Geological Facies Recovery Based on Weighted $\ell _1$-Regularization

Abstract

A weighted compressed sensing (WCS) algorithm is proposed for the problem of channelized facies reconstruction from pixel-based measurements. This strategy integrates information from: (i) image structure in a transform domain (the discrete cosine transform); and (ii) a statistical model obtained from the use of multiple-point simulations (MPS) and a training image. A method is developed to integrate multiple-point statistics within the context of WCS, using for that a collection of weight definitions. In the experimental validation, excellent results are reported showing that the WCS provides good reconstruction for geological facies models even in the range of [0.3–1$\%$] pixel-based measurements. Experiments show that the proposed solution outperforms methods based on pure CS and MPS, when the performance is measured in terms of signal-to-noise ratio, and similarity perceptual indicators.

Appendices

Appendix A: The CS Recovery Theorem

For the problem presented in Sect. 2.1, the following result can be adopted to obtain performance guarantees for the recovery of ${\bar{x}}$ from ${\bar{y}}$ using sparsity as an assumption on ${\bar{x}}$.

Theorem 1

(Candès and Plan 2011, Theorem 1.2) For any $k<n$ and $\beta >0$, with probability $1-6/n - 6e^{-\beta }$, the solution to Eq. (4) satisfies that

$$\begin{aligned} \left| \left| {\hat{x}} -{\bar{x}} \right| \right| _{\ell _2} \le \frac{C (1+\alpha )}{\sqrt{k}} \cdot \left| \left| {\bar{z}} - {\tilde{z}}_k \right| \right| _{\ell _1}, \end{aligned}$$

(9)

provided that $m\ge C_\beta \cdot \mu (U) k \log n$, where

$$\begin{aligned} \mu (U) = n \cdot \max _{i,j \in \left\{ 1,\ldots ,n\right\} } \left| U_{i,j}\right| ^2, \end{aligned}$$

(10)

${\tilde{z}}_k\in {\mathbb {R}}^n$ is the best k-term approximation of ${\bar{z}}$, $\alpha =\sqrt{\frac{(1+\beta )k\mu \log n \log m \log ^2 k}{m}}$, $C_\beta =(1+\beta )C_0$, and C and $C_0$ numerical constants.

Thus, if the signal is k-sparse in the transform domain U, that is, $\left| \left| {\bar{z}} - {\tilde{z}}_k \right| \right| _{\ell _1} =0$, and the number of measurements satisfies that $m \ge C_\beta \cdot \mu (U) k \log n$, then ${\hat{x}}={\bar{x}}$ from (9). Therefore, the sparsity assumption on ${\bar{z}}$ and the use of a critical number of measurements (that scales like $O(k \log n)$) guarantees perfect recovery for the $\ell _1$-minimizer scheme in (4).

Appendix: Basis Selection

From Theorem 1, there are two elements to consider to address the problem of selecting U optimally for the recovery of ${\bar{x}}$ from ${\bar{y}}$. On the one hand, U should offer a good sparse representation of the facies images, in the sense that the approximation error $\left| \left| {\bar{z}} - {\tilde{z}}_k \right| \right| _{\ell _1} \approx 0$ for a small fraction of transform coefficients, that is for $k<<n$. On the other hand, the number of required measurements in (10) is proportional to $\mu (U)$, which is an indicator of how coherent (or orthogonal) is U with respect to the canonical (pixel-based) basis. Only looking at this second criterion, to minimize the number of measurements it is required a value of $\mu (U)$ close to 1, which is known to be the minimum value (coherent) that can be obtained for any pair of bases A and U. Integrating the two criteria, there is a tradeoff between the compressibility quality of U, which is the ability of U to represent ${\bar{z}} \equiv U^{\dagger } {\bar{x}}$ with a small proportion of coefficients, and how incoherent is U with respect to the pixel-based domain. More precisely using the result in Theorem 1, for a basis U, a signal ${\bar{x}}$ and a target reconstruction error $\epsilon >0$, the following object can be defined (Calderon et al. 2015)

$$\begin{aligned} k({\bar{x}},U,\epsilon ) \equiv \min \left\{ 1\le k\le n: \text {such that } \frac{C (1+\alpha )}{\sqrt{k}} \cdot \sigma _k(U^{\dagger } {\bar{x}})_{\ell _1}\le \epsilon \right\} , \end{aligned}$$

(11)

where $\sigma _k({\bar{z}})_{\ell _1}=\left| \left| {\bar{z}} - {\tilde{z}}_k \right| \right| _{\ell _1}$ is the approximation error introduced in (9). From Theorem 1, $k({\bar{x}},U,\epsilon )$ represents the critical number of significant (transform) coefficients (with respect to U) needed to have a reconstruction error smaller than $\epsilon$ [Eq. (9)].

Then, considering a collection of orthonormal bases $\left\{ U_j, j\in J\right\}$, the optimal one from Theorem 1 is the one that requires the minimum number of measurements to achieve an error $\epsilon$, that is, the solution of

$$\begin{aligned} U^*(\epsilon ,{\bar{x}})=\arg \min _{U_j, j\in J} k({\bar{x}},U_j,\epsilon )\cdot \mu (U_j). \end{aligned}$$

(12)

It is shown in (Calderon et al. 2015, Sect. 5) that over a rich collection of Wavelet bases, the discrete cosine transform (DCT) by far offers the best tradeoff between compressibility and incoherence to recover channelized facies from pixel-based measurements for the classical CS algorithm in (4). This was shown for different channelized facies models and reconstruction errors. Therefore, the DCT was the basis adopted in this work.