Robust slope estimation

Another important factor in plane-wave orthogonal polynomial transform is the local slope calculation. The accuracy of the slope estimation affects performance of the flattening operation and the following OPT. In this part, we will introduce a robust slope estimation method that is based on the Hilbert transform (Liu et al., 2015).

Rearranging equation 10 we get

$$\sigma = -\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial t}}.$$ (15)

Equation 15 can be further derived such that

$$\begin{split} \sigma = -\frac{\frac{\partial u}{\partial x}}{... ... [ H_{DX}[ F_x u ] ]}{F_t^{-1} [H_{DT}[ F_t u ] ]}, \end{split}$$ (16)

where $H_{DX}$ is frequency response function of the partial derivative in the $x$ direction, and $H_{DT}$ is frequency response function of the partial derivative in the $t$ direction. $F_x$ and $F_t$ denote the Fourier transform along the $x$ and $t$ directions, respectively. It can be straightforwardly derived that

$$\sigma =-\frac{\mathcal{H}_x(u)}{\mathcal{H}_t(u)},$$ (17)

where $\mathcal{H}_x(u)$ denotes the Hilbert transform of $u$ along $x$ direction and $\mathcal{H}_t(u)$ denotes the Hilbert transform of $u$ along $t$ direction.

Figure 4 shows a slope calculation test. We calculate the slope from the noisy data using the traditional PWD method and the robust slope calculation method, respectively. As a comparison, an accurate slope estimation from the clean data using the PWD algorithm is used to evaluate the robustness of different slope estimation approaches in the case of noise. Figure 4a shows the clean data, and Figure 15 shows the slope estimated from the clean data using the PWD algorithm, which is deemed to be the accurate slope. Figure 4b shows the noisy data by adding some Gaussian white noise. Figure 4d shows the slope calculated using the robust slope estimation. It is salient that the slope estimated from the noisy data is fairly close to the accurate slope field. However, using the traditional PWD algorithm, it is difficult to obtain an acceptable slope estimation from the noisy data, as can be seen from the result shown in Figure 4e. From this test, we conclude that the robust slope estimation can be used to obtain robust slope estimation performance even in the presence of strong random noise.

It is worth mentioning that, by equations 10 and 15, we do not consider the spatial gradient of amplitude. In the case of smooth spatial amplitude change (e.g., small spatial gradient), the slope estimation method also works, since the calculation is done locally and the small spatial gradient almost has no influence. However, in the case of sharp spatial amplitude change (e.g., large spatial gradient), the method cannot be adopted. This drawback can be hopefully overcome in the future work. In addition, the problem of spatial gradients of amplitude and the implications for non-plane wave solutions was mentioned in Wielandt (1993).

synth,synth-n,dip,dip2,dip1
Figure 4. Slope calculation test. (a) Clean data. (b) Noisy data. (c) Slope calculated from the clean data using PWD algorithm. (d) Slope calculated from the noisy data using the robust slope calculation algorithm. (e) Slope calculated from the noisy data using the PWD algorithm.