Propagation of Elastic Waves in Anisotropic Media

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Laboratory Measurements

Laser Doppler Interferometer Device

In this thesis, an investigation on the same samples and seismic data set as Malehmir et al. (2011, 2013) presented is carried out to better understand the role of anisotropy on the generation of reflections from the ÖDZ. Two samples, A and B (Figure 3.1c and 3.1d respectively), from the deformation zone are used (the locations where the samples are taken are shown in Figure 4.1). Sample A is a mafic rock rich in phlogopite and amphibolite with coarse grains of quartz. The sample shows a clear indication of folding and strong deformation (Malehmir et al., 2011).
Figure 3.1: (a) Schematic experimental setup of the LDI device, 1: sample, 2: source of elastic waves, 3: Laser Doppler Interferometer, 4: laser beams (emitted and backscattered), 5: reflective film, 6: acquisition system, and 7: pulse generator (modified from Lebedev et al., 2011). (b) Experimental setup (from Malehmir et al., 2013). (c) Sample A and (d) sample B, which are used in this study.
Sample B is relatively felsic in composition and shows clear indications of lamination in the horizontal direction (Malehmir et al., 2011).
Figure 3.2: (a) Velocities of the particle motion on the surface of the sample B shown in Figure 3.1d at direction x, y and z against time. (b) Hodogram showing clear S-wave splitting of S1 and S2 registered by the LDI device on sample B. A time window between 17.42 to 18.50 micro-s is used to produce the hodogram.
LDI setup is shown in Figure 3.1a (Lebedev et al., 2011), The time-dependent displacement of a particular point on the sample surface in three independent directions was measured and transformed to cartesian coordinates to achieve three-component (3C) data (Figure 3.2a).
The anisotropy of these two samples are clear, as Malehmir et al. (2011) showed. In the exam-ple shown in Figure 3.2, we clearly observe shear-wave splitting (S1 and S2), which is already an indication of anisotropy in the sample.

Inversion

Different experiments and theoretical investigations have been carried out on different materi-als attempting to estimate full elastic stiffness tensor. Vestrum (1994) used a generalized linear inversion method to calculate all the elements of the tensor. He assumed no symmetry and ob-tained all components of the tensor. He carried out his experiment for both group and phase velocities. But here, I assumed that phase velocities were measured, which is a reasonable guess according to the size of the sample and the source. Moreover, the formulation here were ex-tracted from Vestrum (1994).
Here, a modified version of Vestrum’s original code is deployed. The code is relatively straight-forward and uses no assumption about the symmetry of the sample in order to recover all the 21 components of the elastic stiffness tensor. Using LDI device, we measured phase velocities (P, S1 and S2) on the sample surface (Figure 3.1). At least a few shear-wave points are needed for the inversion to stabilize the result.
The code starts with an elastic stiffness tensor as an initial guess, after linearizing the prob-lem around the initial guess; using a linear least-square inversion method, it finds out a new guess in which the average squared phase velocity error (calculated with the measured ones) is minimized. This is done in an iterative manner until it reaches to a stage that the average squared error stops decreasing. In the inversion part, Christoffel’s equation (Equation 2.8) is used for the forward modelling and in order to control validity of the results, velocities with more than 1 percent error to the measured ones will be eliminated in each step; velocities are computed from the forward modeling.
The ultimate goal of the inversion code is to minimize the difference between the observed phase velocities and the calculated ones via: δvi = αji∆i, (3.1) where Cj is the jth element of stiffness matrix (j varies between 1 and 21) and using the matrix notation: δvi = viobs − vicalc, αji = ∂vj ∆i = ∆Cj, ∂Cj
The difference between the observed and calculated phase velocities (δvi) can be calculated with: −→ e e + λI]−1 e −→ (3.2) ∆ = [αTα αT δv. where λ is a small scalar quantity, which is added to diagonal of the matrix to be inverted in order to damp the solution and stabilize the inversion process.

Discussion

For sample A, only 12 measurements of P-waves and 3 measurements of both S-waves were used, which are not enough to obtain a unique solution. Some elements of sample A seem prob-lematic like the elements below one in the matrix 3.4. Moreover, as it is shown in Figure 3.1c, sample A is fairly folded and strongly deformed. So, the inversion might have failed in this case due to the complicated form of anisotropy and lack of sufficient data.
For sample B, 18 measurements of P-waves and 3 measurements of both S-waves were used, which are enough to obtain a unique solution. The results for sample B, more or less, are con-sistent with the previous studies (Malehmir et al., 2013). As it is shown in Table 3.2, there is a small difference between C44 and C55 that made this result Orthorhombic but it is sufficiently near to the previous study result. Hence, I decided not to rely on sample A for the next section. This explains why results from sample B are only shown in the next section. Please note that the nearest class of anisotropy for sample B is Orthorhombic.
The new results and those from previous study (Malehmir et al., 2013) for sample B are shown in Table 3.2. Please note the classes of symmetry in both case. This thesis shows slightly dif-ferent results from the previous study but they are reasonably close to each other.
And the phase velocities for sample B in m/s is:
• If Isotropic, according to the equation 2.19: VP =5834 and VS=4602.
• If TI, according to the equations 2.20 to 2.22: VP =5497, VSV =4871 and VSH =4144.
• If TI, according to the Thomsen’s approximation for weak anisotropy via the equations 2.30 to 2.32:
VP ≈5809, VSV ≈4507 and VSH ≈4167.
• If TI, according to the Berryman’s approximation for stronger anisotropy via the equations 2.33 and 2.34:
VP ≈5791 and VSV ≈4528.
• If Orthorhombic, according to equations 2.23 and 2.24: VP =6393 and VSV =3541.
Looking at the values of maximum P- and S-wave velocities in the Table 3.1 and comparing them with the above results show that the assumption of Orthorhombic as a class of symmetry for sample B is reasonable (note that the result of different TI approximations for VS is fairly high but the Orthorhombic approximation seems convincing in both velocities).Thus, the LDI inversion results for sample B suggest Orthorhombic anisotropy and anisotropy parameters ob-tained here will be used in the next chapter for reprocessing the actual seismic data using an anisotropic processing approach.

