1. Introduction
The last decade has seen a great increase in hydrocarbon production from unconventional reservoirs. However, there are still many challenges in predicting their production potential. The quantity of hydrocarbons produced depends on the distribution and quantity of fractures present in the reservoir. Fracture identification and monitoring can be done by analyzing microseismic data [
1]. With the goal of improving the ability to track the fracture distribution, the amount of seismic data acquired during reservoir monitoring has been increasing.
Recent developments in reservoir monitoring use fiber optic cables to record both low frequency strain and seismic waves in a technique called distributed acoustic sensing (DAS) [
2]. DAS provides a new and unique view of the reservoir by recording strain rate data with a high sample rate and dense spatial sampling. Higher recording rates and more receiver positions increase the amount of microseismic activity recorded while monitoring the reservoir. Recording microseismic events with DAS significantly increases the memory requirement of the monitoring data. Thus, data processing methods that can compress the data and reduce its noise would complement the new instrumental developments in reservoir monitoring.
Signal processing algorithms are often based on the transformation of signal into a new domain. Early examples of compression involve discrete cosine transform [
3] and wavelet transforms [
4,
5,
6] which use cosine functions or wavelets to represent the data in order to reduce the memory requirements. Reduced-rank methods, which approximate the noiseless seismic data using low rank matrices and tensors, have been used in noise reduction [
7] while also reconstructing missing data [
8,
9]. In recent years, dictionary learning has seen wide application in seismic data. Because of its ability to provide a compact and informative representation of seismic signals, it has been used for both noise reduction [
10,
11,
12] and data compression [
13,
14]. Unfortunately, the drawback of the dictionary learning applications is the computation time that comes with learning and updating the dictionary. There have been successful efforts to reduce the associated computational times [
15], but the computational cost of applying dictionary learning to microseismic DAS recordings is still to great. Thus far, dictionary learning methods have been applied to data collected by geophones, which can be very large but are still much smaller than data obtained by a single fiber-optic cable which can sample strain thousands of times a second on hundreds, even thousands of receiver locations. Since DAS is used for microseismic monitoring, the great amounts of data would need to be processed in real time, which puts a constraint on the computation time of processing methods.
To achieve computationally efficient compression and denoising, we created a new data decomposition method by improving and further developing ideas developed as a part of the local SVD [
16]. Similarly to the previously mentioned reduced-rank methods, local SVD applies SVD to a window in seismic data and represents the data using a small number of singular vectors. What makes local SVD unique is the process of shifting the columns of the window to maximize their correlation prior to applying SVD. This allows singular vectors to capture the signal in seismic data with high accuracy, while ignoring most of the noise. Once the data in the window is processed with SVD, the window is moved to the next location. This process is repeated until column shifting and SVD have been applied to every part of the data matrix. The path of the moving window as well as the number of singular vectors used at each location are predetermined. Local SVD can enhance a seismic data set even if it contains multiple wave arrivals. However, local SVD struggles to capture seismic signals if waves with different dips are interfering or present in the same window. The “dip” of the wave refers to the slope of the wave in the matrix, or how much the row position of a wave changes as we move from one column to its adjacent column.
Some of the problems encountered with local SVD were resolved with the development of structure-oriented singular value decomposition (SOSVD) [
17]. By using plane wave destruction [
18], SOSVD can identify several dominant slopes at each window location. While using plane wave destruction provides a noticeable improvement, numerical artifacts can still appear at the intersection of the waves with different slopes.
Our method is inspired by the SOSVD and local SVD, and it also shifts the columns of the matrix before applying SVD. However, in order to avoid numerical artifacts, we use different processes to determine how columns should be shifted. Specifically, we do not use a moving window with a predetermined path. Instead, we use a geometric mean filter that adaptively chooses which elements to use in the geometric mean. We apply the geometric mean filter to seismic data in order to highlight areas which contain wave arrivals. The windows from the matrix to which we apply SMD depend on the results of the geometric mean filter. This allows us to use more singular vectors to describe areas with multiple wave arrivals, and fewer singular vectors to describe areas dominated by noise.
Additionally, local SVD and SOSVD use SVD results solely to denoise seismic data. They use them to reconstruct denoised version of seismic data as soon as they obtain them and do not discuss the compression achieved by storing SVD results. In our work, instead of reconstructing the data immediately, we store the SVD results. The final product of our algorithm is the collection of SVD results, that can later be reconstructed into the denoised version of original data. By doing this, our algorithm can be used for data compression as well as noise suppression. We call our new method the shifted-matrix decomposition, or SMD for short.
It should also be noted that DAS has seen increasing use outside geophysical exploration. Distributed acoustic sensing has been used to record signals from earthquakes and volcanic events [
19]. Because of its unprecedented spatial and temporal resolutions, DAS is expected to see increasing use in earthquake monitoring, imaging of faults and many other geologic formations, and hazard assessment [
20]. The growing potential of DAS application outside of geophysical exploration, adds importance to our method, which we believe will play an integral role in the processing of DAS data.
