(d) are the reconstructed displacements when using the correct medium parameters in the inversion. Figures 6(e) and 6(f) are the reconstructed displacements when using a homogeneous medium with $ V_p = 1500 \frac{m}{s}$ , $ V_s = 700 \frac{m}{s}$ and $ \rho = 1.5 \frac{gr}{cm^3}$ (i.e. - about 15% too slow). Figures 6(g) and 6(h) are the reconstructed displacements when using a homogeneous medium with $ V_p = 2000 \frac{m}{s}$ , $ V_s = 1000 \frac{m}{s}$ and $ \rho = 2.0 \frac{gr}{cm^3}$ (i.e. - about 15% too fast). Note that the displacements at the receivers have indeed been matched.

lu1-0 lu1-1 lu1-2 lu1-3 lu1-4 lu1-5 lu1-6 lu1-7
lu1-0,lu1-1,lu1-2,lu1-3,lu1-4,lu1-5,lu1-6,lu1-7
Figure 6.
Synthetic and reconstructed displacements of land data. (a) synthetic vertical displacement, (b) synthetic horizontal displacement, (c) reconstructed vertical displacement with correct velocity, (d) reconstructed horizontal displacement with correct velocity, (e) reconstructed vertical displacement with 15% too slow velocity, (f) reconstructed horizontal displacement with 15% too slow velocity, (g) reconstructed vertical displacement with 15% too fast velocity, (h) reconstructed horizontal displacement with 15% too fast velocity.
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Figures 7(a)-8(d) are the result of applying the Helmholtz separation operator to the reconstructed displacement fields, to calculate the P and S-waves. The finite-difference approximation to the Helmholtz operator uses the displacement value at the receivers (which have been matched), and also the values one depth level below the receivers. These values are not matched by the inversion, and therefore unless the medium parameters are correct - they will not be an accurate representation of the true field values there. This will cause the vertical derivative to be inaccurate as well. The question then is: how detrimental are such inaccuracies to the P/S separation result?

Figure 7(a) is the forward modeled and separated synthetic P-wave recording. Figure 7(b) is the reconstructed P recording, when using the correct medium parameters in the inversion. Figure 7(c) is the reconstructed P when using a $ 15\%$ too slow homogeneous medium, and Figure 7(d) is for a $ 15\%$ too fast homogeneous medium. Figure 7(e) is a summation of the forward modeled and separated P and S recording, and is useful for estimating the separation quality for each scenario.

In Figure 7(b), the reconstructed P-wave is almost identical to the forward modeled P-wave, while the S-wave is barely visible. This indicates that the reconstructed displacement fields were a good approximation of the forward modeled displacement fields. In Figures 7(c) and 7(d) the medium parameters are incorrect, and the S-wave is more visible. However, the separation quality is still reasonable, when compared to Figure 7(e). This indicates that though the medium parameters were off by $ 15\%$ , the displacement fields were a good approximation to the forward modeled fields one depth step below the receivers (as well as at the receivers themselves). Therefore, the vertical derivatives of the displacement fields reasonably approximate the true derivatives, and the P separation (equation 3) is effective.

Figure 8(a) is the synthetic S-wave recording. Figure 8(b) is the reconstructed S recording, when using the correct medium parameters in the inversion. Figure 8(c) is the reconstructed S when using a $ 15\%$ too slow homogeneous medium, and Figure 8(d) is for a $ 15\%$ too fast homogeneous medium. Figure 8(e) is the sum of the forward modeled P and S. As for the previous set of figures, we can see that the separation quality does decrease as the medium parameters vary from the true ones, but the separation is still of reasonable quality even when the parameters are off by a large percentage.

There is an additional effect of the inversion modeling process, which is visible in this set of figures and particularly in Figure 8(b). As a result of the edges of the receiver line, a diffraction is generated at both ends during the adjoint propagation, and this diffraction is recorded at the virtual-source line. It can be seen in the virtual-source functions, in Figures 5(a)-5(b). The recording of this diffraction event is then repropagated during the forward propagation stage of the inversion, generating an additional diffraction event off the edges of the virtual-source line. This diffraction event was not part of the original wavefield. Since it arrives at the same time as the first P-wave, it is not visible in Figures 7(b)-7(d), but it is visible in Figures 8(b)-8(d), even when using the correct velocity for the inversion.

lp1-0 lp1-2 lp1-4 lp1-6 lp1-8L
lp1-0,lp1-2,lp1-4,lp1-6,lp1-8L
Figure 7.
Synthetic and reconstructed P-wave of land data. (a) Synthetic P, (b) reconstructed P with correct velocity, (c) reconstructed P with 15% too slow velocity, (d) reconstructed P with 15% too fast velocity, (e) sum of synthetic P and S-waves. All figures are identically clipped.
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lp1-1 lp1-3 lp1-5 lp1-7 lp1-8
lp1-1,lp1-3,lp1-5,lp1-7,lp1-8
Figure 8.
Synthetic and reconstructed S-wave of land data. (a) Synthetic S, (b) reconstructed S with correct velocity, (c) reconstructed S with 15% too slow velocity, (d) reconstructed S with 15% too fast velocity, (e) sum of synthetic P and S-waves. All figures are identically clipped.
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Next: Synthetic land data 2 Up: Inversion results Previous: Inversion results

2012-05-10