Autoregressive modeling is used to estimate the multi-dimensional spectrum of aliased data. A region of spectral support is determined by identifying the location of peaks in the estimated spectrum. This information is used to pose a Fourier reconstruction problem that inverts for the few dominant wavenumbers that are required to model the data. Synthetic and real data examples are used to illustrate the method. In particular, we show that the proposed method can accurately reconstruct aliased data and data with gaps. The method provides a unifying thread between band-limited and sparse Fourier reconstruction, and f-x and f-k interpolation methods.
Spitz (1991) showed how one could extract prediction filters from spatial data at low frequencies to reconstruct aliased spatial data. This idea was expanded by Naghizadeh and Sacchi (2007) and used to reconstruct data with irregular distribution of traces on a grid. The latter is named Multi-Step Auto-Regressive (MSAR) reconstruction. The MSAR reconstruction method is a combination of a Fourier reconstruction method (Liu and Sacchi, 2004) and f-x interpolation (Spitz, 1991). MSAR can be summarized as follows:
- The low frequency (unaliased) portion of data is reconstructed using Minimum Weighted Norm Interpolation (MWNI) (Liu and Sacchi, 2004).
- Prediction filters of all frequencies are extracted from already regularized low frequency spatial data.
- The estimated prediction filters are used to reconstruct the missing spatial samples in the aliased portion of the spectrum.
1) and 2) are estimation stages and 3) is the reconstruction stage. In this paper we propose a new and robust method to solve the reconstruction stage. In the original formulation of MSAR the reconstruction stage uses prediction filters harvested from low frequencies to reconstruct spatial data in the aliased portion of the spectrum (Spitz, 1991). In this article we propose to use the autoregressive (AR) spectrum of the data to define a region of spectral support. Once the region of spectral support (areas of unaliased energy in the f-k plane) is defined we turn the reconstruction problem into a Fourier reconstruction algorithm that solves for unknown spectral components using the least-squares method (Duijndam et al., 1999). Synthetic examples illustrate that the proposed method can handle gaps and extrapolation problems much better than our original formulation of MSAR (Naghizadeh and Sacchi, 2007). EXAMPLES Synthetic example
The first example is a 2D spatial data set composed of three dipping planes which are aliased in both spatial directions. The data set contains 1200 traces distributed on a 30×40 regular spatial lattice. Figure 1a shows the original data. The top view is a time slice at 0.65 (s), the front view is the 21st slice in the Y direction, and the side view is 17th slice in the X direction. Figure 1b shows the f-k panel of the data from the front view of Figure 1a. The f-k representation shows the presence of aliasing as well as random noise in the original data. A data set with missing traces was simulated by first eliminating every other X slices of data and, in addition, by randomly eliminating 75% of the remaining traces (Figure 1c).