We organize the paper as follows: First we give an overview of singular value decomposition and present two simple examples that show advantages and drawbacks of its application to seismic data. Next, we demonstrate the improvements achieved by shifting the traces before extracting singular vectors. In the following subsection, we describe in detail each step of the SMD algorithm. The SMD algorithm depends on several parameters. The optimal values of said parameters are determined in a machine learning stage following the algorithm description subsection. In the training stage we use seismic field data obtained from marine seismic gathers, which has a large amount of interference between coherent waves, and noise which can be difficult to differentiate from signal. After training on marine seismic gathers, SMD provides accurate results on other seismic data as well as marine seismic gathers, which allows us to skip the training stage in future applications. To confirm the accuracy of SMD, we reproduce synthetic data from [
17], and compare results of SMD to results of local SVD and SOSVD. The SMD is then tested on real seismic data. While SMD is primarily developed for application to microseismic data recorded by DAS, we currently do not have access to such data. Instead, we apply SMD to field data obtained from marine seismic gathers [
21]. The results of applying SMD to field data are used to reconstruct a denoised version of the data as well as to estimate the elastic wave velocity. Finally, we discuss possible future applications of SMD and how its results could be used in signal detection during seismic monitoring.
4. Discussion
In addition to seismic data compression and noise reduction, SMD also provides a new method of seismic data analysis. Rather than a list of displacements distributed in space and time to describe incoming waves, we have pairs of singular vectors paired with shift vectors. In an ideal scenario, for each arriving wave the column vector represents the waveform, the row vector represents the amplitudes at different receiver locations and the shift vector represents relative arrival times. Even though the ideal case is rarely achieved, we believe that the results of SMD provide an excellent advance in the realm of seismic analysis especially with its application of machine learning. With SMD, identifying features that can be used for building models is facilitated when noise-reduced data is represented in the SMD-compressed format.
The application of machine learning to data compression was explored in [
24], which applied SVD to synthetic data and developed a model for estimating source location and orientation. However, SMD may provide greater opportunities for machine learning application.
It is instructive to provide an example of how the SMD algorithm can help estimate physical properties such as elastic wave velocities. To this end, herein we predict the average acoustic velocity () in our model (seawater) by analyzing the results of the SMD algorithm. Specifically, we use curvature of the shift vectors, and the zero offset time to estimate the average velocity of the reflected waves. Since we have a controlled source, a wave’s zero offset time is simply it’s row position in the first column of the data matrix multiplied by the time difference between consecutive recordings (). Since the row position of a wave can be determined from the shift vector and the first element of the column vector, our wave velocity estimation is obtained solely from the data stored in SMD results.
To derive the formula for wave velocity we must make several assumptions about our surroundings. First, we assume that the reflecting surface (the ocean floor) is horizontal, and that the depth of the ocean floor (
) is significantly larger than the horizontal distance from source to the receiver (
). This gives us the expression for the total distance travel by the reflected waves (
d):
Since we are considering the average acoustic velocity (
) in our model, we can rewrite the expression (
11) in terms of the reflected wave travel-time (
T):
The zero offset time
is defined as travel-time
T at zero horizontal distance (
):
Taking the second derivative of the expression (
12) with respect to horizontal position
x gives:
Using expression (
13) to substitute
with
in expression (
14) gives:
Expression (
15) can be used to calculate the acoustic velocity if we can express both
and
in terms of SMD results.
In
Section 2.2.2 it is explained that the row position (
) of a wave in any column
j is equal to the sum of the
element of the shift vector (
) and the waveform row (
):
In
Section 2.2.3 it is explained that the waveform row (
) can be obtained from the first element of the column vector (
) and the predetermined length of the recorded waveform (
):
Using expressions (
16) and (
17), the zero offset time (
) can be described with the row position of the wave in the first column (
) multiplied by the time difference between consecutive recordings (
):
The expression (
18) will be used to obtain the zero offset time from SMD results.
If we assume that the relative arrival times were accurately recorded by the shift vector, we can make the following substitution:
where
is the distance between adjacent receivers. In the expression (
19),
is the second derivative of an element in the shift vector
s with respect to the position (
j) of the element in the shift vector.