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Real Seismic Data

Background

As it was mentioned earlier including anisotropy has an important role in exploration seismol-ogy and physical property studies are an essential prerequisites to design any seismic survey for crustal seismic imaging (Helbig and Thomsen, 2005; Tsvankin et al., 2010; Malehmir et al., 2013). Figure 4.1 shows the geological map of the study area and the location of the seismic profiles acquired in the previous study. The acquisition parameters and processing steps from previous study by Malehmir et al. (2011) without considering anisotropy are shown in Table 4.1. Malehmir et al. (2011) implied that the velocity anisotropy plays an important part in the reflections from the deformation zones in the study area. So, we want to introduce anisotropy to the same processing steps of the real seismic data acquired over the major deformation zone (the Österbybruk Deformation Zone) to examine if by introducing anisotropy, the seismic im-age of the ÖDZ improves. This, then, can somewhat support if anisotropy is playing a role in the generation of the reflections from the ÖDZ.
To simplify the result and prior to highlighting the effect of anisotropy from the lab data to the real seismic data, I take your attention in following:
• The importance of different frequencies used in the lab and in the seismic data should not be underestimated and its contribution (and to what extent) should further be investigated using real field 3C data;
• One or a few samples (five samples in this study were used) may not be representative of the whole deformation zone thus our results provide first order information about the anisotropy until more data and experiments are conducted;
• In this study, the nearest form of anisotropy is assumed, orthorhombic here, to the in-version result for the data processing to be able to use the Tsvankin’s dimensionless anisotropic parameters and the non-hyperbolic moveout equation (Equation 2.39).

Results

Knowing the limitation mentioned in section 4.1, we can use the Tsvankin’s extension of the Thomsen’s parameters for orthorhombic media, which was briefly described in section 2.7, to in-troduce anisotropy to the processing steps of the Dannemora data via the non-hyperbolic move-out equation (Equation 2.39).
The event marked by arrow in CDP 1621 over the deformation zone, shown in Figure 4.2, is used to compute three dimensionless coefficients (equations 2.35 to 2.37) in order to calcu-late η (Equation 2.38) to use in the non-hyperbolic moveout equation (Equation 2.39). These are the results obtained: ϵ = 0.06, σ = 1.35 and γ = −0.14,
Processing software CLARITAS was used to apply the non-hyperbolic moveout equation as a kind of static shift to the normal moveout that previously had been applied to the data. We can compare the unmigrated stacked section crossing the deformation zone before (Figure 4.3) and after (Figure 4.4) introducing anisotropy parameters into the processing steps.
To highlight the effect of introducing anisotropy to the processing steps, we shall take a closer look at the unmigrated stacked section crossing the deformation zone before (Figure 4.5a) and after (Figure 4.5b) introducing anisotropy parameters into the processing steps. Please note partial improvement in the circled areas.
Finally, in Figure 4.6 you can see portions of the unmigrated stacked section crossing the de-formation zone before (Figure 4.6a and 4.6c) and after (Figure 4.6b and 4.6d) introducing the anisotropy parameters into the processing steps (this study). Again please note partial improve-ments marked on the section by the red arrows.

Discussion

As it is briefly shown in Figures 4.3 – 4.6, introducing anisotropy to the processing steps at least partially improves the unmigrated stacked section crossing the deformation zone. Moreover, If we see the velocity versus azimuth plot, we might see further support for the presence of anisotropy in the Dannemora data. In Figure 4.8 each dot represents a P-wave direct arrival velocity measured over the deformation zone with respect to its azimuth. The red line is a best fitted curve over the whole data using an 8-order sinusoidal line with the following details:
Figure 4.8: P-wave direct arrival velocity measured over the deformation zone with respect to the azimuth. Please note that each dot represent one P-wave arrival velocity and the red line is a fitted curve using MATLAB curve fitting application. The graph may suggest an NE-SW anisotropy direction that correlate well with the strike of the ÖDZ observed in the geological map.
General model Sin8:
f(x) = a1sin(b1x + c1) + a2sin(b2x + c2) + a3sin(b3x + c3) + a4sin(b4x + c4)+ a5sin(b5x + c5) + a6sin(b6x + c6) + a7sin(b7x + c7) + a8sin(b8x + c8).

Table of contents :

1 Introduction 
2 Background 
2.1 General
2.2 Anisotropic Form of Hook’s Law
2.3 Propagation of Elastic Waves in Anisotropic Media
2.4 Voigt Notation
2.5 Common Anisotropy Classes
2.6 Phase Velocities for Two Anisotropy Classes
2.7 Thomsen’s notation for weak anisotropy in TI media
2.8 Tsvankin’s extended Thomsen’s parameters for orthorhombic media
3 Laboratory Measurements 
3.1 Laser Doppler Interferometer Device
3.2 Inversion
3.3 Results
3.4 Discussion
4 Real Seismic Data 
4.1 Background
4.2 From The Laboratory to the Real Data
4.3 Results
4.4 Discussion
5 Conclusions 
6 References

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