^{1}

^{2}

Commonly, seismic data processing procedures, such as stacking and prestack migration, require the ability to detect bad traces/shots and restore or replace them by interpolation, particularly when the seismic observations are noisy or there are malfunctioned components in the recording system. However, currently available trace/shot interpolation methods in the spatial or Fourier domain must deal with requirements such as evenly sampled traces/shots, infinite bandwidth of the signals, and linear seismic events. In this paper, we present a novel method, termed the E-S (eigenspace seismic) method, using principal component analysis (PCA) of the seismic signal to address the issue of reliable detection or interpolation of bad traces/shots. The E-S method assumes the existence of a correlation between the observed seismic entities, such as trace or shot gathers, making it possible to estimate one of these entities from all others for interpolation or seismic quality control. It first transforms a trace (or shot) gather into an eigenspace using PCA. Then in the eigenspace, it treats every trace as a point with its loading scores of PCA as its coordinates. Simple linear, bilinear, or cubic spline 1 dimensional (1D) interpolation is used to determine PCA loading scores for any arbitrary coordinate in the eigenspace, which are then used to construct an interpolated trace for the desired position in physical space. This E-S method works with either regular or irregular sampling and, unlike various other published methods, it is well-suited for band-limited seismic records with curvilinear reflection events. We developed related algorithms and applied these to processed synthetic and offshore seismic survey data with or without simulated noises to demonstrate their performance. By comparing the interpolated and observed seismic traces, we find that the E-S method can effectively assess the quality of the trace, and restore poor quality data by interpolation. The successful processing of synthetic and real data using the E-S method presented in this approach will be widely applicable to seismic trace/shot interpolation and seismic quality control.

It is necessary to evaluate the quality of observed seismic records, and to detect and interpolate bad traces or shots [

A significant advancement in seismic data interpolation came with the appearance of the f-x method [

The f-k method as discussed in [

The 2 dimensional (2D) prediction error filter (PEF) as discussed in [

Reference [

The method given by [

References [

Reference [

Recently, References [

Reference [

In this paper, we present the novel method, termed Eigenspace Seismic (E-S), for seismic data interpolation and quality evaluation, using PCA, without assuming linear seismic events, infinite bandwidth, or regular spatial sampling. Currently, there is no published example showing this method and the applications demonstrated in our study. Though many examples exist for seismic data processing using PCA [

In the E-S method, we treat a seismic trace as a trace vector and arrange the trace vectors of any type of trace gather (e.g. common shot, common receiver, common mid-point, common offset, etc.) a trace matrix with its traces as rows, without losing any generality. For a shot gather, traces of a shot are concatenated to be a shot vector, after interpolation, we decipher the shot vector in the same way with the concatenation procedure. We fulfill the PCA of trace or shot matrix among their rows using the singular value decomposition (SVD) method to preserve energy and have high calculation performance.

In the following sections, we will first introduce the method and its data processing procedures (Section 2), and then test its performance using both a synthetic example and a trace and shot gather of offshore seismic data (Section 3). We discuss the working principal, possible improvement, and conclusions in Section 4.

The E-S method handles a whole trace, or shot, as an inseparable entity. A trace gather spans a vector feature space that originates from its correlation structure and has spatial metrics among these entities. However, this raw feature space is sparse and we can transform it into an eigenspace with linear properties using PCA. In the eigenspace, we can reconstruct an entity, such as a trace at any observation point, or construct a virtual entity, such as a virtual trace at an unknown location, using 1D interpolation of their coordinate trajectories in the eigenspace. Comparing the reconstructed entities with the observed ones enables us to evaluate seismic quality, and the construction of virtual entities is used to fill gaps for seismic trace/shot interpolation.

