# Effects of wrong adjoints for RTM in TTI media.

Mathias Louboutin1*, Philipp Witte2 and Felix J. Herrmann1
1. Georgia Insitute of Technology, department of Computational Science and Engineering.
2. Seismic Laboratory for Imaging and Modelling (SLIM), University of British Columbia

## Summary:

In order to obtain accurate images of the subsurface, anisotropic modeling and imaging is necessary. However, the twenty-one parameter complete wave-equation is too computationally expensive to be of use in this case. The transverse tilted isotropic wave-equation is then the next best feasible representation of the physics to use for imaging. The main complexity arising from transverse tilted isotropic imaging is to model the receiver wavefield (back propagation of the data or data residual) for the imaging condition. Unlike the isotropic or the full physics wave-equations, the transverse tilted isotropic wave-equation is not not self-adjoint. This difference means that time-reversal will not model the correct receiver wavefield and this can lead to incorrect subsurface images. In this work, we derive and implement the adjoint wave-equation to demonstrate the necessity of exact adjoint modeling for anisotropic modeling and compare our result with adjoint-free time-reversed imaging.

## Theory

Transverse-isotropic wave equations provide a kinematically accurate acoustic representation of the physics with a reduced number of parameters in comparison to the general elastic wave equation (five parameters instead of 21). We refer to Thomsen (1986), Alkhalifah (2000) and L. Zhang et al. (2005) for a detailed derivation and justification of the formulation and concentrate here on the derivation of the adjoint TTI wave-equation and the its relevance for imaging. In a TTI medium, the governing equations with the conventional physical parametrization (squared slowness $$m(\mathbf{x})$$, Thomsen parameters $$\epsilon(\mathbf{x}), \delta(\mathbf{x})$$ and tilt and azimuth $$\theta(\mathbf{x}), \phi(\mathbf{x})$$) are given by (Thomsen, 1986, Y. Zhang et al. (2011), Duveneck and Bakker (2011)): \begin{aligned} &m(x) \frac{d^2 p(\mathbf{x},t)}{dt^2} - (1+2\epsilon(\mathbf{x}))(G_{\bar{x}\bar{x}} +G_{\bar{y}\bar{y}} )p(\mathbf{x},t) \\ &- \sqrt{(1+2\delta(\mathbf{x}))}G_{\bar{z}\bar{z}} r(\mathbf{x},t) = q, \\ &m(x) \frac{d^2 r(\mathbf{x},t)}{dt^2} - \sqrt{(1+2\delta(\mathbf{x}))}(G_{\bar{x}\bar{x}} +G_{\bar{y}\bar{y}} ) p(\mathbf{x},t) \\ &- G_{\bar{z}\bar{z}} r(\mathbf{x},t) = q, \\ \end{aligned} \label{TTIfwd} where $$G_{\bar{z}\bar{z}}, G_{\bar{x}\bar{x}}, G_{\bar{y}\bar{y}}$$ are the rotated second order differential operators that depend on the tilt, azimuth and the conventional (isotropic) spatial derivatives $$\frac{d}{dx}, \frac{d}{dy}$$ and $$\frac{d}{dz}$$. As discussed in Y. Zhang et al. (2011) and Duveneck and Bakker (2011), we consider the discrete finite-difference representations of the three differential operators $$G_{\bar{z}\bar{z}}, G_{\bar{x}\bar{x}}, G_{\bar{y}\bar{y}}$$ to be self-adjoint to ensure numerical stability. For example, we can choose: \begin{aligned} G_{\bar{x}\bar{x}} &= D_{\bar{x}}^T D_{\bar{x}} \\ D_{\bar{x}} &= cos(\theta(\mathbf{x}))cos(\phi(\mathbf{x}))\frac{d}{dx} + cos(\theta(\mathbf{x}))sin(\phi(\mathbf{x}))\frac{d}{dy} - sin(\theta(\mathbf{x}))\frac{d}{dz}. \end{aligned} \label{rot} The rotated finite-difference operators contain the angles of the TTI symmetry axis, which are assumed to be spatially varying as described in equation $$\ref{rot}$$. The squared slowness and Thomsen parameters are spatially varying as well, under the assumption that $$\epsilon > \delta$$. With the TTI system of equations as defined above, we consider the standard zero-lag cross-correlation imaging condition for reverse-time migration for the case of couple equations: $$$I = \sum_{t=1}^{n_t} ( \ddot{p} p_a + \ddot{r} r_a). \label{TTIgrad}$$$ In the above expression, $$p_a, r_a$$ are the two components of the adjoint (TTI) wavefields and $$\ddot{p}, \ddot{r}$$ are the second order time derivatives of the two forward wavefields. Note, that the gradient is computed as the sum of the separate correlation of the two components, not the correlation of the sum of the components. Furthermore, the adjoint wavefields $$p_a, r_a$$ cannot be computed by forward propagating the time-reversed shot records, but require solving the actual adjoint TTI wave equations. Even though the spatially varying finite-difference operators $$G$$ are self-adjoint by construction, the overall system itself is not self-adjoint, since the operators are multiplied with terms that contain the spatially varying Thomsen parameters. The correct adjoint system of equations corresponding to the TTI forward wave equations (equation $$\ref{TTIfwd}$$) is given by: \begin{aligned} &m(x) \frac{d^2 p_a(\mathbf{x},t)}{dt^2} - \\ &(G_{\bar{x}\bar{x}} +G_{\bar{y}\bar{y}})((1+2\epsilon(\mathbf{x}))p_a(\mathbf{x},t) + \sqrt{1+2\delta(\mathbf{x})} r_a(\mathbf{x},t)) = q_a, \\ &m(x) \frac{d^2 r_a(\mathbf{x},t)}{dt^2} - G_{\bar{z}\bar{z}}(\sqrt{1+2\delta(\mathbf{x})} p_a(\mathbf{x},t) + r_a(\mathbf{x},t)) = q_a, \\ \end{aligned} \label{TTIadj} where $$q_a$$ is the adjoint source, which, for RTM, are the observed shot records. Compared to the forward TT wave equations, the adjoint system is fundamentally different. While the forward system consists of two coupled PDEs that are merely two rotated and weighted acoustic wave-equations, the corresponding adjoint system consists of two fully decoupled horizontal/vertical equations. To illustrate the differences of the true adjoint wave equations and the time-reversed equations, we compare the impulse response of the subsurface for both cases by back-propagating a time-reversed Ricker wavelet, injected at a single receiver location.

