Using Diffusion Maps for Latent Space Analysis of Seismic Waveforms of Incised Valleys in the Anadarko Basin, Oklahoma, USA

In this work we present a new method for latent space mapping based upon inter-point similarities. This method, diffusion maps, has a number of nice qualities compared with previous methods including the fact that it is based upon inter-point similarities rather than a Euclidean space. We then demonstrate application of this approach to mapping an incised valley system from the Anadarko Basin, Oklahoma, USA. Multi-dimensional data is commonly encountered in attribute analysis where the desire is to combine several attributes with complementary properties. As geophysicists, we tend to view attribute spaces higher than four dimensional as being undesirable as they cannot be visualized using common color models such as ARGB space. Furthermore, mathematical considerations such as the Curse of Dimensionality (Bellmann, 1957) make working in lower dimensional space necessary. (Guo, Marfurt, Liu, & Dou, 2006) discussed an unsupervised learning method for doing dimensionality reduction of attributes using Principal Component Analysis (PCA). While this method has been shown to be useful in a broad range of applications, they are limited in their ability to capture non-linear structure in multidimensional attribute space. Retaining this non-linearity is important as data sets generally present a heterogeneous mix of latent processes that, taken as a whole, are unlikely to be well represented by a single lowdimensional linear manifold. A number of non-linear methods of manifold learning have been applied to seismic attributes and waveform modeling (Wallet & Marfurt, 2008) . Self organizing maps (SOM) is the best known of these approaches, and it is available in a number of commercial products. (Wallet & Perez, 2009) also demonstrated the use of a statistical method, generative topographical maps (GTM). (Wallet & Perez, 2009) applied diffusion maps to the problem of modeling well log data. They noted that the high computational demands of diffusion maps was an impediment to scaling to reasonable sized seismic problems. In this paper, we present an estimation method that deals with this problem.