To simplify the descriptions, we will start from the E-S method using a common mid-point gather. Without losing any generality, we assume the seismic profile direction is x and acknowledge its m recorded traces with the trace sampling length (n). If we assume a trace at the line coordinate i (x = i) as a vector s i (i = 0, m − 1), then we can express this trace gather as an m * n matrix S:

S = ( s 0 , s 1 , s 2 , ⋯ , s m − 1 ) T (1)

where T means transpose of a matrix. All traces or rows of matrix S will span a vector-space [

S = U Σ V T (2)

where S, U, ∑, and V are the seismic trace matrix, left eigenvector matrix, middle diagonal singular value matrix, and right eigenvector matrix with dimensions m * n, m * r, r * r, and r * n (r is the rank of matrix S), respectively. The columns of V T are the bases of the constructed eigenspace and S has been mapped into the eigenspace having the rows of U as its loading scores or coordinates in the eigenspace, and the diagonal entries of ∑ as its spectrum reflecting its energy contribution. Note that rows and columns of U and V T are unit (length) and mutually orthogonal, we can write any trace s_{i} as a row of S as:

s i = Σ i u i v i T = Σ i ( u i × v i T ) ( i = 0 , m − 1 ) (3)

where s i , Σ k , u k , ×, and v k T are i^{th} row of S, i^{th} entry of ∑, i^{th} row of U, outer product, and i^{th} column of V T , respectively.

Rows of matrix U are the coordinates of the entities, such as PCs of traces, in the eigenspace. According to Equation (3), seismic traces can be reconstructed in the eigenspace using V T as its base and U as its “loading scores” (coordinates in the eigenspace) and its spectral density ∑ when the trace number i (i = 0, m − 1) or row of S is limited to be an integer. If we relax i to be a real number x (distance along the seismic profile), such as 0.5 which means it locates in the middle between the first and second trace, then we can deduce the following equation from Equation (3):

s x = u x × ( Σ i v i T ) (4)

We define the constructed s_{x} as a virtual trace at position x. During this procedure, V T , the base of eigenspace, and ∑, the energy distribution on every base of eigenspace, do not change. To calculate s x , we estimate u x along the coordinate trajectories of u i (Equation (3)) using a 1D interpolation method. Practically, we use the simple linear, bilinear or cubic spline 1D interpolation method to estimate u x ( 0.0 ≤ x ≤ m ) from U. Currently, we do not use the E-S method for extrapolation outside the range of traces.

We can revise the E-S method for detection and restoration of bad traces if we compare any observed trace with its reconstructed trace from any other observations using the E-S method. We use the residual mean square (rms) to evaluate the difference between the observed and reconstructed trace:

r m s i = ( S i − S ′ i ) ⋅ ( S i − S ′ i ) T n − 1 (5)

where S ′ i is the reconstructed trace at current trace location i (x = i) from the trace gather not including the i^{th} trace.

Therefore, we start to detect bad traces from looping through all, except the first and last, traces in a trace gather by interpolating a new trace at the current looping trace location from the whole trace gather excluding the current one. Then we calculate and plot the rms of the observed and interpolated virtual ones. By examining the rms chart, we mark traces with rms bigger than our expected rms value, estimated from visual inspection of the rms chart, as bad traces and we replace them with the interpolated ones. To process noisy records with many bad traces, we repeat this procedure iteratively and pick only one trace as a bad trace in each iteration.

We test the E-S method with a simulated trace gather by convolving a synthetic reflectance coefficient profile, representing an unconformity and rift basin horizons, with a predefined Ricker wavelet that has frequency characteristics commonly used in offshore seismic surveys. We also test the E-S method on a stacked trace gather from an offshore seismic survey that has horizons with steep dip, curved, onlap, offlap, and chaotic reflection of basin floor fans.

To simulate a synthetic seismic profile, we draw a simulated stratigraphic profile with horizontal and curved interfaces and assign a constant reflectance to these horizons. Then we convolve this reflection trace gather with a predefined Ricker wavelet having 0.064 s duration, 100 Hz center frequency, and 2 ms sampling interval. We display the final simulated profile of 0.4 s and 100 traces in

From

we find the residual is close to zero and the E-S method has performed very well in the interpolation

Next, we test the E-S interpolation method using a section of offshore seismic survey from the east coast of Canada, with 4 ms sampling interval, and show the results in Figures 2-8.

Figures 2-4 display the input traces (a), interpolation results (b), residuals (c), and rms (d) for interpolation using every second, third and fourth trace, respectively.