Figure 1 shows that the impulse responses of the time-reversed and adjoint TTI systems are drastically different. As expected, the amplitudes along the wave-fronts differ, since the adjoint wavefield consists of a purely vertical and a purely horizontal components, while the time-reversed wavefields are a combination of both. The difference plots in the bottom row of Figure 1 show that the amplitudes of wave fronts differ throughout the full domain and will therefore lead to different results, once correlated for an RTM image. Furthermore, we plot two traces of the wavefields that were extracted at the locations indicated by the red lines (Figure 2). While the amplitude difference in the snapshots may appear only minor, the trace plots reveal that the amplitude differences are in fact substantial. The vertical trace shows that the amplitude differs up to a factor of five and that the amount of energy in the two wavefield components seems to be flipped (i.e. $$p_a$$ contains more energy than $$r_a$$). The most problematic differences can be observed in the horizontal trace comparison, where wee see that the energy of the wavefields is flipped between the left and right side of the wavefield (with respect to the source), which means the subsurface illumination of the time-reversed wavefield in incorrect.

## Reverse Time Migration

We finally show a Reverse-Time-Migration (RTM) example on the 2007 BP TTI dataset. We compare in this example the images obtained with the true adjoint wavefields $$\eqref{TTIadj}$$ and the time-reversed “adjoint” that propagates backward in time Equation $$\ref{TTIfwd}$$ for the “adjoint” modeling. We use a 20 Hz Ricker wavelet and back propagate the muted data as in Sun et al. (2016) without any extra processing or filtering.

The two images do look similar, both contain the salt body and the anticline and complex events on the left. However there are a number of substantial differences between these two images that require a closer look. We show on Figure 4 three parts of the final image overlaid with the background velocity model. The three selected areas are highlighted in Figure 3 and are within the part of the model with a non zero tilt angle.

The first difference to notice on these zoomed images, is that the salt boundaries are imaged at incorrect positions with the adjoint-free time-reversed method. The bottom flat layer is too deep while the vertical channel is shifted to the left. On the second zoomed in part, we observe that the image obtained with the true adjoint is more focused and highlights more reflectors aligned with the velocity variations while the time-reversed one is missing a lot of reflectors on the left of the anticline. We also see that that the reflectors matching the high velocity at the bottom are blurry and too deep with the time-reversed method. Finally, on the last image, both methods focus at the correct positions, however, the time-reversed image is unfocused and has missing or blurred reflectors. While these differences may look minor, the shifts in space are in the order of hundred of meters and are non negligible errors for interpretation.

## Conclusions

In this work, we detailed the main differences between adjoint-free time-reversal imaging and imaging with proper adjoints in a transverse tilted isotropic medium. We demonstrated on a realistic dataset that time-reversal does not image the subsurface correctly and that rigorous adjoint modeling is necessary for anisotropic imaging. Even though the time-reversed image perhaps seems good by itself, we showed that the subsurface events actually do not align with the velocity once overlaid. Such a displacement with good focusing could lead to incorrect interpretation. On the other hand, we demonstrated improved imaging accuracy with adjoint modeling that places subsurface events at the correct locations and creates a more focused image in several areas such as the anticline region.

Finally, adjoint modeling provides one extra tool that would not be possible with time-reverse imaging. While reverse-time migration does provide good images, complex model such as the one we presented here, requires a least-squares solution to image areas with poor illumination or steep events such as the flanks of the salt body. These least-square methods require an exact adjoint to stay stable and converge. With the correct adjoint implemented, the next step is to incorporate our simulator in a least-square imaging workflow to improve the final image.

## Acknowledgements

This research was carried out within Georgia Institute of Technology, School of Computational Science and Engineering. This research was carried out as part of the SINBAD II project with the support of the member organizations of the SINBAD Consortium.

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