The value of
is determined by fitting a parabola to first
elements of the shift vector. The parameter
need not have an exact value. While we want to use enough elements from the shift vector to confidently fit a parabola, we also want to only use data from the receivers close to the source in order to follow the
condition. Therefore, the parameter
is set to 50. Once we fit the parabola, the value of
is estimated to be the quadratic coefficient
, multiplied by 2:
By plugging the expressions (
18)–(
20) into expression (
15) we estimate the acoustic velocity. However, the values of a shift vector can be affected by interfering waves, or noise. To minimize the error from those sources, we estimate the velocity based on the first
pairs of shift vectors and column vectors. Similar to
, the
parameter does not need to be set to any specific value. In this example, the parameter
is set to 5. The average acoustic velocity
is estimated as the weighted average of the results from the first
extracted pairs of a of shift vector and a column vector. Each term is weighted by the quality of the parabolic fit, which is equal to the inverse of the error norm
e:
The formula for
, defined in expression (
21), was applied to 20 marine field data files, each recording a seismic response from a unique and controlled seismic source. For each file, we used the formula from expression (
21) to estimate the velocity. The average estimated velocity was
and the standard deviation among the results was
. Considering the ocean depth being about 3 km, the correct average acoustic velocity was likely about
. If we assume the correct average wave velocity experienced by the reflected waves was
, then the average and the median error from the 20 velocity estimates were
and
, respectively.
The experiment in this subsection proves that there is a strong correlation between the shift vector and relative arrival times of the wave at different receivers. We were able to use that correlation to estimate the velocity of the waves without relying on any information about the medium through which the waves were traveling. We believe that there is also a strong correlation between the column vector and the waveform, as well as between the row vector and the relative amplitudes of the waveform at different receivers. However, proving these correlations will require further testing.
In this example, SMD was applied to seismic data from a controlled source. In such cases we can always be certain that there are coherent waves present. However, we are planning a wide range of applications for SMD. During microseismic monitoring, often recorded by DAS, we might want to analyze coherent waves that are not coming from controlled sources. Therefore, when applying SMD to data from microseismic monitoring we do not know in advance whether the data contains signals of interest. For that reason, we developed a method for recognising coherent signal in noisy data, that also relies solely on the results from SMD.
We created a method that can differentiate between noisy data and data containing signal, by measuring the curvature in the first several shift vectors. Applying SMD to noisy data without coherent waves returns a set of completely unrelated shift vectors. On the other hand, applying SMD to data containing coherent waves, usually coming from the same source, returns a set of shift vectors most of which have similar curvatures. Therefore, the method differentiates between noisy data and data containing coherent waves, by calculating the standard deviation among the curvatures from the first 10 extracted shift vectors. To test this method, we applied SMD separately to two subsets of data from each of the 20 previously mentioned data files. The first subset contains only the recordings of noise, and the second subset contains recordings of both the coherent waves and the preceding noise (
Figure 14).
For each subset, we use expression (
20) to estimate the curvatures of the first 10 extracted shift vectors and then we compute the standard deviation of the 10 curvature values. The results are presented in
Figure 15, in which the standard deviation for each subset without coherent waves is presented in blue, and the standard deviation for each subset containing coherent waves is presented in red. Due to shift vector curvature having random values when SMD results represent noise, the standard deviation among shift vector curvatures should be greater for the data subsets which contain only noise. As can be seen in
Figure 15, the standard deviation among shift vectors is consistently lower for subsets containing coherent waves, which are presented in red. This confirms that we can differentiate between data containing only noise and data containing signal, by using our described method.
It is important to emphasize that this signal detection method does not contain any information regarding the amplitude of the waves. It is entirely dependent on the curvature of the recorded shift vectors, which is a unique property. This means that this method, while accurate, can also be used to complement other signal detection methods, all of which rely on other properties in seismic data (such as STA/LTA for example, which relies on the amplitudes of the waves). Furthermore, this signal detection method only requires the first 10 shift vectors, which can be acquired very quickly compared to the to the processing time of the SMD. Therefore, when applying SMD to data obtained during seismic monitoring, we suggest first running a quick version of SMD that only extracts the first 10 shift vectors in order to run signal detection. If the presence of coherent waves is confirmed, SMD may continue to fully process the data.
The analysis of SMD results that was conducted in this subsection was not based on machine learning but instead on our understanding of the correlations between compressed data and attributes of the recorded waves. In the future, those strong correlations will be used with machine learning to train algorithms to accurately infer various properties of the source and the surrounding velocity model directly from compressed data. Additionally, application of data from marine surveys is difficult because of large amounts of interference between arriving waves. Thus, the SMD method may be more effective in different circumstances, such as in unconventional reservoirs monitored by distributed acoustic sensing.
Thus far, we only considered application of SMD to two-dimensional seismic data, obtained by a single line of receivers. However, receivers may often be distributed over an area, in a large number of lines. In such case, SMD may still be applied individually, to the data collected from each line. However, there may be a lot of redundancy between SMD results from each of the lines of receivers. Similar redundancies can also occur by applying SMD to multiple files from different times, obtained by a same set of receivers. In future work, we will take advantage of the fact that SMD results may be similar for batches of data collected from different receivers or during different times. Dictionary learning could be applied to SMD results to further compress the data, although we can not yet recommend a method for learning the dictionary. Future work will therefore likely include further compression by applying dictionary learning to SMD results from multiple data files or multiple receiver lines.