Trace^{#} | 3369 | 3389 | 3409 | 3429 | 3449 | 3469 | 3479 | All traces |
---|---|---|---|---|---|---|---|---|

Mean of errors | −4.8 | −2.3 | 3.0 | 7.4 | −8 | 5.2 | −1.4 | 2.4 |

rms | 12.1% | 12.9% | 14.4% | 16.3% | 14.2% | 18.9% | 13.1% | 100% |

eigenbases, 90.3% on the first 31, and 99.1% for the first 60. The stronger the correlation that exists among traces, the faster the energy for every eigenbase (thin solid line in

We plot the spatial loading scores (and 1D interpolation trajectories) and eigenbase in

In

To test the performance of the E-S method for bad trace detection, we add normal distribution noise n ( μ , σ ) , as shown in

s n o i s e = s + 2.333 ∗ n ( μ , σ ) (6)

where s and s n o i s e are the original seismic trace, and the seismic trace with added noise. The normal noise n has mean μ and standard deviation σ . We estimate μ and σ from the same trace where we add noise and 2.333 is a constant to control the signal to noise ratio.

We interpolate traces at any trace position, except the first and last, using the trace gather with added noises excluding current trace. Then we calculate the rms for each interpolated trace and display it as shown in

We use 120 shots from a seismic survey offshore east coast of Canada to test the performance of the E-S method for bad shot detection and shot interpolation. Every shot has 16 traces (receivers) and each trace has 1751 samples with 2 ms recording rate. The procedure is similar to the above example, in that noise was added to some shots, interpolation was used to produce a virtual shot using other shots at every shot location, and rms was calculated and plotted to detect bad shots. We display the rms chart example of these operations in

In this paper, we propose a novel method termed Eigenspace Seismic (E-S) method to improve seismic data quality and restore bad recordings, using PCA to interpolate in eigenspace.

We demonstrate the effectiveness of the E-S method for trace interpolation by processing synthetic and offshore seismic survey data to restore original data from decimated data sets with 1/2, 1/3, and 1/4 of the original traces. We also show its possible usage to clarify stratigraphic structures such as onlapping units and basin floor fans, in the interpolated trace gather by augmenting the number of original traces by a factor of 5 using the E-S method. The method works best for sparsely recorded or densely recorded seismic data with many bad components.

By analyzing the features of the E-S method using SSA, we can evaluate its energy distribution, features of eigenbases and the smoothness of loading scores (or coordinates in the eigenspace) trajectories that are helpful in the success of E-S method. The major advantage of the E-S method comes from the fact that it handles a whole trace or shot as an inseparable entity and transforms the 2D interpolation into a 1D problem in the eigenspace.

The computational cost of the E-S method is not an issue for trace interpolation and trace quality control, but it does become an issue for shot-gather interpolation and quality control. The state-of-art fast PCA method [

The E-S method does not assume the signal is stationary so that it may be effective for nonstationary noises commonly associated with marine data acquisition, such as shark strikes, swell noise, or nonlaminar flow around cable instrumentation. Theoretically, the E-S method is based on coherency of a processed trace so that it will be efficient to process seismic data having coherent signals. The effectiveness of the E-S method depends on the distribution of these noises on the PCs. There are some methods specifically designed to solve the energy concentration problems, such as factor analysis [

Currently, we use simple 1D spline interpolation methods in the spatial loading score (U) interpolation in the E-S method. In the future, it should be possible to use other advanced 1D interpolation methods for the U estimation to get better E-S results. There are many potential applications of the E-S method to seismic data or other datasets, such as the construction of a 3D velocity volume from irregularly distributed 3D velocity observations [

Natural Resources Canada through the UNCLOS Program of the Geological Survey of Canada, Lands and Minerals Sector of Natural Resources Canada supported this work. Natural Resources Canada through the Public Safety Geoscience Program partially supported this work. The authors wish to thank Mary-Lynn Dickson at the Geological Survey of Canada (Atlantic) for helpful discussions, Patrick Potter for preparing the offshore marine seismic reflection data and for constructive discussion, and John Shimeld for his critical internal review and suggestions. We also thank the Editors and two anonymous reviewers of the journal for their helpful suggestions for the improvement of the manuscript.

The authors declare no conflicts of interest regarding the publication of this paper.

Li, Q.M. and Dehler, S.A. (2019) Seismic Data Quality Control and Interpolation Using Principal Component Analysis. International Journal of Geosciences, 10, 950-966. https://doi.org/10.4236/ijg.2019.1010054