Automated spectral recomposition with application in stratigraphic interpretation

Introduction

Frequency-domain seismic attributes can be useful in stratigraphic and hydrocarbon reservoir characterization (Li et al., 2011; Castagna et al., 2003). If the seismic response can be captured at each frequency subset, the reservoir interval of interest can then be scrutinized in greater detail. Spectral decomposition is a technique that was developed at Amoco in the 1990's (Partyka et al., 1999). Various time-frequency analysis methods have been employed for frequency decomposition since then. Dilay and Eastwood (1995) applied short-time Fourier transform, which unfortunately suffers from a time-frequency resolution limit (Chakraborty and Okaya, 1995). Liu (2006) and Chen et al. (2008) applied spectral decomposition in the time domain by decomposing the input seismogram into constituent wavelets and then summing the Fourier spectra of individual wavelets. This approach experiences difficulties when the frequency range is large, because it relies on the accuracy of wavelet decomposition, whose residuals commonly introduce bias into ``frequency gathers'' (Chen et al., 2008). Liu et al. (2011) and Liu (2006) implemented spectral decomposition by time-frequency analysis using an iterative inversion framework with the help of local attributes.

Tomasso et al. (2010) defined frequency recomposition in seismic forward modeling as an estimation of components of the seismic spectrum. They showed how to make forward seismic models by recomposing single-frequency models into a multi-frequency model. Different from all previous approaches, spectral recomposition models and reconstructs the seismic spectrum instead of decomposing it. However, Tomasso et al. (2010) manually picked component frequencies and amplitudes, which may not be accurate and highly depends on personal experience. In this paper, we propose spectral recomposition using separable nonlinear least-squares estimation (Golub and Pereyra, 1973), which simultaneously and automatically estimates both linear and nonlinear parts of the Ricker wavelet spectrum. It provides an accurate and direct estimation of amplitudes and peak frequencies of various Ricker wavelets.

A problem is separable if the model can be represented as a linear combination of functions that have a nonlinear parametric dependence. A separable least-squares estimation fits frequencies and amplitudes with large variations, and provides computing confidence, as well as prediction and calibration intervals. The Gauss-Newton algorithm, a method of minimizing the residual sum of squares, is effective both when residuals are small and when measurement errors are additive and the data set is large (Osborne, 2007). An analogous method was used previously by Browaeys and Fomel (2009) for fitting von Kármán distributions, and by Liu and Fomel (2010) for fitting a single Ricker wavelet.

We represent a seismic spectrum as the sum of different Ricker components and use the Gauss-Newton method to fit it with a sum of Ricker wavelet spectra so as to estimate the peak frequency and amplitude of each component. On a field data example from the Gulf of Mexico, we show that automatic spectral recomposition can improve seismic stratigraphic interpretation and help in seismic attribute studies.

Automated spectral recomposition with application in stratigraphic interpretation