Smoothing Three-Dimensional Manifold Data, with Application to Tectonic Fault Detection

Abstract

Three-dimensional manifold data arise in many contexts of geoscience, such as laser scanning, drilling surveys and seismic catalogs. They provide point measurements of complex surfaces, which cannot be fitted by common statistical techniques, like kriging interpolation and principal curves. This paper focuses on iterative methods of manifold smoothing based on local averaging and principal components; it shows their relationships and provides some methodological developments. In particular, it develops a kernel spline estimator and a data-driven method for selecting its smoothing coefficients. It also shows the ability of this approach to select the number of nearest neighbors and the optimal number of iterations in blurring-type smoothers. Extensive numerical applications to simulated and seismic data compare the performance of the discussed methods and check the efficacy of the proposed solutions.

Appendices

Appendix 1: Proof of Results (14)

The proof of Eq. (14) need assumptions similar to those of Comaniciu and Meer (2002). The Gaussian kernel $K(\cdot )$ in (11) is based on the profile function $k(z)=\exp (-z/2)$, which is positive, decreasing and convex. Letting $c=1 / \Big ( (2 \pi )^{m/2} N \beta ^m \Big )$, the expression in (14, i) can then be written as

$$\begin{aligned} \Big [ \hat{f}_N ( \hat{\varvec{y}}_i^{(t+1)} ) - \hat{f}_N ( \hat{\varvec{y}}_i^{(t)} ) \Big ] = c \sum _{j=1}^N \bigg [ k \Big ( \Big \Vert \, \hat{\varvec{y}}_i^{(t+1)} - {\mathbf{x}}_j \Big \Vert ^2 / \beta ^2 \Big ) - k \Big ( \Big \Vert \, \hat{\varvec{y}}_i^{(t)} - {\mathbf{x}}_j \Big \Vert ^2 / \beta ^2 \Big ) \bigg ] , \end{aligned}$$

(30)

for all i, t. Convex functions have the property that $[k(z_2)-k(z_1)]\ge -k'(z_1) (z_1 - z_2)$ for all $z_i , z_2 > 0$, and for negative exponentials $-k'(z)=k(z)$. Thus, the above expression satisfies the inequality

$$\begin{aligned} (30) \ge \frac{c}{2 \beta ^2} \sum _{j=1}^N k \Big ( \Big \Vert \, \hat{\varvec{y}}_i^{(t)} - {\mathbf{x}}_j \Big \Vert ^2 / \beta ^2 \Big ) \cdot \Big ( \Big \Vert \, \hat{\varvec{y}}_i^{(t)} - {\mathbf{x}}_j \Big \Vert ^2 - \Big \Vert \, \hat{\varvec{y}}_i^{(t+1)} - {\mathbf{x}}_j \Big \Vert ^2 \Big ) . \end{aligned}$$

(31)

By defining $\hat{k}_{ij}^{(t)} = k \Big ( \Big \Vert \, \hat{\varvec{y}}_i^{(t)} - {\mathbf{x}}_j \Big \Vert ^2 / \beta ^2 \Big )$, the weights $\hat{w}_{ij}^{(t)} = \hat{k}_{ij}^{(t)} / \sum _j \hat{k}_{ij}^{(t)}$ and the constants $C_t = c \sum _j \hat{k}_{ij}^{(t)} / 2 \beta ^2$, the above can be written as

$$\begin{aligned} (31)= & {} C_t \sum _{j=1}^N \hat{w}_{ij}^{(t)} \, \Big ( \Big \Vert \, \hat{\varvec{y}}_i^{(t)} - {\mathbf{x}}_j \Big \Vert ^2 - \Big \Vert \, \hat{\varvec{y}}_i^{(t+1)} - {\mathbf{x}}_j \Big \Vert ^2 \Big ) ,\nonumber \\\ge & {} C_t \sum _{j=1}^N \hat{w}_{ij}^{(t)} \, \Big \Vert \, \hat{\varvec{y}}_i^{(t)} - \hat{\varvec{y}}_i^{(t+1)} \Big \Vert ^2 = C_t \, \Big \Vert \, \hat{\varvec{y}}_i^{(t+1)} - \hat{\varvec{y}}_i^{(t)} \Big \Vert ^2 , \end{aligned}$$

(32)

where $C_t > 0$. Inequality (32) follows as in the proof of Theorem 1 of Li et al. (2007); thus, combining (30) and (32) one has

$$\begin{aligned} \Big [ \hat{f}_N ( \hat{\varvec{y}}_i^{(t+1)} ) - \hat{f}_N ( \hat{\varvec{y}}_i^{(t)} ) \Big ] \ge C_t \, \Big \Vert \, \hat{\varvec{y}}_i^{(t+1)} - \hat{\varvec{y}}_i^{(t)} \Big \Vert ^2 \ge 0 , \end{aligned}$$

(33)

for all i, t. This implies that $\Big \{ \hat{f}_N ( \hat{\varvec{y}}_i^{(t)}) \Big \}$ is non-decreasing in t, but since it is bounded as k(z), it converges for all i, as stated by Eq. (14, i).

Also the result (14, ii) follows from (33), by noting that $\min _t C_t > 0$ and $\hat{\varvec{y}}_i^{(t)}$ is bounded. In fact, having $\min _t \Big \{ \hat{f}_N ( \hat{\varvec{y}}_i^{(t)}) \Big \} = \hat{f}_N ({\mathbf{x}}_i) > 0$ (from the initial condition in (11)), this implies the boundedness of $\hat{\varvec{y}}_i^{(t)}$ (otherwise it would be $\min _t \hat{f}_N = 0$). As a consequence, there exists $D>0$ such that $\max _j \Vert \, \hat{\varvec{y}}_i^{(t)} - {\mathbf{x}}_j \Vert ^2 \le D$ and therefore $C_t \ge c \, k(D/\beta ^2) / 2 \beta ^2 = C > 0$. Using C in (33), the result (14, i) implies (14, ii).

Fig. 12

Application of other MS methods to the data of Fig. 1 with $\beta =1.33$. a, b Hessian method (12). c, d Method of Ozertem and Erdogmus (2011)

Appendix 2: Results of MS Estimates

This appendix reports the manifold estimates of the data in Fig. 1 obtained with the algorithm (13) with the full Hessian matrix (12), and the method of Ozertem and Erdogmus (2011) based on the Hessian estimation of the inverse covariance. The two estimators use the same bandwidth $\beta =0.133$, identified in Fig. 4(b)(d). Results in Fig. 12 show that the first algorithm has a significant bias in two-dimensional at the corners, whereas the second reduces the three-dimensional sphere to a disk.