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Wavefield Reconstruction Using Wavelet Transform CWP-930 Wavefield reconstruction using wavelet transform Iga Pawelec1, Paul Sava1 & Michael Wakin2 1Center for Wave Phenomena, Colorado School of Mines 2 SINE Center for Research in Signals and Networks, Colorado School of Mines ABSTRACT We propose wavelet-based data reconstruction to interpolate land data with large dy- namic range without amplitude processing or windowing. We test two approaches us- ing compressive sampling to recover full unaliased data: sparsity promoting recon- struction by `1 minimization and projection onto convex sets. Unlike the Fourier do- main, the wavelet domain provides a good representation of non-stationary signals and allows to rebuild data of high dynamic range with relatively small percentage of all coefficients. We solve an `1 minimization problem to find a sparse representation of full data in the wavelet domain and compare it with results of a wavelet-domain POCS algorithm. Tests on synthetic and field data reveal that both approaches can recover missing data highly coherent with the existing data, while taking advantage of the full dynamic range of the data. Key words: interpolation, wavelets, compressive sensing, land acquisition 1 INTRODUCTION vided in several categories, including prediction error filters (PEFs) (Spitz, 1991), tensor completion (Kreimer et al., 2013), Modern techniques of seismic data analysis and inversion, rank reduction (Chen et al., 2016) and deep learning (Wang such as AVO/AVA, full waveform inversion (FWI) and least et al., 2019). However, the most widely used interpolation squares reverse time migration (LSRTM), operate on big data techniques are transform-based approaches. Such methods are volumes. However, multi-azimuth, long offset dense surveys well-studied in the context of data aliasing and irregular sam- are expensive to acquire and sometimes access restrictions pling and rely on data representation in a transform domain do not permit acquisition over certain areas, resulting in data to recover missing information. Although different transforms gaps. Seismic data reconstruction plays a key role in both sce- have been used, including Radon transform (Kabir and Ver- narios, allowing to fill-in missing data before advanced pro- schuur, 1995; Yu et al., 2007; Wang et al., 2010) and wavelet cessing and inversion are performed. or seislet transforms (Yu et al., 2007; Gan et al., 2015), the Mosher et al. (2017) show that a novel approach, utiliz- Fourier transform remains the most popular choice because it ing ideas from compressive sensing, can significantly speed up is easy to interpret and fast to compute. Liu and Sacchi (2004) acquisition without jeopardizing data quality. The underlying develop a framework for data recovery based on weighted idea is to randomize receiver placement and shot timing ac- norm minimization, using spectral weights bootstrapped from cording to compressive sampling rules and solve a large scale FK representation of data. This framework can be extended to regularization problem, recovering a full data volume from re- five dimensions (Trad, 2009). The Fourier domain is also used duced measurements. To recover full data, one must develop in the projection onto convex sets (POCS) method described a strategy for dense data recovery that takes into account ac- by Abma and Kabir (2006). To deal with problems of non- quisition geometry: parameters such as source and receiver lo- uniform sampling and aliasing artifacts in the Fourier domain, cations, maximum gap size and timing of the shots have to Xu et al. (2005, 2010) propose an antileakage version of the be considered. For example, most conventional 3D acquisition Fourier transform. One downside of Fourier-based approaches geometries have regular but poor sampling in at least one di- is that data have to be windowed for non-stationarity and, as rection (Trad, 2009) and one has to contend with aliasing and a consequence, only local information is used for interpola- sometimes big data gaps due to access restrictions. In contrast, tion. Another attractive transform for seismic data interpola- compressive sensing surveys would have deliberate irregular tion, gaining significant popularity, is the curvelet transform sampling and often simultaneous shooting. Thus, data recov- (Hennenfent et al., 2010; Herrmann, 2010; Naghizadeh and ery strategy needs to be tuned with particular geometry restric- Sacchi, 2010). Curvelets provide an optimally sparse represen- tions in mind. tation of seismic wavefields (Candes` and Demanet, 2005), but Current techniques for infilling missing data can be di- their redundancy implies that for a dataset of size N, as many 2 I. Pawelec, P. Sava & M. Wakin as 7 × N curvelet coefficients have to be computed, depend- Historically, seismic data have been acquired on a reg- ing on the chosen number of scales, which can be prohibitively ular grid or have been regularized after acquisition - a prag- expensive for large 3D datasets. matic choice, since many processing and imaging algorithms In this paper, we present a seismic data reconstruction require regular spacing. However, such acquisition is limited strategy exploiting wavelet domain sparsity under randomized by the Nyquist - Shannon sampling theorem (Candes` et al., acquisition. We discuss the features of seismic signals in the 2006a) which dictates a sampling rate of at least two points wavelet domain, review the theory highlighting the favorable per wavelength for successful recovery of a non-aliased sig- recovery conditions and present data reconstruction results for nal. Furthermore, the number of sensors needed to record good synthetic and field data with two strategies: projection onto quality, unaliased land data on a regular grid is exceedingly convex sets (POCS) and `1 norm minimization. Wavelet trans- high. The advent of compressive sensing (CS) (Candes` et al., form is fast to compute and represents well large dynamic 2006b) opened new, exciting possibilities for signal recon- range in data. The POCS approach iteratively restores missing struction from incomplete information. Hennenfent and Her- data, but its success hinges on developing a good thresholding rmann (2008) and Herrmann (2010) examine randomized ac- strategy. `1 optimization is less robust to acquisition geometry, quisition using much fewer sensors than a regular-grid sur- but does not need any thresholding. vey and achieve comparable data density and quality. Mosher et al. (2017) demonstrate that compressive sensing can be suc- cessfully applied to field seismic acquisition. The main re- quirement for data reconstruction is sparse representation in a 2 CHALLENGES IN DATA RECONSTRUCTION known transform domain. We discuss sparse recovery in more detail in the following section. Several challenges have to be addressed for successful data The curvelet domain is optimal for representing wave reconstruction: the presence of data aliasing, the pattern of phenomena (Candes` and Demanet, 2005). The curvelet trans- missing traces and size of data gaps, and the dynamic range form divides the frequency plane into dydaic bands which are of seismic data. These challenges are more prominent for land then split into overlapping angular wedges doubling in every seismic acquisition due to the highly complex heterogeneous other dydaic scale. The curvelet transform is highly redundant: shallow subsurface, which traps a large portion of energy re- there is no unique representation of a signal in the curvelet do- leased by the seismic source and produces slowly propagating main and the number of curvelet coefficients is much larger surface waves (Keho and Kelamis, 2012). In the following, we than the number of data points. This feature of the curvelet discuss these challenges in more detail and explain how differ- transform is favorable for denoising and finding sparse sig- ent transforms handle them. nal representation, at the expense of increased storage require- In land seismic data, aliasing of surface waves can be ments, which makes curvelets a memory-expensive choice for especially severe due to the much slower velocities of sur- large datasets. face waves compared with the body waves. Figure 1 shows the same land data record sampled at different trace intervals The wavelet transform on the other hand offers a good (coarse sampling results from discarding a portion of the full middle ground between the frequency and curvelet domains. data) and the corresponding frequency spectra. Aliasing oc- Wavelets provide a so called multiresolution approximation curs in this example even at 2.5m sampling interval, which is and in 2D are sensitive to three directions: horizontal, vertical 10 times finer than what would commonly be used in big land and diagonal. Although the wavelet representation of wave- surveys. Severe aliasing makes it difficult to use surface waves fields is less sparse than a curvelet representation, the wavelet for characterizing the shallow subsurface or to remove the sur- transform can be orthogonal, providing a unique representa- face waves from the seismic record entirely. tion of the signal and preserving its total energy. Another ad- One way to overcome the aliasing problem is by data re- vantage of the wavelet transform is its computational speed construction exploiting prediction error filters (PEF) (Spitz, surpassing even that of a Fast Fourier Transform, thus making 1991). The underlying idea is that filter coefficients derived wavelets suitable for analysis of large datasets. from low, unaliased frequencies can be used to interpolate Wavelets are
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